Handbook of Teichmuller Theory, Volume III

IRMA Lectures in Mathematics and Theoretical Physics 17 Edited by Christian Kassel and Vladimir G. Turaev Institut de R...

Author: Athanase Papadopoulos

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IRMA Lectures in Mathematics and Theoretical Physics 17 Edited by Christian Kassel and Vladimir G. Turaev

Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 rue René Descartes 67084 Strasbourg Cedex France

IRMA Lectures in Mathematics and Theoretical Physics Edited by Christian Kassel and Vladimir G. Turaev This series is devoted to the publication of research monographs, lecture notes, and other material arising from programs of the Institut de Recherche Mathématique Avancée (Strasbourg, France). The goal is to promote recent advances in mathematics and theoretical physics and to make them accessible to wide circles of mathematicians, physicists, and students of these disciplines. Previously published in this series: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18

Deformation Quantization, Gilles Halbout (Ed.) Locally Compact Quantum Groups and Groupoids, Leonid Vainerman (Ed.) From Combinatorics to Dynamical Systems, Frédéric Fauvet and Claude Mitschi (Eds.) Three courses on Partial Differential Equations, Eric Sonnendrücker (Ed.) Infinite Dimensional Groups and Manifolds, Tilman Wurzbacher (Ed.) Athanase Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature Numerical Methods for Hyperbolic and Kinetic Problems, Stéphane Cordier, Thierry Goudon, Michaël Gutnic and Eric Sonnendrücker (Eds.) AdS/CFT Correspondence: Einstein Metrics and Their Conformal Boundaries, Oliver Biquard (Ed.) Differential Equations and Quantum Groups, D. Bertrand, B. Enriquez, C. Mitschi, C. Sabbah and R. Schäfke (Eds.) Physics and Number Theory, Louise Nyssen (Ed.) Handbook of Teichmüller Theory, Volume I, Athanase Papadopoulos (Ed.) Quantum Groups, Benjamin Enriquez (Ed.) Handbook of Teichmüller Theory, Volume II, Athanase Papadopoulos (Ed.) Michel Weber, Dynamical Systems and Processes Renormalization and Galois Theories, Alain Connes, Frédéric Fauvet and Jean-Pierre Ramis (Eds.) Handbook of Pseudo-Riemannian Geometry and Supersymmetry, Vicente Cortés (Ed.) Strasbourg Master Class on Geometry, Athanase Papadopoulos (Ed.)

Volumes 1–5 are available from Walter de Gruyter (www.degruyter.de)

Handbook of Teichmüller Theory Volume III Athanase Papadopoulos Editor

Editor: Athanase Papadopoulos Institut de Recherche Mathématique Avancée CNRS et Université de Strasbourg 7 Rue René Descartes 67084 Strasbourg Cedex France

2010 Mathematics Subject Classification: Primary 30-00, 32-00, 57-00, 32G13, 32G15, 30F60; secondary 11F06, 11F75, 14D20, 14H15, 14H60, 14H55, 14J60, 20F14, 20F28, 20F38, 20F65, 20F67, 20H10, 30C62, 30F20, 30F25, 30F10, 30F15, 30F30, 30F35, 30F40, 30F45, 53A35, 53B35, 53C35, 53C50, 53C80, 53D55, 53Z05, 57M07, 57M20, 57M27, 57M50, 57M60, 57N16.

ISBN 978-3-03719-103-3 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Foreword This Handbook is growing in size, reflecting the fact that Teichmüller theory has multiple facets and is being developed in several directions. In this new volume, as in the preceding volumes, there are chapters that concern the fundamental theory and others that deal with more specialized developments. Some chapters treat in more detail subjects that were only briefly outlined in the preceding volumes, and others present general theories that were not treated there. The study of Teichmüller spaces cannot be dissociated from that of mapping class groups, and like in the previous volumes, a substantial part of the present volume deals with these groups. The volume is divided into the following four parts: • The metric and the analytic theory, 3. • The group theory, 3. • The algebraic topology of mapping class groups and moduli spaces. • Teichmüller theory and mathematical physics. The numbers that follow the titles in the first two parts indicate that there were parts in the preceding volumes that carry the same titles. This Handbook is also a place where several fields of mathematics interact. For the present volume, one can mention the following: partial differential equations, one and several complex variables, algebraic geometry, algebraic topology, combinatorial topology, 3-manifolds, theoretical physics, and there are several others. This confluence of ideas towards a unique subject is a manifestation of the unity and harmony of mathematics In addition to the fact of providing surveys on Teichmüller theory, several chapters in this volume contain expositions of theories and techniques that do not strictly speaking belong to Teichmüller theory, but that have been used in an essential way in the development of this theory. Such sections contribute in making this volume and the whole set of volumes of the Handbook quite self-contained. The reader who wants to learn the theory is thus spared some of the effort of searching into several books and papers in order to find the material that he needs. For instance, Chapter 4 contains an introduction to arithmetic groups and their actions on symmetric spaces, with a view towards comparisons and analogies between this theory and the theory of mapping class groups and their action on Teichmüller spaces. Chapter 5 contains an introduction to abstract simplicial complexes and their automorphisms. Chapter 9 contains a concise survey of group homology and cohomology, and an exposition of the Fox calculus, having in mind applications to the theory of the Magnus representation of the mapping class group. Chapter 10 contains an exposition of the theory of Thompson’s groups in relation with Teichmüller spaces and mapping class groups. The same chapter contains a review of Penner’s theory of the universal

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Foreword

decorated Teichmüller space and of cluster algebras. Chapter 10 and Chapter 14 contain an exposition of the dilogarithm, having in mind its use in the quantization theory of Teichmüller space and in the representation theory of mapping class groups. Chapter 11 contains a section on the intersection theory of complex varieties, as well as an introduction to the theory of characteristic classes of vector bundles, with applications to the intersection theory of the moduli space of curves and of its stable curve compactification. Chapter 13 contains an exposition of Lp -cohomology, of the intersection cohomology theory for projective algebraic varieties and of the Hodge decomposition theory for compact Kähler manifolds, with a stress on applications to Teichmüller and moduli spaces. Finally, let us mention that several chapters in this volume contain open problems directed towards future research; in particular Chapter 4 by Ji, Chapter 5 by McCarthy and myself, Chapter 7 by Korkmaz, Chapter 8 by Habiro and Massuyeau, Chapter 9 by Sakasai, Chapter 10 by Funar, Kapoudjian and Sergiescu, and Chapter 13 by Ji and Zucker. Up to now, sixty different authors (some of them with more than one contribution) have participated to this project, and there are other authors, working on volumes in preparation. I would like to thank them all for this fruitful cooperation which we all hope will serve generations of mathematicians. I would like to thank once more Manfred Karbe and Vladimir Turaev for their interest and their care, and Irene Zimmermann for the seriousness of her work. Strasbourg, April 2012

Athanase Papadopoulos

Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Introduction to Teichmüller theory, old and new, III by Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part A. The metric and the analytic theory, 3 Chapter 1. Quasiconformal and BMO-quasiconformal homeomorphisms by Jean-Pierre Otal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Chapter 2. Earthquakes on the hyperbolic plane by Jun Hu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter 3. Kerckhoff’s lines of minima in Teichmüller space by Caroline Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Part B. The group theory, 3 Chapter 4. A tale of two groups: arithmetic groups and mapping class groups by Lizhen Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Chapter 5. Simplicial actions of mapping class groups John D. McCarthy and Athanase Papadopoulos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Chapter 6. On the coarse geometry of the complex of domains by Valentina Disarlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Chapter 7. Minimal generating sets for the mapping class group of a surface by Mustafa Korkmaz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 Chapter 8. From mapping class groups to monoids of homology cobordisms: a survey Kazuo Habiro and Gwénaël Massuyeau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

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Contents

Chapter 9. A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces by Takuya Sakasai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 Chapter 10. Asymptotically rigid mapping class groups and Thompson’s groups Louis Funar, Christophe Kapoudjian and Vlad Sergiescu . . . . . . . . . . . . . . . . . . . . . 595 Part C. The algebraic topology of mapping class groups and their intersection theory Chapter 11. An introduction to moduli spaces of curves and their intersection theory by Dimitri Zvonkine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667 Chapter 12. Homology of the open moduli space of curves by Ib Madsen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717 Chapter 13. On the Lp -cohomology and the geometry of metrics on moduli spaces of curves by Lizhen Ji and Steven Zucker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 Part D. Teichmüller theory and mathematical physics Chapter 14. The Weil–Petersson metric and the renormalized volume of hyperbolic 3-manifolds by Kirill Krasnov and Jean-Marc Schlenker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 Chapter 15. Discrete Liouville equation and Teichmüller theory by Rinat M. Kashaev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821 Corrigenda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857

Introduction to Teichmüller theory, old and new, III Athanase Papadopoulos

Contents 1

2

3

4

Part A. The metric and the analytic theory, 3 . . . . . . . . . . . . . . . . . 1.1 The Beltrami equation . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Earthquakes in Teichmüller space . . . . . . . . . . . . . . . . . . . 1.3 Lines of minima in Teichmüller space . . . . . . . . . . . . . . . . . Part B. The group theory, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mapping class groups versus arithmetic groups . . . . . . . . . . . . 2.2 Simplicial actions of mapping class groups . . . . . . . . . . . . . . 2.3 Minimal generating sets for mapping class groups . . . . . . . . . . . 2.4 Mapping class groups and 3-manifold topology . . . . . . . . . . . . 2.5 Thompson’s groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Part C. The algebraic topology of mapping class groups and moduli spaces 3.1 The intersection theory of moduli space . . . . . . . . . . . . . . . . 3.2 The generalized Mumford conjecture . . . . . . . . . . . . . . . . . . 3.3 The Lp -cohomology of moduli space . . . . . . . . . . . . . . . . . Part D. Teichmüller theory and mathematical physics . . . . . . . . . . . . 4.1 The Liouville equation and normalized volume . . . . . . . . . . . . 4.2 The discrete Liouville equation and the quantization theory of Teichmüller space . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 4 9 11 11 15 17 18 23 27 27 28 30 32 33 34

Surveying a vast theory like Teichmüller theory is like surveying a land, and the various chapters in this Handbook are like a collection of maps forming an atlas: some of them give a very general overview of the field, others give a detailed view of some crowded area, and others are more focussed on interesting details. There are intersections between the chapters, and these intersections are necessary. They are also valuable, because they are written by different persons, having different ideas on what is essential, and (to return to the image of a geographical atlas) using their proper color pencil set. The various chapters differ in length. Some of them contain proofs, when the results presented are new, and other chapters contain only references to proofs, as it is usual in surveys. I asked the authors to make their texts accessible to a large number of readers. Of course, there is no absolute measure of accessibility, and the response depends on the sound sense of the author and also on the background of the reader. But in principle

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all of the authors made an effort in this sense, and we all hope that the result is useful to the mathematics community. This introduction serves a double purpose. First of all, it presents the content of the present volume. At the same time, reading this introduction is a way of quickly reviewing some aspects of Teichmüller theory. In this sense, the introduction complements the introductions I wrote for Volumes I and II of this Handbook.

1 Part A. The metric and the analytic theory, 3 1.1 The Beltrami equation Chapter 1 by Jean-Pierre Otal concerns the theory of the Beltrami equation. This is the partial differential equation N D @; @ (1.1) where W U ! V is an orientation preserving homeomorphism between two domains U of V of the complex plane and where @ and @N denote the complex partial derivativations

1 @ @ @ D i 2 @x @y

and

N D 1 @ C i @ : @ 2 @x @y

N If is a solution of the Beltrami equation (1.1), then D @=@ is called the complex dilatation of . Without entering into technicalities, let us say that the partial derivatives @ and N of are allowed to be distributional derivatives and are required to be in L2 .U /. @ loc The function that determines the Beltrami equation is in L1 .U /, and is called the Beltrami coefficient of the equation. The Beltrami equation and its solution constitute an important theoretical tool in the analytical theory of Teichmüller spaces. For instance, the Teichmüller space of a surface of negative Euler characteristic can be defined as some quotient space of a space of Beltrami coefficients on the upper-half plane. As a matter of fact, this definition is the one commonly used to endow Teichmüller space with its complex structure. The classical general result about the solution of the Beltrami equation (1.1) says that for any Beltrami coefficient satisfying kk1 < 1, there exists a quasiconformal homeomorphism D f W U ! V which satisfies a.e. this equation, and that f is unique up to post-composition by a holomorphic map. There are several versions and proofs of this existence and uniqueness result. The first version is sometimes attributed to Morrey (1938), and there are versions due to Teichmüller (1943), to Lavrentieff (1948) and to Bojarski (1955). In the final form that is used in Teichmüller theory, the result is attributed to Ahlfors and Bers, who published it in their paper Riemann’s

Introduction to Teichmüller theory, old and new, III

3

mapping theorem for variable metrics (1960). This result is usually referred to as the Measurable Riemann Mapping Theorem. Note that in the case where is identically zero, the Beltrami equation reduces N D 0, and the result follows from the classical to the Cauchy–Riemann equation @ Riemann Mapping Theorem. Ahlfors and Bers furthermore showed that the correspondence 7! f is holomorphic in the sense that if t is a family of holomorphically parametrized Beltrami coefficients on the open set U , with t being a parameter in some complex manifold, then the map t 7! f t .z/ (with a proper normalization) is holomorphic for any fixed z 2 U . This result was used as an essential ingredient in the construction by Bers of the complex structure of Teichmüller space. Indeed, considering the elements of Teichmüller space as equivalence classes of solutions f of the Beltrami equation with coefficient , the complex structure of Teichmüller space is the unique complex structure on that space satisfying the above parameter-dependence property. Chapter 1 is an account of recent work on the Beltrami equation. It contains a proof of the Measurable Riemann Mapping Theorem. While the original work on the Beltrami equation, as developed by Morrey, Bojarski and Ahlfors–Bers uses hard analysis (Calderon–Zygmund theory, etc.), the proof presented here should be more accessible to geometers. The existence part in this proof was recently discovered by Alexey Glutsyuk. It concerns the case where the Beltrami coefficient is of class C 1 . The general case can be deduced by approximation. After presenting Glutsyuk’s proof, Otal surveys a substantial extension of the theory of the Beltrami equation, namely, the extension to the case where kk1 D 1. It seems that such an extension was first studied by Olli Lehto in 1970, with several technical hypotheses on the set of points in U where kk1 D 1. The hypotheses were substantially relaxed later on. A major step in this direction was taken by Guy David who, in 1988, proved existence and uniqueness of the solution of the Beltrami equation with kk1 D 1, with satisfying a logarithmic growth condition near the subset fjj D 1g of U . This general version of the Beltrami equation led to many applications, in particular in complex dynamics. There have been, since the work of David, several improvements and variations. In particular, Ryazanov, Srebro & Yakubov introduced in 2001 a condition where the dilatation function K of , defined by K D 1Cj.z/j , is bounded a.e. by a function 1j.z/j which is locally in the John–Nirenberg space BMO.U / of bounded mean oscillation functions . (We recall that since, in the hypothesis of David’s Theorem, kk1 D 1 instead of kk1 < 1, the dilatation function K is not necessarily in L1 .) In this case, the quasiconformal map f W U ! V provided by the theorem is not quasiconformal in the usual sense, and it is called a BMO-quasiconformal homeomorphism (which explains the title of Chapter 1). The chapter also contains some useful background material on quasiconformal maps, moduli and extremal length that is needed to understand the proofs of the results presented.

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1.2 Earthquakes in Teichmüller space After the chapter on the existence and uniqueness of solutions of the Beltrami equation, Chapter 2, written by Jun Hu, surveys another existence and uniqueness result, which is also at the basis of Teichmüller theory, namely, Thurston’s Earthquake Theorem. The setting here is the hyperbolic (as opposed to the conformal) point of view on Teichmüller theory. The earthquake theorem says that for any two points in Teichmüller space, there is a unique left earthquake path that joins the first point to the second. A “global” and an “infinitesimal” version of this theorem are presented in their most general form, and a parallel is made between this generalization and the general theory of the Beltrami equation and its generalization that is reviewed in Chapter 1. Earthquake theory has many applications in Teichmüller theory. Some of them appear in other chapters of this volume, e.g. Chapter 3 by Series and Chapter 14 by Krasnov and Schlenker. Before going into the details of Chapter 2, let us briefly review the evolution of earthquake theory. The theory originates from the so-called Fenchel–Nielsen deformation of a hyperbolic metric. We recall the definition. Given a hyperbolic surface S containing a simple closed geodesic ˛, the time-t left (respectively right) Fenchel–Nielsen deformation of S along ˛ is the hyperbolic surface obtained by cutting the surface along ˛ and gluing back the two boundary components after a rotation, or shear, “to the left” (respectively “to the right”) of amount t . The sense of the shear (left or right) depends on the choice of an orientation on the surface but not on the choice of an orientation on the curve ˛. The amount of shearing is measured with respect to arclength along the curve.1 The precise definition needs to be made with more care, so that while performing the twist, one keeps track of the homotopy classes of the simple closed geodesics that cross ˛. In more precise words, the deformation is one of marked surfaces. In particular, the surface obtained from S after a complete twist (a Dehn twist), as an element of Teichmüller space, is not the element we started with, because its marking is different. The next step is to shear along a geodesic which is not a simple closed curve. For instance, one can shear along an infinite simple geodesic, that is, a geodesic homeomorphic to the real line. Making such a definition is not straightforward, unless the geodesic is isolated in the surface (for instance, if it joins two punctures, or two points on the ideal boundary). An earthquake deformation is a generalization of a Fenchel– Nielsen deformation where, instead of shearing along a simple closed geodesic, one performs a shearing along a general measured geodesic lamination. Here, the amount of shearing is specified by the transverse measure of the lamination. On order to make such a definition precise, one can define a time-t left (respectively right) earthquake deformation along a measured geodesic lamination as the limit of a sequence of time-t left (respectively right) earthquake deformations associated to weighted simple 1 There is another normalization which is useful in some contexts, where the amount of shear is t length.˛/. In this case, one talks about a normalized earthquake.


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closed curves ˛n , as this sequence converges, in Thurston’s topology on measured lamination space, to the measured geodesic lamination . Although this definition is stated in a simple way, one cannot avoid entering into technicalities, because one has to show that the result does not depend on the choice of the approximating sequence ˛n . In any case, it is possible to make a definition of a time-t left (or right) earthquake along a general measured lamination . For a fixed , varying the parameter t , one obtains a flow on the unit tangent bundle to Teichmüller space: at each point, and in each direction (specified by the measured geodesic lamination) at that point, we have a flowline. This flow is called the earthquake flow associated to . Earthquake deformations were introduced by Thurston in the 1970s, and the first paper using earthquakes was Kerckhoff’s paper Nielsen Realization Problem, published in 1983, in which Kerckhoff gave the solution of the Nielsen Realization Problem. The solution is based on the convexity of geodesic length functions along earthquake paths, and on the “transitivity of earthquakes”, that is, the result that we mentioned above on the existence of earthquakes joining any two points in Teichmüller space. The transitivity result is due to Thurston. Kerckhoff provided the first written proof of that result as an appendix to his paper. A few years later, Thurston developed a much more general theory of earthquakes, in a paper entitled Earthquakes in two-dimensional hyperbolic geometry (1986). This included a new proof of the transitivity result. In that paper, earthquake theory is developed in the setting of the universal Teichmüller space, that is, the space parametrizing the set of complete hyperbolic metrics on the unit disk up to orientation-preserving homeomorphisms that extend continuously as the identity map on the boundary of the disk. (Note that without the condition on homeomorphisms extending as the identity map on the boundary, all hyperbolic structures on the disk would be equivalent.) We recall by the way that the universal Teichmüller space was introduced byAhlfors and Bers in the late 1960s.2 One reason for which this space is called “universal” is that there is an embedding of the Teichmüller space of any surface whose universal cover is the hyperbolic disk into this universal Teichmüller space. The universal Teichmüller space also appears as a basic object in the study of the Thompson groups, surveyed in Chapter 10 of this volume. By lifting the earthquake deformations of hyperbolic surfaces to the universal covers, the earthquake deformation theory of any hyperbolic surface can be studied as part of the earthquake deformation theory of the hyperbolic disk. The deformation theory of the disk not only is more general, but it is also a convenient setting for new developments; for instance it includes quantitative relations between the magnitude 2 There is a relation between the universal Teichmüller space and mathematical physics, which was foreseen right at the beginning of the theory; see Bers’s paper Universal Teichmüller space in the volume Analytic methods in Mathematical Physics, Indiana University Press, 1969, pp. 65–83. In that paper, Bers reported that J. A. Wheeler conjectured that the universal Teichmüller space can serve as a model in an attempt to quantize general relativity. A common trend is to call Diff C .S1 /=PSL.2; R/ the physicists universal Teichmüller space and QS.S1 /=PSL.2; R/ the Bers universal Teichmüller space. Here, Diff C .S1 / denotes the group of orientationpreserving homeomorphisms of the circle and QS.S1 / its group of quasi-symmetric homeomorphisms, of which we talk later in this text.

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of earthquake maps and distortions of homeomorphisms of the circle, as we shall see below. Thurston’s 1986 proof of existence and uniqueness of left (respectively right) earthquakes between hyperbolic structures in the setting of the universal Teichmüller space is based on a convex hull construction in the hyperbolic plane. In Thurston’s words, this proof is “more elementary” and “more constructive” than the previous one. One may also note here that in 1990, G. Mess gave a third proof of the earthquake theorem that uses Lorentz geometry. In Mess’s words, this proof is “essentially Thurston’s second (and elementary) proof, interpreted geometrically in anti-de Sitter space”.3 We finally note that Bonsante, Krasnov and Schlenker gave a new version of the earthquake theorem, again using anti-de Sitter geometry, which applies to surfaces with boundary. Their proof relies on the geometry of “multi-black holes”, which are 3-dimensional anti-de Sitter manifolds, topologically the product of a surface with boundary and an interval. These manifolds were studied by physicists. In that case, given two hyperbolic metrics on a surfaces with n boundary components, there are 2n right earthquakes transforming the first one into the second one.4 The anti-de Sitter setting has similarities with the quasi-Fuchsian setting; that is, the authors consider an anti-de Sitter 3-manifold which is homeomorphic to the product of a surface times an interval, and the two boundary components of that manifold are surfaces that are naturally equipped with hyperbolic structures. Now we must talk about the notion of quasi-symmetry, which is closely related to the notion of quasiconformality. Consider the circle S1 D R=2Z. An orientation-preserving homeomorphism h W S1 ! S1 is said to be quasi-symmetric if there exists a real number M 1 such that for all x on S1 and for all t in 0; =2Œ, we have ˇ

ˇ

ˇ h.e i.xCt/ / h.e ix / ˇ 1 ˇ M: ˇˇ ˇ M h.e ix / h.e i.xt / /

(1.2)

The notion of quasi-symmetric map was introduced by Beurling and Ahlfors in 1956, in a paper entitled The boundary correspondence under quasiconformal mappings. The main result of that paper says that every quasiconformal homeomorphism of the unit disk D 2 extends to a unique homeomorphism of the closed disk D 2 , that the induced map on the boundary S1 D @D 2 is quasi-symmetric and that conversely, any quasi-symmetric map of S1 is induced by a quasiconformal map of D 2 . Like the notion of quasiconformality, the notion of quasi-symmetry admits several generalizations, including an extension to higher dimensions and an extension to mappings between general metric spaces. The latter was studied by Tukia and Väisälä. 3 Mess’s work on that subject is reviewed and expanded in Chapter 14 of Volume II of this Handbook by Benedetti and Bonsante. 4 The number 2n corresponds to the various ways in which a geodesic lamination can spiral around the boundary components of the surface.


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The space of quasi-symmetric maps of the circle considered as the boundary of the hyperbolic unit disk is an important tool in the theory of the universal Teichmüller space. Using the correspondence between the set of quasiconformal homeomorphisms of the open unit disk and the set of quasi-symmetric homeomorphisms of the boundary circle and making a normalization, the universal Teichmüller space can be identified with the space of quasi-symmetric homeomorphisms of S1 that fix three points. Thurston noted in his 1986 paper that the fact that any quasiconformal homeomorphism of the circle extends to a homeomorphism of the disk establishes a one-to-one correspondence between the universal Teichmüller space and the set of right cosets PSL.2; R/nHomeo.H2 /. We now recall that the quasiconformal distortion of a homeomorphism of the hyperbolic disk can be defined in terms of distortion of quadrilaterals in that disk. Analogously, the quasi-symmetry of a homeomorphism h of the circle can be defined in terms of distortion of cross ratios of quadruples of points on that circle. The parallel between these two definitions hints to another point of view on the relation between quasi-symmetry and quasiconformality. Any one of the definitions of a quasi-symmetric map of the circle leads to the definition of a norm on the set QS.S1 / of quasi-symmetric maps. One such norm is obtained by taking the best constant M that appears in Inequality (1.2) defining quasi-symmetry. Another norm is obtained by taking the supremum over distortions of all cross ratios of quadruples. More precisely, given a homeomorphism h W S1 ! S1 , one can define its cross ratio norm by the formula ˇ ˇ ˇ cr.h.Q// ˇ ˇ ˇ; khkcr D sup ˇ ln ˇ Q

cr.Q/

where Q varies over all quadruples of points on the circle and cr.Q/ denotes the cross ratio of such a quadruple. A homeomorphism is quasi-symmetric if and only if it has finite cross ratio norm. Now we return to earthquakes. Thurston calls relative hyperbolic structure on the hyperbolic disk a homotopy class of hyperbolic structures in which one keeps track of the circle at infinity. A left earthquake, in the setting of the universal Teichmüller space, is a transformation of a relative hyperbolic structure of the hyperbolic disk D 2 that consists in cutting the disk along the leaves of a geodesic lamination and gluing back the pieces after a “left shear” along each component of the cut-off pieces. The map thus obtained from D 2 to itself is a “piecewise-Möbius transformation”, in which the domain pieces are the complementary components of a geodesic lamination on D 2 , where the comparison maps fj B fi1 between any two Möbius transformations fi and fj defined on two such domains is a Möbius transformation of hyperbolic type whose axis separates the two domains and such that all the comparison Möbius transformations translate in the same direction. Such a piecewise-Möbius transformation defined on the unit disk is discontinuous, but it induces a continuous map (in fact, a homeomorphism)

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of the boundary circle. From this boundary homeomorphism one gets a new relative hyperbolic structure on the unit disk. Thurston proved the following: (1) Any two relative hyperbolic structures can be joined by a left earthquake. (2) There is a well-defined transverse measure (called the shearing measure) on the geodesic lamination associated to such a left earthquake. This transverse measure encodes the amount of earthquaking (or shearing) along the given lamination. (3) Two relative hyperbolic structures obtained by two left earthquakes with the same lamination and the same transverse measure are conjugate by an isometry. Thurston also introduced the notion of a uniformly bounded measured lamination, and of an associated uniformly bounded earthquake. Here, the notion of boundedness refers to a norm (which is now called Thurston’s norm) on transverse measures of geodesic laminations of the disk. Specifically, the Thurston norm of a transverse measure of a geodesic lamination is defined by the formula k kTh D sup .ˇ/; where the supremum is taken over all arcs ˇ of hyperbolic length 1 that are transverse to . Thurston proved that for any given uniformly bounded measured geodesic lamination , there exists an earthquake map having as a shearing measure. Thurston’s arguments and techniques have been developed, made more quantitative, and generalized in several directions, by Gardiner, Lakic, Hu and Šarić. A result established by Hu (2001) says that the earthquake norm of a transverse measure of a lamination of the unit disk and the cross ratio distortion of the circle homeomorphism h induced by earthquaking along are Lipschitz-comparable; that is, we have 1 khkcr kkTh C khkcr ; C with C being a universal constant. This is a more explicit version of a result of Thurston saying that a transverse measure is Thurston bounded if and only if the induced map at infinity h is quasi-symmetric. In their work entitled Thurston unbounded earthquake maps (2007), Hu and Su obtained a result that generalizes Thurston’s result from bounded to unbounded earthquake measures, with some control on the growth of the measures at infinity, that is, on the measure of transverse segments that are sufficiently close to the boundary at infinity of the hyperbolic disk. As the authors put it, this result can be compared to the result by David on the generalized solution of the Beltrami equation, reported on in Chapter 1 of this volume, in which the L1 -norm of the Beltrami coefficient is allowed to be equal to 1, with some control on its growth near the set where this supremum is attained. In any case, if is a geodesic lamination and a bounded transverse measure on , then the pair .; / defines an earthquake map. Introducing a non-negative real parameter t , we get an earthquake curve E t induced by .; / and a corresponding 1-parameter family of homeomorphisms h t of the circle, also called an earthquake


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curve. The differentiability theory of earthquakes is then expressed in terms of the differentiability of the associated quasi-symmetric maps. For each point x on the circle, the map h t .x/ is differentiable in t and satisfies a certain non-autonomous ordinary differential equation which was established and studied by Gardiner, Hu and Lakic. This differentiable theory is then used for establishing a so-called infinitesimal earthquake theorem. The theory uses the notion of Zygmund boundedness. A continuous function V W S1 ! C is said to be Zygmund bounded if it satisfies jV .e 2 i.Ct/ / C V .e 2 i.t / / 2V .e 2 i /j M jt j for some positive constant M . The reader will notice that this definition of Zygmund boundedness has some flavor of quasi-symmetry. The infinitesimal earthquake theorem can be considered as an existence theorem establishing a one-to-one correspondence between Thurston bounded earthquake measures and normalized Zygmund bounded functions. Hu showed that the cross-ratio norm on the set of Zygmund bounded functions and the Thurston norm on the set of earthquake measures are equivalent under this correspondence. Chapter 2 of the present volume is an account of Thurston’s original construction and of the various developments and generalizations that we mentioned. The chapter includes a proof of Thurston’s result on the transitivity of earthquakes, an algorithm for finding the earthquake measured geodesic lamination associated to a quasi-symmetric homeomorphism of the circle, a presentation of the David-type extension to non-bounded earthquake measures, an exposition of a quantitative relation between earthquake measures and cross ratio norms, and an exposition of the infinitesimal theory of earthquakes.

1.3 Lines of minima in Teichmüller space Chapter 3, by Caroline Series, is a survey on lines of minima in Teichmüller space. These lines were introduced by Kerckhoff in the early 1990s. Their study involves at the same time properties of Teichmüller geodesics and of earthquakes. Let us first briefly recall the definition of a line of minima. Let S be a surface of finite type. For any measured lamination on S , let l W T .S / ! R be the associated length function on the Teichmüller space T .S / of S . Consider now two laminations and that fill up S in the sense that for any measured lamination on S, we have i.; / C i.; / > 0. Kerckhoff noticed that for any t 2 .0; 1/, the function .1 t /l C t l W T .S / ! R

(1.3)

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has a unique minimum He proved this fact using the convexity of geodesic length functions along earthquakes, and the existence of an earthquake path joining any two points in Teichmüller space. For any t 2 .0; 1/, let M t D M..1 t /; t / denote the unique minimum of the function defined in (1.3). The set of all such minima, for t varying in .0; 1/, is a subset of Teichmüller space called the line of minima of and , and is denoted by L.; /. It is known that for any two points in Teichmüller space there is a line of minima joining them, but it is unknown whether such a line is unique. In 2003, Díaz and Series studied limits of certain lines of minima in the compactified Teichmüller space equipped with its Thurston boundary, T .S / [ PML.S /. They showed that for any line of minima .M t / t2.0;1/ associated to two measured laminations and such that is uniquely ergodic and maximal, the point M t converges as t ! 0 to the point Œ in Thurston’s boundary. They also showed that, at the opposite extreme, if is a rational lamination in the sense that is a weighted sum of closed P geodesics, D N iD1 ai ˛i , then the limit as t ! 0 of M t is equal to the projective class Œ˛1 C C ˛N ; that is, the point M t converges, but its limit is independent of the weights ai . In particular, this limit is (except in the special case where all the weights are equal) not the point Œ. Thus, if and are arbitrary, then the projective class of in Thurston’s boundary is not always the limit of M t as t ! 0. There is a formal analogy between these results and results obtained by Howard Masur in the early 1980s on the limiting behavior of some geodesics for the Teichmüller metric. We also note that Guillaume Théret, together with the author of this introduction, obtained analogous results on the behavior of stretch lines. These lines are geodesic for Thurston’s asymmetric metric. The fact that such results hold for lines of minima has a more mysterious character than in the cases of Teichmüller geodesics and of stretch lines, because up to now, unlike Teichmüller lines and stretch lines, lines of minima are not associated to any metric on Teichmüller space. Series made a study of lines of minima in the context of the deformation theory of Fuchsian groups. She established a relation between lines of minima and bending measures for convex core boundaries of quasi-Fuchsian groups. This work introduced the use of lines of minima in the study of hyperbolic 3-manifolds. Series showed (2005, based on a previous special case studied by herself and Keen) that when the Teichmüller space T .S/ is identified with the space F .S / of Fuchsian groups embedded in the space of quasi-Fuchsian groups Q.S /, a line of minima can be interpreted as the intersection with F .S/ of the closure of some pleating variety in Q.S /. This theory involves the complexification of Fenchel–Nielsen parameters, which combines earthquaking and bending, and it also involves a notion of complex length, defined on quasi-Fuchsian space by analytic continuation of the hyperbolic length function. More recently (2008), Choi, Rafi and Series discovered relations between the behavior of lines of minima and geodesics of the Teichmüller metric. They obtained a combinatorial formula for the Teichmüller distance between two points on a given line of minima, and they proved that a line of minima is quasi-geodesic with respect to the Teichmüller metric. The latter means that the distance between two points on a


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line of minima, with an appropriate parametrization, is uniformly comparable (in the sense of large-scale quasi-isometry) to the Teichmüller distance between these points. The proof of that result is based on previous work by Rafi. It involves an analysis of which closed curves get shortened along a line of minima, and the comparison of these curves with those that get shortened along the Teichmüller geodesic whose horizontal and vertical projective classes of measured foliations are the classes of the measured geodesic laminations and associated to the line of minima. Summing up, the account that Series makes of lines of minima in Chapter 3 includes the following topics: (1) The limiting behavior of lines of minima in Teichmüller space compactified by Thurston’s boundary. (2) The relation between lines of minima and quasi-Fuchsian manifolds. (3) The relation between lines of minima and the geodesics of the Teichmüller metric.

2 Part B. The group theory, 3 2.1 Mapping class groups versus arithmetic groups In Chapter 4 Lizhen Ji gives a survey of the analogies and differences between mapping class groups and arithmetic groups, and between Riemann’s moduli spaces and arithmetic locally symmetric spaces. This subject is vast and important, in particular because a lot of work done on mapping class groups and their actions on Teichmüller spaces (and other spaces) was inspired by results that were known to hold for arithmetic groups and their actions on associated symmetric spaces. Let us start with a few words on of arithmetic groups. This theory was initiated and developed by Armand Borel and Harish-Chandra. It is easy to give some very elementary examples of arithmetic groups: Z, Sp.n; Z/, SL.n; Z/ and their finite-index subgroups. But the list of elementary examples stops very quickly, and in general, to know whether a certain group that arises in a certain algebraic or geometric context is isomorphic or not to an arithmetic group is a highly nontrivial question. Important work has been done in this direction. A famous theorem due to Margulis, described as the “super-rigidity theorem”, gives a precise relation between arithmetic groups and lattices in Lie groups. Interesting examples of arithmetic groups are some arithmetic isometry groups of hyperbolic space found by E. B. Vinberg, in the early 1970s. Several analogies between mapping class groups and arithmetic groups were already highlighted in the late 1970s by Thurston, Harvey, Harer, McCarthy, Mumford, Morita, Charney, Lee and many other authors. Several questions on mapping class groups were motivated by results that were known to hold for arithmetic groups, sometimes with the hope that some property of arithmetic groups will not hold for mapping class groups, implying that the latter are not arithmetic.

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There are several fundamental properties that are shared by arithmetic groups and mapping class groups. For instance, any group belonging to one of these two classes is finitely presented, it has a finite-index torsion free subgroups, it is residually finite and virtually torsion free, it has only finitely many conjugacy classes of finite subgroups, its virtual cohomological dimension is finite, and it is a virtual duality group in the sense of Bieri and Eckmann. Furthermore, every abelian subgroup of a mapping class group or of an arithmetic group is finitely generated with torsion-free rank bounded by a universal constant, every solvable subgroup of such a group is of bounded Hirsch rank, it is Hopfian (that is, every surjective self-homomorphism is an isomorphism) and co-Hopfian (every injective self-homomorphism is an isomorphism), it satisfies the Tits alternative (every subgroup is either virtually solvable or it contains a free group on two generators), and there are several other common properties. For mapping class groups, all these properties were obtained in the 1980s, gradually and by various people, after the same properties were proved for arithmetic groups. The question of whether mapping class groups are arithmetic appeared explicitly in a paper by W. Harvey in 1979, Geometric structure of surface mapping class groups, at about the same time where mapping class groups started to become very fashionable. In the same paper, Harvey also asked whether these groups are linear, that is, whether they admit finite-dimensional faithful representations in linear groups. In 1984, Ivanov announced the result that mapping class groups of surfaces of genus 3 are not arithmetic. Harer provided the first written proof of this result in his paper The virtual cohomological dimension of the mapping class group of an orientable surface (published in 1986). The fact that a mapping class group cannot be an arithmetic subgroup of a simple algebraic group of Q-rank 2 follows from the fact that any normal subgroup of such an arithmetic group is either of finite index or is finite and central. The mapping class group does not have this property since it contains the Torelli group, which is normal and neither finite nor of finite index. Harer solved the remaining case (Q-rank 1) by showing that the virtual cohomological dimension of a mapping class group does not match the one of an arithmetic group. Goldman gave another proof of this fact, at about the same time Harer gave his proof. Ivanov published a proof that the mapping class group is not arithmetic in 1988. Despite the non-arithmeticity result, several interesting properties of mapping class groups that were obtained later on were motivated by the same properties satisfied by arithmetic groups, or more generally, by linear groups. Some of these properties can be stated in terms that are identical to those of arithmetic groups. For instance, Harer proved a stability theorem of the cohomology for mapping class groups of surfaces with one puncture as the genus tends to infinity, and he showed that mapping class groups are virtual duality groups. Harer and Zagier obtained a formula for the orbifold Euler characteristic of Riemann’s moduli space of surfaces with one puncture, and Penner obtained the result for n 1 punctures. The formula involves the Bernoulli numbers, as expected from the corresponding formula in the theory of arithmetic groups. Other properties can be stated in similar, although not identical, terms for mapping class groups and arithmetic groups.


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One of the most important general properties shared by arithmetic groups and mapping class groups, which gives the key to most of the results obtained, is the existence of natural and geometrically defined spaces on which both classes of groups act. The actions often extend to actions on various compactifications and boundaries, on cell-decompositions of the spaces involved, and on a variety of other associated spaces. In parallel to the fact that mapping class groups are not arithmetic, one can mention that Teichmüller spaces (except if their dimension is one) are not symmetric spaces in any good sense of the word. Likewise, moduli spaces are not locally symmetric spaces. Meanwhile, one can ask for Teichmüller spaces and moduli spaces several questions about properties that can be shared by symmetric spaces, for instance, regarding their compactifications or, more generally, bordifications. Borel–Serre bordifications of symmetric spaces were used to obtain results on the virtual cohomological dimension and on the duality properties of arithmetic groups. Similar applications were found for mapping class groups using Borel–Serre-like bordifications of Teichmüller space, which are partial compactifications. Lizhen Ji, in Chapter 4, makes a catalogue of the various compactifications of Teichmüller space and moduli space. He describes in detail the contexts in which these compactifications arise, and the known relations between the various compactifications. He discusses the question of when a compactification of moduli space can be obtained from a compactification of Teichmüller space, and he points out various analogies between the compactifications of Teichmüller space and moduli space on the one hand and those of symmetric spaces and locally symmetric spaces on the other hand. He addresses questions such as what is the analogue for moduli space of a Satake compactification of a locally symmetric space, in particular, of the quotient of a symmetric space by an arithmetic group. As we already mentioned, the question of the extent to which mapping class groups are close to being arithmetic is still an interesting question. One can mention the realization of an arithmetic group as a subgroup of a Lie group, that is inherent in the definition of an arithmetic subgroup, leading naturally to the question of the realization of mapping class groups as discrete subgroups of Lie groups. There are two instances where the mapping class group of a surface is arithmetic, namely, the cases where the surface is the torus or the once-punctured torus. In both cases, the mapping class group is the group PSL.2; Z/. The Teichmüller space in that case is the corresponding symmetric space, namely, the upper-half plane H2 . Furthermore, this identification between the Teichmüller space with H2 is consistent with the complex structures of the two spaces and the Teichmüller metric on the upperhalf plane coincides with the Poincaré metric. The action of the mapping class group on the Teichmüller space corresponds to the usual action of PSL.2; Z/ on H2 by fractional linear transformations. Lizhen Ji makes in Chapter 4 a list of notions that are inherent in the theory of arithmetic groups and that have been (or could be) adapted to the theory of mapping class groups. This includes the notions of irreducibility, rank, congruence subgroup,

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parabolic subgroup, Langlands decomposition, existence of an associated symmetric space, Furstenberg boundaries and Tits buildings encoding the asymptotic geometry, reduction theory, the Bass–Serre theory of actions on trees, and there are many others. All these questions from the theory of arithmetic groups gave already rise to very rich generalizations and developments that were applied to the study of mapping class groups and their actions on various spaces. The curve complex is an important ingredient in the study of mapping class groups. It was introduced as an analogue for these groups of buildings associated to symmetric spaces and locally symmetric spaces. Curve complexes turned out to be useful in the description of the large-scale geometry and the structure at infinity of mapping class groups and of Teichmüller spaces. Volume IV of this Handbook will contain a survey by Lizhen Ji, entitled Curve complexes versus Tits buildings: structures and applications, that explores in great detail the relation between curves complexes and Tits buildings. Another topic of interest in both theories is the study of fundamental domains. It is well known that producing a good fundamental domain for an action and understanding its geometry gives valuable information on the quotient space. An idea that appears in the survey by Lizhen Ji is to make a relation between Minkowski reduction theory and mapping class group actions on Teichmüller spaces, from the point of view of producing intrinsically defined fundamental domains. In a generalized form, reduction theory can be described as the theory of finding good fundamental domains for group actions. This theory was developed by Siegel, Borel and HarishChandra and others. Gauss worked out the reduction theory for quadratic forms. We recall in this respect that the theory of quadratic forms is related to that of moduli spaces by the fact that H2 D SL.2; R/=SO.2/ is also the space of positive definite quadratic forms of determinant 1. Poincaré polyhedra and Dirichlet domains are examples of good fundamental domains. The Siegel domain for the action of SL.2; Z/ on the hyperbolic plane is a prototype for both theories, arithmetic groups and mapping class groups. The upper-half plane H2 is the space of elliptic curves in algebraic geometry, and at the same time it is the Teichmüller space of the torus equipped with the mapping class group action. In the case where there is no obvious good fundamental domain, one may try to find rough fundamental domains. In the sense used by Ji in this survey, this means that the natural map from the fundamental domain to the quotient space is finite-to-one. Finding a good fundamental domain, or even a rough fundamental domain, in the case where the quotient is non-compact, is not an easy matter. Motivated by reduction theory, Ji addresses the question of the existence of various kinds of fundamental domains (geometric, rough, measurable, etc.), and of studying finiteness and local finiteness properties of such domains in relation to questions of finite generation and of bounded generation, and other related questions on group actions.


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2.2 Simplicial actions of mapping class groups Chapter 5, written by John McCarthy and myself, is a survey of several natural actions of extended mapping class groups of surfaces of finite type on various simplicial complexes. The earliest studies of actions of mapping class groups on combinatorial complexes that gave rise to substantial results are the actions on the pants complex and on the cut system complex. These studies were done by Hatcher and Thurston in the mid 1970s, at the time Thurston was developing his theory of surface homeomorphisms. This work paved the way for a theory that included a variety of other simplicial actions of mapping class groups. The curve complex was introduced slightly later (in 1977) by Harvey. While the main motivation of Hatcher and Thurston for studying the actions on the pants complex and the cut system complex was to get a finite presentation of the mapping class group, the original motivation of Harvey in studying the curve complex was to construct some boundary structure for Teichmüller space. After the curve complex was introduced, several authors studied it from various points of view. Ivanov proved in the 1990s the important result stating that (except for a few surfaces of low genus and small number of boundary components) the simplicial automorphism group of the curve complex coincides with the natural image of the extended mapping class group in that group.5 Later on, Ivanov used this action to give a new and more geometric (as opposed to the original analytic) proof of the celebrated theorem obtained by Royden in 1971 saying that (again, except for a few surfaces of low genus and small number of boundary components) the natural homomorphism from the extended mapping class group to the isometry group of the Teichmüller metric is an isomorphism. Ivanov’s proof is based on a relation between the curve complex and some boundary structure of Teichmüller space, a relation that was already suspected by Harvey. Masur and Minsky (1996) studied the curve complex, endowed with its natural simplicial metric, from the point of view of large-scale geometry. They showed that this complex is Gromov hyperbolic. Klarreich (1999) identified the Gromov boundary of the curve complex with a subspace of unmeasured lamination space UML, that is, the quotient space of measured lamination space obtained by forgetting the transverse measure. The Gromov boundary of the curve complex is the subspace of UML consisting of minimal and complete laminations. Here, a measured lamination is said to be complete if it is not a sublamination of a larger measured lamination, and it is called minimal if there is a dense leaf (or, equivalently, every leaf is dense) in its support. Now we mention results on the other complexes. 5 Ivanov’s original work did not include the case of surfaces of genus 0 and 1, and this was completed by Korkmaz. The work of Korkmaz also missed the case of where the surface S is a torus with two holes, which was completed by Luo. Luo also gave an alternative proof of the complete result.

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The pants graph is the 1-skeleton of the Hatcher–Thurston pants complex. The hyperbolicity of the pants graph was studied by Brock and Farb (2006). Brock (2003) proved that the pants graph of S is quasi-isometric to the Teichmüller space of S endowed with its Weil–Petersson metric. Margalit (2004) proved that (again, with the exception of a few surfaces of low genus and small number of boundary components) the simplicial automorphism group of the pants graph coincides with the natural image of the extended mapping class group in that automorphism group. Other complexes with vertex sets being homotopy classes of compact subsets of the surface that are invariant by the extended mapping class group action were studied by various authors. We mention the arc complex, the arc-and-curve complex, the ideal triangulation complex, the Schmutz graph of non-separating curves, the complex of non-separating curves6 , the complex of separating curves, the Torelli complex, and there are other complexes. All these actions were studied in detail, and each of them presents interesting features. The study of mapping class group actions on simplicial complexes is now a large field of research, which we may call the subject of “simplicial representations of mapping class groups”. The aim of Chapter 5 is to give an account of some of the simplicial actions, with a detailed study of a complex that I recently introduced with McCarthy, namely, the complex of domains, together with some of its subcomplexes. The complex of domains is a flag simplicial complex which can be considered as naturally associated to the Thurston theory of surface diffeomorphisms. The various pieces of the Thurston decomposition of a surface diffeomorphism in Thurston’s canonical form, which we call the thick domains and annular or thin domains, fit into this flag complex. Unlike the curve complex and the other complexes that were mentioned above and for which, for all but a finite number of exceptional surfaces, all simplicial automorphisms are geometric (i.e. induced by surface homeomorphisms), the complex of domains admits non-geometric simplicial automorphisms, provided the surface has at least two boundary components. As a matter of fact, if the surface has at least two boundary components, then the simplicial automorphism group of the complex of domains is uncountable. The non-geometric automorphisms of the complex of domains are associated to certain edges of this complex that are called biperipheral, and whose vertices are represented by biperipheral pairs of pants and biperipheral annuli. A biperipheral pair of pants is a pair of pants that has two of its boundary components on the boundary of the surface. A biperipheral annulus is an annulus isotopic to a regular neighborhood of the essential boundary component of a biperipheral pair of pants. The complex of domains can be projected onto a natural subcomplex by collapsing each biperipheral edge onto the unique vertex of that edge that is represented by a regular neighborhood of the associated biperipheral curve. In this way, the computation of the simplicial automorphism group of the complex of domains is reduced to the computation of the simplicial automorphism group of this subcomplex, called the 6 The one-skeleton of the complex of non-separating curves is different from the Schmutz graph of nonseparating curves.


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truncated complex of domains. With the exception, as usual, of a certain finite number of special surfaces, the simplicial automorphism group of the truncated complex of domains is the extended mapping class group of the surface. From this fact, we obtain a complete description of the simplicial automorphism group of the complex of domains. Besides the interesting fact that the automorphism groups of most of the complexes mentioned are isomorphic to extended mapping class groups, it turns out that the combinatorial data (links of vertices, links of links of vertices, etc.) are sufficient, in many cases, to reconstruct the topological objects that these vertices represent. Thus, in many ways, the combinatorial structure of the complexes “remembers” the surface and the topological data on the surface that were used to define the complexes. This is another theme of Chapter 5, and it is developed in detail in the case of the complex of domains and the truncated complex of domains. In Chapter 6, Valentina Disarlo studies the coarse geometry of the complex of domains D.S/ equipped with its natural simplicial metric. She proves that for any subcomplex X.S/ of D.S/ containing the curve complex C.S /, the natural simplicial inclusion C.S/ ! X.S/ is an isometric embedding and a quasi-isometry. She also proves that with the exception of a few surfaces of small genus and small number of boundary components, the arc complex A.S / is quasi-isometric to the complex P@ .S / of peripheral pairs of pants, and she gives a necessary and sufficient condition on S for the simplicial inclusion P@ .S/ ! D.S/ to be a quasi-isometric embedding. She then applies these results to the study of the arc and curve complex AC.S /. She gives a new proof of the fact that AC.S/ is quasi-isometric to C.S /, and she discusses the metric properties of the simplicial inclusion A.S / ! AC.S /.

2.3 Minimal generating sets for mapping class groups Chapter 7 by Mustafa Korkmaz is a survey on generating sets of minimal cardinality for mapping class groups of surfaces of finite type. Three types of generating sets are considered: Dehn twists, torsion elements and involutions. Let us first discuss the case of orientable surfaces. It is well known that Dehn twists generate the mapping class group. Such generators were first studied by Dehn in the 1930s, who showed that a finite number of them suffice. Humphries (1979) found a minimal set of Dehn twist generators. Maclachlan (1971) showed that the mapping class group is generated by a finite number of torsion elements, and he used this fact to deduce that moduli space is simply connected. McCarthy and Papadopoulos (1987) showed that the mapping class group is generated by involutions. Luo (2000), motivated by the case of SL.2; Z/ and by work of Harer, showed that torsion elements of bounded order generate the mapping class group of a surface with boundary, except in the special case where the genus of the surface is 2 and the number of its boundary components is of the form 5k C 4 for some

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integer k. In this exceptional case, Luo showed that the torsion elements generate a subgroup of index 5 of the mapping class group. Brendle and Farb (2004) solved Luo’s question in the case of closed orientable surfaces, by showing that there is a finite generating set of involutions whose cardinality does not depend on the genus. Besides surveying minimal generating sets, Korkmaz provides some background material on the set of relations between Dehn twist elements in mapping class groups. Mapping class groups of non-orientable surfaces are also discussed in Chapter 7. In this case, the mapping class group is defined as the group of all homotopy classes of homeomorphisms (there is no orientation involved). Lickorish (1963) showed that Dehn twists generate a subgroup of index two in this mapping class group, and he produced a system of generators for it: Dehn twists along two-sided curves and the isotopy class of a homeomorphism called a “cross-cap slide”, and supported on a Klein bottle embedded in the surface. Chillingworth (1969) showed that the mapping class group is generated by finitely many elements. Korkmaz (2002) extended Chillingworth’s result to the case of surfaces with boundary. Motivated by the work done in the orientable case, Szepietowski obtained results on involutions in mapping class groups of non-orientable surfaces. He showed that the mapping class group of a closed non-orientable surface is generated by four involutions. The chapter ends with some open questions.

2.4 Mapping class groups and 3-manifold topology Chapter 8 by Kazuo Habiro and Gwénaël Massuyeau, and Chapter 9 by Takuya Sakasai concern relations between mapping class groups and 3-manifolds. The two chapters are complementary to each other. In each of them, the authors study a monoid that arises in 3-manifold topology and that is an extension of the mapping class group. The elements of this monoid are called homology cobordisms by Habiro and Massuyeau, and homology cylinders by Sakasai.7 The results of these two chapters especially apply to a surface S D Sg;1 , that is, a compact oriented surface of genus g 1 with one boundary component.8 The mapping class group D g;1 in this context is defined as the group of isotopy classes of orientation-preserving homeomorphisms that fix the boundary pointwise. The basepoint of the fundamental group 1 .S/ is chosen on the boundary, and in this way the mapping class group acts naturally on 1 .S /. This fundamental group is free on 2g generators, and by a result attributed to Dehn, Nielsen and Baer, the natural homomorphism ! Aut.1 .S// is injective. Thus, we have a natural monomorphism 7 Habiro and Massuyeau call homology cylinder an object that is more special than the homology cylinder in the sense of Sakasai. Likewise, Sakasai uses the term homology cobordism in a different sense than the one used by Habiro and Massuyeau, namely, he uses it in association with an equivalence relation involving 4-manifolds. This is a very unfortunate inconsistency in the mathematics literature. There was no obvious way to make things uniform in this Handbook, and I decided to leave the authors stick to the terminology used in the papers referred to in their contribution. 8 We note however that most of the constructions in Chapter 8 by Habiro and Massuyeau also apply to closed surfaces.


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from the mapping class group of S into the automorphism group of a free group. This is an instance of the general fact that the theory of free groups is much more present in the study of mapping class groups of surfaces with boundary than in that of closed surfaces. The theory of the monoid of homology cobordisms (respectively homology cylinders) is based on surgery techniques that were introduced by Goussarov and Habiro independently in the second half of the 1990s.9 The aim of these techniques was to prove general properties of finite type invariants for 3-manifolds and for links in these manifolds. We recall that the expression “finite type invariant” in 3-manifold theory refers to invariants that behave polynomially with respect to some surgery (that is, cut-andpaste) operations. Examples of such invariants are the cohomology ring of a manifold (which is a degree-one finite type invariant), the Rochlin invariant for closed spin 3-manifolds (also degree-one finite type invariant) and the Casson invariants (degree-two invariants). The Johnson and Morita theories of the Torelli group of surfaces also involve finite type invariants of 3-manifolds. It seems that Ohtsuki was the first to introduce the notion of finite type invariant, in the setting of integral homology spheres, and he constructed the first examples. Goussarov and Habiro extended this notion to all 3-manifolds, and they developed the necessary techniques to study the general case. The surgery techniques introduced by Goussarov and Habiro are called clover and clasper techniques respectively.10 These theories are essentially equivalent to each other. They originate in a surgery theory called Borromean surgery, due to Matveev. Clasper calculus can also be seen as a topological analogue of commutator calculus in groups. Like Matveev’s surgery, clasper surgery does not affect the homology of the underlying 3-manifold. Using the techniques they introduced independently, Goussarov and Habiro obtained results similar to each other. These techniques were also used to obtain a topological interpretation of “Jacobi diagrams”, which may be compared to Feynman diagrams and which appear in the theory of universal finite type invariants. The Johnson homomorphisms, the Magnus representation and several other algebraic notions that pertain to mapping class group theory extend to the setting of the homology cobordism (respectively homology cylinder) monoid. To present in more precise terms the chapter by Habiro and Massuyeau, we recall a few definitions. A cobordism of a surface Sg;1 is a pair .M; m/ where M is a compact connected oriented 3-manifold and where m W @ Sg;1 Œ1; 1 ! @M is an orientation-preserving homeomorphism. The homeomorphism m is regarded as a parametrization of @M , and the two inclusions of Sg;1 into @M obtained by restricting m to the upper and lower factor of Sg;1 Œ1; 1 allow one to talk about the top and the bottom boundary of M . 9 Habiro wrote his thesis, on this theory, in 1997. Goussarov did not publish much. He passed away in a drowning accident in 1999. In both cases, the first papers on the theory appeared in print around the year 2000. 10 It seems that the word clasper is the one that is mostly used today, and in this introduction we shall use it.

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Two cobordisms .M; m/ and .M 0 ; m0 / of the same surfaces are said to be homeomorphic if there exists an orientation-preserving homeomorphism f W M ! M 0 such that f j@M B m D m0 . Composition of cobordisms .M; m/ and .M 0 ; m0 / of S is defined by gluing the bottom boundary of M 0 to the top boundary of M . A homology cobordism .M; m/ of S is a cobordism whose top and bottom inclusions induce isomorphisms between the homology groups H .S / and H .M /. Homology cobordisms are stable under composition. This operation makes the set of homology cobordisms (up to the homeomorphism relation defined above) a monoid, called the homology cobordism monoid and denoted by C .S /. The unit in that monoid is the homology cobordism .S Œ1; 1; Id f1g; Id f1g). The definition of homology cobordism is due to Goussarov and Habiro (independently), and it was used by Garoufalidis and Levine in the study of finite-type invariants of 3-manifolds. The recent developments in the theory of homology cobordisms are due to Garoufalidis and Levine, Habiro, Massuyeau, Meilhan, Habegger, Sakasai, Morita, and there are certainly other authors. Denoting as before by .S/ the mapping class group of S , there is an embedding .S/ ! C .S / obtained by the mapping cylinder construction, in which the 3-manifold M is defined as the product S Œ1; 1, the top boundary homeomorphism being the given element of .S/ and the lower boundary homeomorphism being the isotopy class of the identity map of S. The map .S/ ! C .S/ is not surjective. Surgery along claspers provides examples of homology cylinders that are not obtained as images of elements of the mapping class group. Since we are dealing with the homology of the surface, the Torelli subgroup of the mapping class group plays a central role in this theory. We recall that the Torelli group is the subgroup of .S/ that consists of the elements that induce the identity on homology. A homology cylinder over S (in the sense of Habiro and Massuyeau) is a cobordism that has the same homology type as the trivial cobordism, that is, .S Œ1; 1; Id/. Like the set of homology cobordisms, the set of homology cylinders is stable under composition, and it forms a submonoid C .S / C .S /. It turns out that the image of the Torelli group .S/ by the embedding .S / ! C .S / is contained in the submonoid C .S / of homology cylinders. The map .S / ! C .S / is injective, and therefore C .S / can be thought of as an extension of the Torelli group. The image of .S / (respectively of .S/) in C.S/ (respectively in C .S /) is the group of units (that is, the group of invertible elements) of C.S/ (respectively of C .S /). The study of the inclusion C .S/ C.S/ can be done using finite type invariants of 3-manifolds, in particular clasper calculus. In Chapter 8, Habiro and Massuyeau present the recent developments in the theory of the monoid C.S/ of homology cobordisms, with special attention given to the


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submonoid C .S/ of homology cylinders, and to its relation to the Torelli group. There is also a strong relation to the Johnson homomorphisms. We recall that the Johnson homomorphisms are defined on a filtration of the mapping class group, and they give a kind of measure of the unipotent part of the action of the Torelli group .S/ on the second nilpotent truncation of 1 .S /. The first Johnson homomorphism was introduced in the early 1980s by Johnson. In 1993, Morita studied in detail a sequence of homomorphisms that extend the first Johnson homomorphism. These homomorphisms are sometimes referred to as the “higher Johnson homomorphisms”.11 In 2005 Garoufalidis and Levine published a paper12 in which the Johnson homomorphisms and their generalization by Morita were extended to the setting of homology cobordisms. There is a Johnson filtration fg .k/gk , and the k-th Johnson homomorphism is a homomorphism g .k/ from g .k/ to a certain finitely-generated free abelian group arising from the k-th graded piece of a certain graded Lie algebra. Chapter 8 also contains a report on an “infinitesimal version” of the Dehn–Nielsen representation, first defined by Massuyeau, and of which the homomorphisms of Johnson and Morita become special cases. This work uses the Malcev Lie algebra of 1 .S / instead of the group 1 .S/ itself. We recall here that by work of Malcev published in 1949, every torsion-free finitely generated nilpotent group can be embedded as a discrete co-compact subgroup of a Lie group. From this, one can associate to any finitely generated group a tower of nilpotent Lie groups. To this tower is then associated a tower of corresponding Lie algebras. Applied to the case where D 1 .S /, this gives a tower whose inverse limit is the Malcev Lie algebra associated to 1 .S /. In 1988, Le, Murakami and Ohtsuki, based on the Kontsevich integral and using surgery presentations in the 3-sphere, constructed an invariant of closed oriented 3-manifolds, which is now called the LMO invariant. This invariant is particularly interesting for the study of homology spheres. In 2008, Cheptea and Habiro–Massuyeau extended the LMO invariant to compact oriented 3-manifolds with boundary. In this work, this extension is presented as a functor defined on a certain cobordism category, which the authors called the LMO functor, and which is a kind of TQFT theory. This cobordism category contains the homology cylinder monoid. In 2009, Habiro and Massuyeau defined the LMO homomorphism on C .S / by restriction of the LMO functor. They obtained and studied a monoid homomorphism, which they called the “LMO homomorphism”, from C .S/ to the algebra of Jacobi diagrams. This homomorphism provides a diagrammatic representation of the monoid C .S /, and it is useful in the study of the action of C .S/ on the Malcev Lie algebra of 1 .S /. It is 11 Chapter 7 of Volume I of this Handbook, written by Morita, is a survey on mapping class groups and related groups, and it contains a section on the Johnson homomorphisms. Let us mention by the way that some ideas that are at the basis of the Johnson homomorphisms can be found in the work of Andreadakis (1965) who introduced and studied the filtration on fAut.Fn /.k/gk that is induced from the action of Aut.Fn / on nilpotent quotients of Fn . 12 The paper, entitled Tree-level invariants of three manifolds, Massey products and the Johnson homomorphism, was published in 2005 in the proceedings of a conference, but it seems that the results were obtained before 2001, when Levine published a paper on related work.

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injective on the image of the Torelli group. It is also a useful tool in the study of the Johnson and Morita homomorphisms, and more generally, in the study of the way the Torelli group embeds into the monoid C .S /. In Chapter 8, Habiro and Massuyeau also report on a filtration, called Y -filtration and defined by clasper surgeries, of the monoid of homology cylinders. The Y filtration is an analogue of the lower central series of the Torelli group. The graded abelian group associated to this filtration is computed (in the case of rational coefficients) diagrammatically using the LMO homomorphism and the clasper calculus. The first quotient of this graded abelian group, that is, the quotient C .S /=Y2 , is computed in a way analogous to the way Johnson computed the abelianization of the Torelli group, that is, using the (first) Johnson homomorphism and the Birman–Craggs homomorphism. The authors in Chapter 8 also report on this generalized Birman–Craggs homomorphism, defined on C .S/. Garoufalidis and Levine introduced a group H .S / whose elements are homology cobordism classes of homology cobordisms. The mapping class group still embeds in the group H .S/. Habiro and Massuyeau present some recent work on this group, and this group is also studied in Chapter 9 by Sakasai. Chapter 9 by Sakasai provides another point of view on the theory of homology cobordisms, which are called there homology cylinders (and we shall adopt from now on the latter terminology). The author reviews the classical theory of the Magnus representation and its extension to the setting of these homology cylinders. The extension of the Magnus representation to homology cylinders was introduced by Sakasai. In this work, Sakasai heavily used various localization and completion techniques of groups and rings that are due to Vogel, Le Dimet, Levine, Cohn and others. These techniques had previously been used in the algebraic theory of knots and links. We recall that the Magnus representation of the mapping class group g;1 is a crossed homomorphism from g;1 into the group GL.2g; ZŒ1 .Sg;1 //. The definition of this representation is usually presented using Fox calculus and Fox derivation. Chapter 9 includes the necessary background on Fox calculus. We recall that a Fox derivation (or Fox derivative) on a free group Fn with a free generating set 1 ; : : : ; n is a map denoted, for i D 1; : : : ; n, by @ W Fn ! ZŒFn : @i This notation and the name “Fox derivative” reflect the fact that Fox derivation satisfies rules which look formally like the rules of partial derivation on differentiable functions. @i D ıij where ıij is the Kronecker delta; there is a “chain For instance, one has @ j rule” for Fox derivatives, a “Leibniz rule” for the Fox derivative of products, and so on. The Fox differential calculus produces matrix representations of free groups of finite rank, of automorphism groups of these free groups, and of subgroups of these automorphism groups.


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In its original form, the Magnus representation is a matrix representation of free groups and their automorphism groups. The Magnus representation, which was first defined as a representation of the automorphism group of a free group, was later on adapted to the setting of the mapping class groups by Morita, and it plays an important role in the study of the Johnson homomorphisms. Using the Dehn–Nielsen–Baer theorem which injects the group g;1 into the group Aut.Fn / by using the natural action of g;1 on the fundamental group of the surface Sg;1 , one obtains the Magnus representation (which is a crossed homomorphism), r W g;1 ! GL.2g; ZŒ1 .Sg;1 //: By restriction and after reduction of the coefficients 1 .S / ! H , where H D 1 =Œ1 ; 1 , one has also a Magnus representation of the Torelli group g;1 (which is a genuine homomorphism) g;1 ! GL.2g; ZŒH /: The Magnus representation of the Torelli group was studied by various authors, with the hope of better understanding that group. Morita was the first who used the Magnus representation g;1 ! GL.2g; ZŒH / defined through Fox derivation, to get results about the mapping class group. Suzuki showed in 2002 that this representation of the Torelli group in GL.2g; ZŒH / is not faithful for g 2. Church and Farb obtained in 2009 that the kernel of this representation is not finitely generated, and that the first homology group of that kernel has infinite rank. Morita proved that the Magnus representation of the mapping class group is symplectic in some twisted sense. Chapter 9 by Sakasai also contains some algebraic background which should be useful for geometers, namely, a quick survey of group homology and cohomology, a short exposition of the Fox calculus and of other concepts and tools that are used in the definitions of the Magnus representation and its various extensions. Furthermore, Sakasai reviews some invariants of homology cylinders that are obtained through the Magnus representation. He also describes several abelian quotients of the monoid and of the homology cobordism groups of homology cylinders.

2.5 Thompson’s groups Chapter 9, by Louis Funar, Christophe Kapoudjian and Vlad Sergiescu, is on Thompson’s groups. These are finitely presented groups that were introduced by Richard Thompson in 1965, originally in connection with certain questions in mathematical logic. The theory of these groups was later on developed in several directions, in relation to word problems, combing properties of groups, Dehn functions, normal form theory, automaticity and to other questions. It also turned out that Thompson’s groups are related to braid groups, to surfaces of infinite type and their mapping class groups, to asymptotic Teichmüller spaces, and to quantization of Teichmüller spaces. In fact,

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Thompson’s groups are in some precise sense mapping class groups of some infinite type surfaces. For all these reasons it seemed natural to have a chapter on Thompson’s groups in this Handbook. First, let us recall the definitions. There are three classes of Thompson’s groups and, classically, they are denoted by F , T and V . The elements of the group F are the piecewise linear homeomorphisms f of the unit interval Œ0; 1 satisfying the following properties: (1) The homeomorphism f is locally linear except at finitely many points which are dyadic rational numbers. (A dyadic rational number is of the form p=2q where p and q are positive integers.) (2) On each subinterval of Œ0; 1 on which f is linear, its derivative is a power of 2. The elements of the group T are the piecewise-linear homeomorphisms f of the circle S 1 D Œ0; 1=0 1 with the following properties: (1) The homeomorphism f preserves the images in S 1 of the set of dyadic rational numbers. (2) The homeomorphism f is differentiable except at a finite set of points contained in the image by the natural projection Œ0; 1 ! S 1 of the dyadic rational numbers. (3) On each interval where f is linear, the derivative of f is a power of 2. The elements V are right-continuous bijections f of S 1 D Œ0; 1=0 1 that have the following properties: (1) The map f preserves the images in S 1 of the set of dyadic rational numbers. (2) The map f is differentiable except at a finite set of points contained in the image of the dyadic rational numbers. (3) On each maximal interval where f is differentiable, f is linear and its derivative is a power of 2. In all cases (F , T and V ), the composition of two bijections in the given class is in the same class. This makes F , T and V groups, the group operation being composition of maps. The three classes of groups turned out to be important. They provided counterexamples to several natural conjectures in group theory. For instance, McKenzie and Thompson used these groups to show that there are finitely presented groups that have unsolvable word problems (1973). The same authors proved that the groups T and V are infinite, simple and finitely presented, providing the first examples of such groups. The group F has a standard two-generator presentation. Brin and Squier proved that this group does not contain any non-abelian free group (1985). There are several open questions about the group F , e.g. whether it is amenable (Geoghegan conjectured in 1979 that the answer is yes). A low-dimensional topologist will surely notice that such piecewise-linear actions with constraints on the nonlinearity set appear in the theory of the action of the mapping


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class group on Thurston’s space of measured foliations. Thurston introduced the notion of piecewise integral projective transformation, as a property satisfied by the action of the mapping class group on measured lamination space. He showed that the Thompson groups F and T have this property. In fact, Thurston was interested in Thompson’s groups in more than one way. He proved that the group T has a representation as a group of C 1 diffeomorphisms of the circle. Ghys and Sergiescu proved later on the stronger result saying that T is conjugate (by a homeomorphism) to a group of C 1 diffeomorphisms of the circle (1987). In 1991, Greenberg and Sergiescu discovered a relation between Thompson’s groups and braid groups, by studying an action of the derived subgroup F 0 of the Thompson group F . Using this action they defined a morphism F 0 ! Out.B1 /, where B1 is the stable braid group, a braid group on a countable number of strands. They deduced the existence of an acyclic extension of F 0 by the stable braid group B1 . In 2001, de Faria, Gardiner and Harvey showed that Thompson’s group F can be realized as a mapping class group of an infinite type surface in the quasiconformal setting. Here, the surface is the complement in the complex plane of a Cantor set, and the Teichmüller space is the space of asymptotically conformal deformations of that surface. In this setting, the marking that defines the elements of Teichmüller space is an asymptotically conformal homeomorphism, meaning that it is quasiconformal and that for every > 0 there exists a compact subset of the surface such that the complex dilatation of the surface is bounded by 1 C on the complement of this compact set.13 The result says that Thompson’s group F admits an embedding into the group of isotopy classes of orientation-preserving homeomorphisms of a surface S0;1 of genus 0 and of infinite topological type. In 2005, Kapoudjian and Sergiescu obtained a similar result for the group T . Whereas Faria, Gardiner and Harvey worked in the quasiconformal setting, Kapoudjian and Sergiescu worked in the topological setting. They introduced in 2004 the notion of asymptotically rigid homeomorphism in the study of Thompson groups, and this notion was extensively used in later works by Funar and Kapoudjian. All Thompson groups have interesting finite and infinite presentations. Some of these presentations use surface homeomorphisms, which make another relation to mapping class groups. There is also a description of each element in the three classes of Thompson’s groups in terms of operations on objects called rooted binary tree pair diagrams. Here, a pair of trees associated to a group element describes the subdivision of the domain and range into subintervals on which the element acts linearly. The tree interpretation makes the Thompson groups related to the so-called Ptolemy groupoids, a category whose objects are marked Farey tessellations and which are also closely related to mapping class groups. 13 We recall that there are various non-equivalent definitions of Teichmüller space in the case of surfaces of infinite type. The quasiconformal setting provides one possible definition, and the hyperbolic setting provides other definitions which in general are not equivalent.

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Based on work of Penner on the universal Teichmüller space, Funar and Kapoudjian showed in 2004 that Thompson’s group T is isomorphic to a “universal mapping class group”, a finitely presented group of mapping classes that are “asymptotically rigid” of the surface S0;1 , which is itself equipped with a certain rigid structure associated to a hexagon decomposition. The universal mapping class group contains all mapping class groups of compact surfaces of genus zero, and it also encodes the mutual relations between these groups. The results are formulated in terms of the Ptolemy groupoid and for this reason the group T also carries the name the Ptolemy– Thompson group. The Ptolemy–Thompson group T is seen as the analogue of the mapping class group of the hyperbolic plane. A dilogarithmic representation of the Ptolemy groupoid induces a representation of the Ptolemy–Thompson group. In the same work, Funar and Kapoudjian discovered a relation between the Ptolemy groupoid and a pants decomposition complex associated to the surface S0;1 of genus 0 and of infinite topological type which generalizes the Hatcher–Thurston pants decomposition complex of compact surfaces. The pants decomposition complex of S0;1 is defined here as an inductive limit of the pants decomposition complexes of compact subsurfaces of S0;1 . To a pants decomposition is associated a hexagon decomposition which is intuitively defined by distinguishing a “visible” and a “hidden” side of S0;1 (this pants decomposition defines the rigid structure that we alluded to above, and this rigid structure is used to define the notion of asymptotic rigidity) and a result by Funar and Kapoudjian says that the Thompson group T is the group of asymptotically rigid mapping classes of that surface which preserve the decomposition into hidden/visible sides. Using the genus-0 infinite-type Hatcher–Thurston complex, and by a method which parallels the work of Hatcher and Thurston on mapping class groups of surfaces of finite type, the authors showed that a certain group defined as the group of asymptotically rigid mapping classes of S0;1 is finitely presented. Funar and Kapoudjian then introduced in 2008 a group T called the braided Ptolemy–Thompson group, which is another extension of T by the stable braid group B1 . They showed that the group T and therefore the group T , are asynchronously automatic, a result that is an analogue of a result by Mosher saying that mapping class groups of surfaces of finite type are automatic. The complete analogue of Mosher’s result is presented as an open problem. By quantization, a projective representation of T (called a dilogarithmic representation) is obtained. The authors of Chapter 10 also present a recent result by Funar and Sergiescu saying that this representation comes from a central extension of T whose class is 12 times the Euler class generator. The relation to clusters is also made. In addition, several extensions of Thompson groups are presented in Chapter 10; for instance, the extension of V by the so-called braided Thompson group of Brin– Dehornoy, its extension by the so-called universal mapping class group and its extension by the asymptotically rigid mapping class group in infinite genus. The authors include all such extensions in a unified setting arising from a functorial algebraic construction, defined on a category whose objects are called cosimplicial symmetric group extensions. This algebraic formalism is also used to describe the action of the


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Grothendieck–Teichmüller group acts on some group completions. Other works on the relation to the Grothendieck–Teichmüller theory were done by Lochak, Nakamura and Schneps. The authors also present the relation between this theory and the theory of the so-called braided Houghton groups, studied by Degenhardt and Dynnikov, and which are also mapping class groups of surfaces of infinite type. We finally note that besides sharing properties with mapping class groups, Thompson’s groups have connections with arithmetic groups.

3 Part C. The algebraic topology of mapping class groups and moduli spaces 3.1 The intersection theory of moduli space Intersection theory is a classical subject in algebraic geometry. Its main object of study is the intersection of subvarieties in an algebraic variety. The theory can be traced back to works by eighteenth century mathematicians on the intersection of hypersurfaces in Rn . The theorem of Bézout, which states that the number of intersection points of two plane algebraic curves is equal to the product of their degrees, is considered as belonging to intersection theory. In the modern theory, intersections are computed in the cohomology ring. The moduli space Mg;n of Riemann surfaces of genus g with n marked points, x g;n , are generalized algebraic together with its Deligne–Mumford compactification M varieties. More precisely, they are algebraic stacks of complex dimension 3g 3 C n. A stack is the analogue, in algebraic geometry, of an orbifold in the analytic setting. In the algebro-geometric setting, Riemann surfaces are called curves (manifolds of complex dimension one) and the elements of the compactification are surfaces with x g;n n Mg;n , that is, a stable curve, nodes called stable curves. An element of M is a singular complex algebraic curve whose singularities are nodes, which are the isolated singularities of the simplest possible kind: the local model of a node is the neighborhood of the origin of the plane algebraic set defined by the equation xy D 0. Topologically, a node is the singularity of two real 2-dimensional disks identified at x g;n is naturally considered as the moduli space their centre. The compactification M of surfaces with nodes. As algebro-geometric orbifolds, the moduli space and its compactification have an intersection theory, with its associated algebro-geometric apparatus on homology and cohomology; it is equipped with complex vector bundles which have their Chern classes, a Grothendieck–Riemann–Roch formula, and so on. Chapter 11 by Dmitry Zvonkine contains a review of the intersection theory of the x g;n , moduli space of curves Mg;n and of its Deligne–Mumford compactification M also called the moduli space of stable curves.

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A strong impetus to the study of the intersection theory of moduli spaces that especially caught the attention of topologists and geometers was given in 1991 by Witten, who conjectured the existence of a generating recursion formula for all interx g;n ; Q/, called -classes. These section numbers of some special elements of H 2 .M classes, 1 ; : : : ; n , also known as tautological classes, are the first Chern classes of some natural line bundles L1 ; : : : ; Ln that are themselves “tautological” in the sense that the fiber at each point is precisely the cotangent line to the corresponding curve or stable curve representing the point. The tautological classes are natural with respect to forgetful maps and attaching maps performed at the level of Riemann surfaces and of stable curves. They generate a subring called the tautological cohomology ring. Witten’s conjecture was proved by Kontsevich in 1992, and several other proofs of this conjecture were given later on. In 2004, Mirzakhani made a relation between the intersection numbers of the -classes and the Weil–Petersson volume of moduli space. In Chapter 11 of this volume, Zvonkine starts by introducing the basic objects in the theory, namely, the moduli space Mg;n of Riemann surfaces of genus g with x g;n , the universal curve n marked points, its Deligne–Mumford compactification M x x Cg;n over Mg;n , and the universal curve Cg;n over Mg;n . He gives a description of x g;n . He then introduces the the smooth orbifold structure of the spaces Mg;n and M x g;n . He gives a wide class of explicit examples tautological cohomology classes on M of tautological classes and he computes intersection numbers between them. The computations are based on the Grothendieck–Riemann–Roch Theorem, and on a study of pull-backs of such classes under attaching and forgetful maps. In order to make the exposition self-contained, the author gives a short introduction to the theory of characteristic classes of vector bundles. He also motivates Witten’s conjecture, which turned out to be a major question for research done in the last two decades. Elements of the proof given by Kontsevich are mentioned. In particular, the string and dilation equations as well as the KdV equations are discussed. In some sense, this chapter complements a chapter by G. Mondello in Volume II of this Handbook which gives a detailed account of the use of ribbon graphs in the intersection theory of moduli space, in relation to the Witten conjecture.

3.2 The generalized Mumford conjecture In Chapter 12, Ib Madsen gives a survey of the proof of a generalized version of the Mumford conjecture which he obtained in joint work with M. Weiss. The original Mumford conjecture states that the stable rational cohomology of the moduli space Mg is a certain polynomial algebra generated by the Mumford–Morita–Miller cohomology classes of even degrees. The conjecture can also be formulated in terms of the cohomology of a classifying space of mapping class groups. The Madsen–Weiss result generalizing Mumford’s conjecture states that a certain map between some classifying spaces which a priori have different natures induces an isomorphism at the level of


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integral homology. The result, obtained in 2002, was published in 2007 in a paper entitled The stable moduli space of Riemann surfaces: Mumford’s conjecture. This solution of the Mumford conjecture is considered as spectacular progress in the question of understanding the homotopy type of moduli space. This question is one of the most fundamental questions in Teichmüller theory. We are still very far from having a response to it, except for some special surfaces. It is in relation to this question that Mumford started in the early 1980s a study of the cohomology ring of moduli space. There is an intimate relation between Riemann’s moduli space and the classifying space of the mapping class group: the rational cohomology rings of the two spaces coincide. It seems that up to now, the only closed orientable surfaces for which we have a complete description of this rational cohomology ring are the surfaces of genus 4. At the other extreme, we have information about the stable cohomology, which can be considered as information about the cohomology ring of moduli spaces of surfaces with very large genus. By definition, the stable rational cohomology ring of moduli space is the direct limit of rational cohomology rings of moduli spaces of a class of surfaces of increasing genus. More concretely, these surfaces are compact with one boundary component, embedded into one another, that is, SgC1;1 is obtained by attaching to Sg;1 , along its boundary component, a torus with two disks removed. Mumford’s conjecture (which appeared in print in 1983) states that the stable rational cohomology of the moduli space Mg is a polynomial algebra generated by certain tautological cohomology classes which Mumford defined in the context of the Chow ring of the Deligne–Mumford compactification of moduli space. The same classes were re-introduced from a more topological point of view by Miller and by Morita, in 1986 and 1987. Miller and Morita defined the tautological cohomology classes as cohomology classes of the classifying space B g of the mapping class group g . These tautological classes, usually denoted by i , are now called Mumford–Morita–Miller classes. Mumford’s conjecture states that the rational cohomology of the stable moduli space is a polynomial algebra generated by the Mumford–Morita–Miller classes i of dimension 2i . Mumford’s conjecture can also be formulated in terms of the cohomology of the classifying space B 1 of the mapping class group. The conjecture seems to have been motivated by a stability result in the context of the Grassmannian of d -dimensional linear subspaces of C n , stating that the cohomology of that space stabilizes as n ! 1 to a polynomial algebra in the Chern classes of the tautological d -dimensional vector bundles. The generalized form of Mumford’s conjecture, proved by Madsen and Weiss, says that the integral cohomology ring of the infinite genus mapping class group is equal to the cohomology ring of a certain space associated with the Pontryagin–Thom cobordism theory. This result is important because the algebraic topology of the space associated with cobordism theory is well understood. This theory had already been used as a basic tool in establishing several major results in geometry, for instance

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Milnor’s construction of exotic spheres and the early proofs of the Atiyah–Singer theorem. In their proof of the generalized conjecture, Madsen and Weiss computed the rational stable cohomology of the mapping class group. But the generalized conjecture may also be used to calculate the mod p cohomology of the stable mapping class group for all primes p, and hence the integral cohomology. The generalized conjecture was formulated by Madsen around the year 2000, after Tillmann discovered, using Harer’s stability theorem, that Quillen’s plus construction applied to the classifying space B 1 of the mapping class group makes this space an infinite loop space. In her work, Tillmann was motivated by string theory, and one can consider these developments as an instance of the fact that ideas in theoretical physics can have a major impact in geometry. The proof by Madsen and Weiss of the generalized Mumford conjecture uses techniques from high-dimensional manifold theory (the Pontryagin–Thom theory of cobordisms of smooth manifolds that we already mentioned) and singularity theory. There is an identification of the rational cohomology of Riemann’s moduli space with what Madsen and Weiss call the embedded moduli space S.2/, the space of differentiable subsurfaces of a high-dimensional Euclidean space. The rational homology isomorphism between the two spaces is obtained by assigning to each differentiably embedded surface its induced Riemann surface structure. The embedded moduli space is then used to classify smooth embedded surface bundles, and one recovers characteristic classes of surface bundles (like the Mumford–Morita–Miller classes) from S.2/ cohomology classes. Chapter 12 also contains a report on a new proof of the generalized Mumford conjecture that was given by Galatius, Madsen, Tillmann and Weiss in 2009. Let us note that in the same year, Eliashberg, Galatius and Mishachev gave another proof of the generalized Mumford conjecture, in a paper entitled “Madsen–Weiss for geometrically minded topologists”. The title indicates that the proof is more geometrical than the original one. It is based on Madsen and Weiss’s original ideas, and it uses a new version of Harer’s stability result which the authors formulate in terms of folded maps. They attribute the idea of such a geometrical proof to Madsen and Tillmann who suggested it in their paper The stable mapping class group and Q.CPC1 /, published in 2001.

3.3 The Lp -cohomology of moduli space Chapter 13 by Lizhen Ji and Steven Zucker concerns the Lp -cohomology of moduli space. The definition of the Lp -cohomology of a non-compact manifold depends on the choice of a metric (usually, a Riemannian metric) on that space. Thus, one has to choose a metric on Teichmüller space. But since Lp -cohomology is a quasi-isometry invariant of the manifold, the results presented here are valid for Teichmüller and moduli spaces with respect to several of their known metrics.


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Before stating the results that are reviewed in this chapter, let us first say a few words on Lp -cohomology. Historically, the theory started with the case p D 2, that is, L2 -cohomology. This theory was developed independently, at the end of the 1970s, by Cheeger and Zucker, as a cohomology theory for non-compact manifolds which is defined in a way parallel to de Rham cohomology, but where one uses, instead of general differential forms, square-integrable forms, with respect to a Riemannian metric on the ambient manifold. As usual, the L2 -theory has an advantage over the general Lp -theory, because L2 norms define Hilbert space structures. It turned out that the L2 -cohomology of a space is related to the intersection cohomology of a suitable compactification of that space. This was first realized by Zucker, who conjectured in 1982 that the L2 -cohomology of an arithmetic Hermitian locally symmetric space is isomorphic to the intersection cohomology of its Baily–Borel compactification, for what is called the middle perversity. Two independent proofs of that conjecture were given by Loojienga in 1988 and by Saper and Stern in 1990, and although it became a theorem, the result is still called the “Zucker conjecture”. Near the end of the 1970s, Cheeger proved that the L2 -cohomology of a Riemannian manifold with cone-like singularities is isomorphic to its intersection cohomology. The theory of Lp -cohomology was developed later on for all p > 1, using the Banach space of Lp -differential forms equipped with a natural Lp -norm. The Lp cohomology of a Riemannian manifold is invariant by bi-Lipschitz diffeomorphisms. For Riemannian manifolds with finite area and cusps, it was expected that an analogue of the Zucker conjecture is true, that is, that the Lp -cohomology coincides with the cohomology of some appropriate compactification. There have been several works in that direction, by Zucker. To what metrics on moduli space does this theory apply? The moduli space Mg;n of algebraic curves of genus g with n punctures carries several complete Riemannian metrics, including the so-called Kähler–Einstein metric, the McMullen metric, the Ricci metric and the Liu–Sun–Yau metric. In contrast with the Weil–Petersson metric, of which we have now a better understanding (but which has the disadvantage of being incomplete), these metrics are still not well studied. But it is known that they are all quasi-isometric and hence they have the same Lp -cohomology. Ji and Zucker showed that for all 1 < p < 1, the Lp -cohomology of moduli space Mg;n is isomorphic to the x DM . This result is ordinary cohomology of its Deligne–Mumford compactification M g;n an analogue of the Zucker conjecture for Hermitian locally symmetric spaces equipped with their Baily–Borel compactification. It also shows that the Lp -cohomology does not depend on p. Ji and Zucker consider this as a rank-one property of moduli space, because in the case of symmetric spaces of rank > 1, the Lp -cohomology in general depends on the value of p. The result concerning the Lp -cohomology of the Weil–Petersson metric is of a different nature. In this case, Ji and Zucker showed that for 4=3 < p < 1, the x DM whereas for 1 p 4=3, Lp -cohomology is isomorphic to the cohomology of M g;n p the L -cohomology is isomorphic to the cohomology of the space Mg;n itself.

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Chapter 13 contains a proof of these results. It also contains survey sections on Lp -cohomology, on the intersection cohomology of projective algebraic varieties, and on the Hodge decomposition of compact Kähler manifolds. This will give the reader a complete and self-contained account of the subject treated. The chapter ends with several open problems concerning the various complete metrics on Teichmüller spaces and moduli spaces. Chapter 13 is somehow in the same spirit as Chapter 4 of this volume in the sense that it presents some analogies between Teichmüller spaces (resp. moduli spaces) and symmetric spaces of non-compact type (resp. non-compact locally symmetric spaces).

4 Part D. Teichmüller theory and mathematical physics This volume had started with two fundamental tools in the deformation theory of Riemann surfaces, namely, the Beltrami equation and the earthquake theorem. We already discussed at length these two tools, the Beltrami equation being at the basis of the analytic deformation theory of Riemann surfaces, while the earthquake theorem is at the basis of the deformation theory of hyperbolic metrics. Now the volume ends with a part on the relation between Teichmüller theory and physics, and the two chapters that constitute this part use a third basic tool in uniformization theory, namely the Liouville equation (1853). We start by recalling the definition. Let h0 be a Riemannian metric on a closed surface S . Any other Riemannian metric which is conformal to h0 can be written as h D e 2 h0 , where is a real-valued function on S. Let 0 be the Laplacian and K0 the Gaussian curvature function on S , both with respect to h0 . The metric h is hyperbolic (i.e. it is a Riemannian metric of constant curvature 1) if and only if it satisfies the following equation (called Liouville equation): 0 K0 D e 2 : In principle, the existence theory of solutions to the Liouville equation can be considered as a precise version of Riemann’s uniformization principle, and it was used by Poincaré in his first attempts to prove the uniformization theorem. But in practice, this approach to uniformization is considered to be too difficult for being useful. The work done on the Liouville equation is mostly due to theoretical physicists, and it is interesting to make this work accessible to mathematicians. Chapter 14 by Kirill Krasnov and Jean-Marc Schlenker and Chapter 15 by Rinat Kashaev should be useful in this respect. The two chapters provide a review of some applications of this equation, from different points of view. Both chapters highlight the connection between the Liouville equation and Teichmüller theory through various recent works.


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The Liouville equation has been extensively used in several domains of theoretical physics, including two-dimensional gravity (Jackiw, 1983), non-critical string theory (Polyakov, 1981), conformal field theory (Belavin, Polyakov & Zamolodchikov, 1984), in the quantization theory of Teichmüller space (Kashaev, 1988 ca. and Teschner 2003 ca.), and more recently in work on N D 2 supersymmetric gauge theories in 4 dimensions (Gaiotto, 2009). Let us mention a few of the developments in theoretical physics. The Liouville equation gives rise to a functional on the moduli space of metrics, called the Liouville functional, which can be defined as Z 1 d vol0 .jrj2 C e 2 2K0 /: SŒh0 ; D 8 In this form, Liouville theory appeared as a tool in non-critical relativistic string theory. Takhtajan and Zograf (1987), showed that the Liouville functional provides a Kähler potential for the Weil–Petersson metric on Schottky space. Several developments followed that discovery, and they are again due to theoretical physicists. In particular, a Liouville action defined as a functional on moduli spaces provided a relation between the Liouville theory and the renormalized volume of hyperbolic manifolds (works of Krasnov, 2000, of Takhtajan & Teo, 2003, and others). One should also mention that the work of Takhtajan & Zograf motivated McMullen in his construction of a Kähler hyperbolic metric on moduli space, and in his discovery of the so-called quasiFuchsian reciprocity law, a duality formula which expresses the fact that the tangent maps at the Fuchsian complex projective structure of the Bers embedding of the two boundary components of a quasi-Fuchsian manifold are adjoint linear operators.

4.1 The Liouville equation and normalized volume Chapter 14 presents the ideas of renormalized volume of hyperbolic 3-manifolds. The relation between the geometry of a hyperbolic 3-manifold and moduli spaces of Riemann surfaces can be conceived most clearly in the case where the manifold is a product S Œ0; 1 of a surface with R, that is, the context of quasi-Fuchsian manifolds. More precisely, these are the complete hyperbolic 3-manifolds that are homeomorphic to the product of a surface with an interval and that contain a non-empty convex set. (We already encountered these manifolds in Chapter 3 of this volume.) A useful ingredient in this relation is the fact that the space of hyperbolic structures on S Œ0; 1 is closely related to the space of projective structures on the boundary. A quasi-Fuchsian 3manifold has infinite volume, but following an idea that is due to theoretical physicists (Witten, 1998), one can define a renormalized volume, using a foliation by equidistant surfaces in the complement of a compact convex subset N of M . The renormalized volume is thereby defined in terms of the asymptotic behavior of the volume of the set of points at distance at most from N as n ! 1. The renormalized volume appears as the constant term of the asymptotic expansion of the volume in terms of the parameter . Krasnov and Schlenker gave an alternative approach to that theory that is based on

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simple differential geometry arguments and that is more suited to mathematicians than the one used by physicists. They obtained a definition of renormalized volume in terms of the volume of the convex manifold N and the total mean curvature of its boundary, and they derived a variational formula which is an analogue of the Schläfli formula for volumes of spherical and hyperbolic tetrahedra. Krasnov and Schlenker also gave an interpretation of renormalized volume as a function on Teichmüller space, and they obtained a new proof of the fact that the renormalized volume of quasi-Fuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a Kähler potential for the Weil–Petersson metric on Teichmüller space. The theory can be developed in the general setting of convex co-compact hyperbolic 3-manifolds where the complex projective structure on the boundary plays a central role, and in which the renormalized volume can be expressed in terms of the Liouville functional at the corresponding projective structure.

4.2 The discrete Liouville equation and the quantization theory of Teichmüller space Chapter 15 by Kashaev is a review of a discrete version of the Liouville equation, interpretating it as a mapping class group dynamical system in the Teichmüller space of an annulus with marked points on its boundary. The theory makes use of Thurston’s shear coordinates on the annulus. The discrete version of the Liouville equation that is reported on in this chapter was first defined on the integer lattice Z2 by Fadeev and Volkov. Kashaev reviews elements of quantum discrete Teichmüller theory and its relation to the usual quantum Teichmüller theory. Both theories are based on the so-called non-compact dilogarithm function, and for the convenience of the reader, the basic properties of this function are reviewed in this chapter.

Part A

The metric and the analytic theory, 3

Chapter 1

Quasiconformal and BMO-quasiconformal homeomorphisms Jean-Pierre Otal Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasiconformal homeomorphisms . . . . . . . . . . . . . . . . . 2.1 The modulus of a family of curves . . . . . . . . . . . . . . 2.2 The geometric definition . . . . . . . . . . . . . . . . . . . 2.3 Compactness property for quasiconformal homeomorphisms 2.4 The analytic definition . . . . . . . . . . . . . . . . . . . . 3 The Morrey–Bojarski–Ahlfors–Bers Theorem . . . . . . . . . . . 4 Dependence of f on . . . . . . . . . . . . . . . . . . . . . . 4.1 The derivative of 7! f . . . . . . . . . . . . . . . . . . 5 The BMO-quasiconformal homeomorphisms . . . . . . . . . . . 5.1 The David Theorem . . . . . . . . . . . . . . . . . . . . . . 5.2 Proof of Theorem 5.6: existence . . . . . . . . . . . . . . . 5.3 Proof of Theorem 5.6: uniqueness . . . . . . . . . . . . . . 5.4 Topological degree of Sobolev maps . . . . . . . . . . . . . 6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Estimates of moduli of annuli . . . . . . . . . . . . . . . . . 6.2 BMO functions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The analytic theory of Teichmüller space is based on the notion of quasiconformal homeomorphisms that is inherent in the definition. Let us recall this definition. Let be a Fuchsian group, i.e. a discrete torsion-free subgroup of PSL(2; R/, that we view as acting on the upper half-plane H D f.x; y/ j y > 0g by Möbius transformations. A quasiconformal deformation of is a pair .; /, where is a representation of into PSL(2; R/ and W H ! H is a quasiconformal homeomorphism from the upper half-space into itself that conjugates and ./. The Teichmüller space T ./ is the quotient of the space of quasiconformal deformations of by the equivalence relation that identifies two quasiconformal deformations .1 ; 1 / and .2 ; 2 / if and only if the extensions of 1 and of 2 to @H are equal on the limit set of .

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By definition, an orientation-preserving homeomorphism W U ! V between open sets in the complex plane is quasiconformal when its distributional derivatives N D @ a.e., for some 2 L1 .U / with kk < 1. N are in L2 .U / and satisfy @ @, @ loc N The equation @ D a.e. is called the Beltrami equation, is called the Beltrami coefficient of . Theorem. Let U C. Then for any 2 L1 .U /, with kk1 < 1, there is an orientation-preserving homeomorphism f W U ! V which is quasiconformal and N D @f a.e. Furthermore f is unique up to the post-composition with satisfies @f a holomorphic homeomorphism. In this general setting, the theorem must be attributed to Charles Morrey [27], Bogdan Bojarski [5] and Lars Ahlfors and Lipman Bers [3]. However, its relevance for the study of Teichmüller space and of Kleinian groups was realized by Ahlfors and Bers (see the comment by Ahlfors in [2], p. 2261 ). In particular, [3] contains also a proof that the map 7! f is “holomorphic” and a computation of the tangent map at the point D 0. This theorem led Bers to show that T ./ is a complex Banach space [10]. The property that 7! f is holomorphic and the explicit value of the tangent map permitted Ahlfors to prove his Finiteness Theorem: when G is a finitely generated subgroup of PSL(2; C/, the quotient of the domain of discontinuity of G by G is a Riemann surface of finite type. In the early 1980s Dennis Sullivan discovered deep connections between Kleinian groups and the iteration of rational maps of the Riemann sphere. For instance, he made a connection between the “Non-wandering domain Conjecture” of Fatou – any component of the Fatou domain of a rational map is eventually periodic – and the Ahlfors Finiteness Theorem. He found a new proof of the last theorem, based on the Beltrami equation that could be translated immediately with the words of rational maps into a solution of the Fatou conjecture. The Beltrami equation has many other applications in complex dynamics (see for instance [9], [28]). In this chapter, we will first present a proof of the existence and uniqueness theorem N D @ in the classical case, i.e. when kk1 is of homeomorphic solutions of @ < 1. We present the beautiful existence proof found by Alexey Glutsyuk [13] when the coefficient W C ! C is C 1 and is invariant by translations with integer coefficients. The general case follows from a classical approximation argument. The approximation procedure is presented with details, as well as the basic properties of quasiconformal homeomorphisms; one reason for this is that we tried to keep the exposition at an elementary level, and another reason is that the same arguments will be used in §5 for studying the Beltrami equation when kk1 D 1. Olli Lehto was perhaps the first to study the Beltrami equation when kk1 D 1 [23], [24]. When satisfies certain conditions, he proved the existence of a solution; 1 Bogdan Bojarski indicated that the paper referenced [4] in this comment should have been B. V. Bojarski, On solutions of elliptic systems in the plane, Doklady Akad. Nauk SSSR (N.S.) 102 (1955), 871–874

Chapter 1. Quasiconformal and BMO-quasiconformal homeomorphisms

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one condition concerns the “bad set” of , i.e. the set of the points z where j.z/j D 1 and another requires the finiteness of a certain integral involving . In 1988, Guy David introduced the class of Beltrami coefficients which satisfy a logarithmic condition: those are the measurable functions W C ! C with kk1 D 1 for which there exist positive constants C and ˛ such that the Lebesgue measure of the set fz j j.z/j > 1 g is C e ˛= . For those , he proved that the equation N D @ a.e. has solutions which are homeomorphisms. Notice first that giving a @ N and @ are defined. A natural meaning to this equation requires that the derivatives @ hypothesis which guarantees this property is that is absolutely continuous on lines (ACL) (cf. Definition 2.10). Theorem ([7]). Let be a Beltrami coefficient which satisfies a logarithmic condition. Then there is an orientation-preserving homeomorphism W C ! C which is ACL N D @ a.e. Furthermore there is only one homeomorphism with and such that @ these properties up to composition with an affine map. This theorem had several important applications in complex dynamics (see for instance [15], [16], [29]). For instance, it is used in [29] to construct a full measure set of in the unit circle f j jj D 1g such that the Julia set of the polynomial z C z 2 is locally connected. In §5, we will focus on the existence and uniqueness theorem of homeomorphic N D @ when kk1 D 1. The original approach by Guy David is a solutions of @ tour de force in real analysis. Now, there are other approaches for existence theorems when kk1 D 1. The hypotheses are often formulated in terms of K D 1Cjj . 1jj (When is the Beltrami coefficient of a homeomorphism which is orientationpreserving, K is called the dilatation function of ). One such hypothesis on K , introduced by Melkana Brakalova and James Jenkins K

in [4], is that e 1Clog K is locally integrable. When this holds and when K satisfies a certain growth condition near 1, they prove existence of an ACL homeomorphism N D @ a.e. of C such that @ Another hypothesis on 2 L1 .U / with kk1 D 1, introduced by Vladimir Ryazanov, Uri Srebro and Eduard Yakubov in [30] is K Q a.e. where Q W U ! Œ1; 1Œ is locally in the John–Nirenberg space BMO.U / (cf. Definition 5.4). A link between this property of and the logarithmic condition of David is provided by Proposition 2.1 in [30], saying that 2 L1 .U /, with kk1 D 1, satisfies locally the logarithmic condition of David if and only if K Q a.e. where Q W U ! Œ1; 1Œ is locally in BMO.U /. Under this assumption, the existence of a homeomorphism which satisfies the Beltrami equation is established in [30]. This is the existence part in the next theorem, the proof of which will occupy §5. Theorem. Let U be a connected open subset of C. Let 2 L1 .U / such that kk1 D 1. Suppose that there exists a function Q W U ! Œ1; 1Œ in BMOloc .U /, such that K Q a.e. Then there exists an orientation-preserving homeomorphism

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N D @ a.e. Furthermore f is unique up f W U ! V which is ACL and satisfies @ to postcomposition with a holomorphic homeomorphism. The construction of the homeomorphic solution of the Beltrami equation in [4] or in [30] is geometric: it begins with estimates of the distortion of the modulus of annuli under quasiconformal maps which imply the equicontinuity of the family of the Q-quasiconformal homeomorphisms, i.e. the ACL homeomorphisms with dilatation bounded by Q a.e. Following the main lines of these two papers, we present a selfcontained proof of existence. In contrast with the case when kk1 < 1, the uniqueness of the solution is really a difficult part of the theorem. One reason is that quasiconformal homeomorphisms (in the classical sense) are in the Sobolev space W 1;2 by definition whereas the ACL homeomorphisms f and g in the theorem are not always in W 1;2 . In order to prove the uniqueness statement, we follow an approach different from [7], issued from [19], p. 254, and Corollary 2.5 in [22]. This approach leads to the following natural question. x C be a closed Jordan domain and let F W x ! C be a continuous map in W 1;1 Let whose Jacobian determinant is 0 a.e. and integrable. Then, is the topological degree x F .@/ > 0? The answer is positive when F is in the of F at any point of F ./ 1;2 Sobolev space W , but the function we consider only belongs to the Orlicz–Sobolev 2 space W 1;L = log L . However, the answer is still positive under this assumption; this is an important result of J. Kauhanen, P. Koskela and J. Malý [22] explained in §5.4. The proof of Theorem 5.12 given here is self-contained except Theorem 5.14, a theorem of Luigi Greco, for which we refer to [14]. Another proof of uniqueness under the same assumptions is given in Theorem 11.5.1 of [19]. The notion of Q-quasiconformal homeomorphism extends to the case where U Rn is an open set and Q W U ! Œ1; 1Œ is a measurable function which is locally in BMO. David’s Theorem has given a new impetus to the study of the Beltrami equation in the plane and also to the study of Q-quasiconformal homeomorphisms in dimension 3. In particular the equicontinuity property of the family of all Q-quasiconformal homeomorphisms evoked above is valid in any dimension [26]. There exist now treatises devoted to the Beltrami equation, and to Q-quasiconformal maps, like [20] in the 2-dimensional case, [19], Chapter 11, and [26] in arbitrary dimensions. In particular, [19] contains existence and uniqueness results for the Beltrami equation which go far beyond the case studied in this chapter. The geometric approach to the study of Q-quasiconformal homeomorphisms is developed in [26]. This chapter grew up from notes for a course on the hyperbolization of 3-manifolds given at CRM Bellaterra during theAutomn of 2002. Athanase Papadopoulos proposed to publish the exposition in the “Handbook of Teichmüller Theory” he is editing. I have chosen to include also the David Theorem in order to (try to) make this result accessible to a larger audience and also because the proofs by Brakalova–Jenkins and Ryazanov– Srebro–Yakubov have put it in the classical framework. During its writing, I benefitted from several interesting discussions with Wladimir Ryazanov and Bogdan Bojarski: I thank them heartly.


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This work was partially supported by Cefipra (Project No 4301) and ANR (Project “Facettes de groupes discrets”).

2 Quasiconformal homeomorphisms 2.1 The modulus of a family of curves Definition 2.1 (The modulus of a quadrilateral). A quadrilateral is a Jordan disc x with four preferred points on its boundary; Q contained in the Riemann sphere C these points cut @Q into two horizontal and two vertical edges. We say that two quadrilaterals are holomorphically equivalent if they are homeomorphic by a map which is holomorphic in the interior of Q and preserves the horizontal/vertical edges. By the Riemann Uniformization Theorem, any quadrilateral Q is holomorphically equivalent to a Euclidean rectangle whose horizontal sides have length a and vertical sides have length b. The ratio a=b only depends on Q; it is called the modulus of Q and denoted m.Q/. Two quadraliterals which are holomorphically equivalent have the same modulus. The modulus of a quadrilateral can also be defined without using the Uniformization Theorem as the modulus of a particular family of curves. Definition 2.2 (The modulus of a family of curves). A family of curves is a set of x Let F be a family of curves. An admissible metric continuous maps W Œ0; 1 ! C. x ! Œ0; 1Œ such that for any curve 2 F which for F is a measurable function RW C is locally rectifiable, the integral . .t //j 0 .t /jdt is 1. The modulus of F is Z m.F / D inf 2 d (2.1) C

where is the Lebesgue measure on C and where the infimum is taken over all the admissible metrics for F . Let Q be a quadrilateral. Let F be the family of curves whose elements are the curves joining the two horizontal sides of Q; then m.Q/ D m.F /. It suffices to check this when Q is a Euclidean rectangle: in this case, it is a consequence of the Cauchy–Schwarz inequality which shows also that the infimum is obtained only when is constant almost everywhere. Particular choices of in Formula 2.1 lead to useful estimates for the modulus of Q like the following Rengel inequality. Denote by sh (resp. by sv ) the Euclidean distance between the horizontal (resp. vertical) sides of Q. Then Area .Q/ sv2 m.Q/ : (2.2) Area .Q/ sh2

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Definition 2.3 (The modulus of an annulus). Let A C be an open annulus. Then A is holomorphically equivalent to a round annulus A.0; r; R/ D fz j r < jzj < Rg; 1 log. Rr /. the modulus of A is m.A/ D 2 The modulus of an annulus A can also be computed as the modulus of particular families of curves. x such that .0/ and .1/ are in – Let F be the family of the curves W Œ0; 1 ! C 1 x distinct connected components of C n A. Then m.A/ D m.F . / – Let G be the family of curves W Œ0; 1 ! A such that Œ0; 1 is a closed curve x n A. Then, m.A/ D m.G /. which separates the two components of C In the Appendix, we collect some estimates of moduli of annuli that we use.

2.2 The geometric definition Definition 2.4 (Geometric definition of a quasiconformal homeomorphism). Let U , V x and W U ! V be an orientation-preserving homeomorphism. be open subsets of C Let K 1. One says that is K-quasiconformal if for any quadrilateral Q U , 1 m.Q/ m.Q/ Km.Q/: K One says that is quasiconformal if it is K-quasiconformal for some K 1. If is a quasiconformal homeomorphism, one defines K./ as the infimum of the K’s such that is K-quasiconformal. The next statements follow from the definition. – If W U ! V is a holomorphic homeomorphism, then is 1-quasiconformal. – If W U ! V is K-quasiconformal, then 1 W V ! U is K-quasiconformal. – If 1 W U ! V is K1 -quasiconformal, and 2 W V ! W is K2 -quasiconformal, then 2 B 1 is K1 K2 -quasiconformal. The property of being K-quasiconformal for an orientation-preserving homeomorphism W U ! V can be defined in an equivalent way using annuli instead of quadrilaterals, saying that W U ! V is K-quasiconformal if for any annulus A U , 1 m.A/ m.A/ Km.A/: K An easy consequence of this definition is that the isolated singularities of a quasiconformal homeomorphism are removable: if W U nfpg ! V is a K-quasiconformal homeomorphism, then extends continuously to a K-quasiconformal homeomorphism defined on U . In particular, if f is a quasiconformal homeomorphism from C to an open subset V of C, then V D C.


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2.3 Compactness property for quasiconformal homeomorphisms x let z1 , z2 , z3 be three points Theorem 2.5. Let U be a connected open subset of C; in U . Let .n / be a sequence of K-quasiconformal homeomorphisms defined on U such that n .zi / D zi . Then there is a subsequence which converges uniformly on compact subsets of U to a homeomorphism which is K-quasiconformal. Proof. The theorem will follow from the next three steps. (i) .n / contains a subsequence which converges uniformly on compact sets of U x to a continuous map W U ! C. x We follow closely an argument from Denote by .:; :/ the spherical distance on C. [25]. By the Ascoli Theorem, it suffices to check that the family fn g is equicontinuous on compact sets in U . Let K be a compact subset of U . There is an r > 0 such that for all z 2 K the disc B.z; r/ is contained in U and does not contain two of the points z1 , z2 , z3 . Let > 0 be smaller than the non-zero spherical distances .zi ; zj /. Choose 1 log ır , the modulus of the annulus A.z; ı; r/, is bigger than 0 < ı < r such that 2 K 2 . Then for any n the modulus of A0 D n A.z; ı; r/ is bigger than 2 . But for any x n A0 contains two of the points n .zi / D zi , say z2 , z 0 2 B.z; ı/ one component of C z3 and the other component contains the two points n .z/ and n .z 0 /. Let be the infimum of the non-zero distances .zi ; zj / and .n .z/; n .z 0 //. By Inequality (6.1) (cf. Appendix 6.1), m.A0 / 2 ; therefore , so that d.n .z/; n .z 0 // for all z 0 2 B.z; ı/ and for all z 2 K. Therefore fn g is equicontinuous. Remark 2.6. Using Claim 6.1 one could prove that the n are Hölder continuous on K, with an exponent depending only on K. By a theorem of Akira Mori this exponent can be taken equal to K1 . Up to extracting a subsequence, we suppose now that .n / converges to uniformly on compact sets in U . (ii) Any point z 2 U has a neighborhood on which is either injective or constant. We argue by contradiction. Suppose that some open disc B U contains three points a1 , a2 , a3 with .a1 / ¤ .a2 / D .a3 /. We may choose B sufficiently small so that it is disjoint, say from the points z2 and z3 . Consider an annulus A B x n A contains a1 and a2 while the other contains a3 and such that one component of C x x n n A is bounded from C n B. Then the spherical diameter of each component of C below independently of n while the distance between these components tends to 0. Therefore, by Inequality (6.3) (cf. Appendix 6.1), m.n A/ ! 0. This contradicts the inequality m.n A/ K1 m.A/ > 0. (iii) If two distinct points a and b of U have the same image by , then is not locally injective near a. Indeed, denote by C.a; / the circle of radius around a. For all > 0 smaller than the spherical distance .a; b/ and for any n, .n .a/; ; n .b// is at least equal

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Jean-Pierre Otal

to the distance between n .a/ and n C.a; /. By letting n tend to 1, the uniform convergence of n gives ..a/; C.a; // D 0. Statements (ii), (iii) and the connectedness of U imply that is injective. By the Invariance of Domain Theorem U is open and W U ! U is a homeomorphism. It remains to show that is K-quasiconformal. One way to see this is to use the following continuity property of the modulus of quadrilaterals. Let .Qn / be a sequence x which converges to a quadrilateral Q in the following sense: for of quadrilaterals in C any > 0 any point on a horizontal (resp. vertical) side of Qn is at distance from the corresponding side of Q, for all sufficiently large n. Then m.Qn / ! m.Q/.

2.4 The analytic definition Definition 2.7. Let be an open subset of C. Let 1 p 1. The Sobolev space W 1;p ./ is the space of measurable maps f W ! C which are locally integrable and whose distributional derivatives @f and @f are functions which are in Lploc ./. @x @y Recall the notation @f D

1 2

@f @x

N D i @f and @f @y

1 2

@f @x

C i @f . @y

Definition 2.8 (Analytic definition of a quasiconformal homeomorphism). Let U and V be open subsets of C and let W U ! V be an orientation-preserving homeomorphism. Let K 1. One says that is K-quasiconformal if (i) 2 W 1;2 .U / and (ii) there is a measurable function 2 L1 .U / with kk1 @ a.e.

K1 KC1

N D such that @

The function 2 L1 .U / with this property is called the Beltrami coefficient of . N D @ a.e. is the Beltrami equation. The equation @ The two definitions of a quasiconformal homeomorphism agree: Theorem 2.9 ([1]). Let U and V be open subsets of C and let W U ! V be an orientation-preserving homeomorphism. Let K 1. Then is K-quasiconformal in the geometric sense if and only if it is K-quasiconformal in the analytic sense. Proof. We will prove only the implication that if is geometrically K-quasiconformal, then it is also in the analytic sense. Suppose that is K-quasiconformal in the geometric sense. We need to show that 2 W 1;2 .U /. For this, we show that is absolutely continuous on lines (ACL). We follow closely the exposition in [1]. Definition 2.10 (ACL homeomorphisms). Let U and V be open subsets of C and let W U ! V be an orientation-preserving homeomorphism. One says that is absolutely continuous on lines (ACL), if for each rectangle R D I J contained


45

in U , the function .x; :/ W J ! C is absolutely continuous for almost all x 2 I and the function .:; y/ W I ! C is absolutely continuous for almost all y 2 J . Recall that a function f W I ! C is absolutely continuous when its total variation is a measure which is absolutely continuous with respect to the Lebesgue measure g is any set of disjoint intervals on I : the criterion to see this is that if fŒai ; bi ; i 2P contained in I with total length smaller than ı, then jf .ai /f .bi /j .ı/ where .ı/ is a function which tends to 0 with ı. Let I J be a rectangle contained in U . Let denote the Lebesgue measure on C. The function y 7! ƒ.y/ D ..I Œ0; y // is increasing and therefore differentiable at almost all y 2 J . (i) If ƒ is differentiable at y0 , then x 7! .x; y0 / is absolutely continuous. To see this, we apply the criterion recalled above for a function being absolutely continuous. Let fŒai ; bi ; i 2 g be disjoint intervals in I . Denote by Riı the rectangle Œai ; bi Œy0 ; y0 C ı . The modulus of Riı can be estimated using Rengel’s inequality (2.2): m.Riı /

di .ı/2 .Riı /

;

where di .ı/ is the Euclidean distance between the vertical sides of Riı . Hence, by the Cauchy–Schwarz inequality P X . di .ı//2 ı : m.Ri / P ı .Ri / But

P

.Riı / ƒ.y0 C ı/ ƒ.y0 /. Therefore 2 X ƒ.y C ı/ ƒ.y / X 0 0 : di .ı/ ı m.Riı / ı

(2.3)

P When ı ! 0, the left hand sideP tends to . j.bi /.ai /j/2 and the upper limit of the right hand side is less than K. jbi ai j/ƒ0 .y0 / since by the K-quasiconformality ij of , m.Riı / Km.Riı / D K jbi a . Therefore, x 7! .x; y0 / is absolutely ı continuous. and @ exist a.e. on U . In One consequence is that the partial derivatives @ @x @y general, this property does not imply that is differentiable a.e. However, for an open map, it does: this is a theorem of Fred Gehring and Olli Lehto (cf. [12], [1], p. 17). The Jacobian determinant of is the measurable function J defined by J .z/ D 2 N j@.z/j2 j@.z/j . If is differentiable at z, denote by Dz the tangent map to at z. Writing the Taylor expansion of near z and using the Rengel inequality one obtains max jDz .e i˛ /j K min jDz .e i˛ /j: ˛

˛

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Jean-Pierre Otal

N Therefore j@.z/j kj@.z/j, where k D z 2 U at which is differentiable.

K1 . KC1

In particular, J .z/ 0 at any point

(ii) 2 W 1;2 .U /. To prove this, we first show that the Jacobian determinant J is locally integrable. Define a measure on U as follows: for any Borel set A, .A/ D ..A//. This measure is finite over compact sets. It can be decomposed relatively to the Lebesgue measure as the sum of two positive measures: one is absolutely continuous with respect to – i.e. it can be written on the form .z/d .z/ – and one is singular with .B.z;r// respect to . The Radon–Nikodym derivative satisfies .z/ D limr!0 .B.z;r// a.e. If this formula holds at a point z where is also differentiable, a Taylor expansion of at z shows that .z/ D J .z/. It follows that for any compact set E U , R J .z/ 2 E J .z/dxdy .E/: therefore J is locally integrable. Since j@.z/j 1k 2 , N are in L2 .U /. Hence, 2 W 1;2 .U /. it follows that @ and @ loc (iii) The quasiconformal homeomorphism W U ! V is absolutely continuous with respect to Lebesgue measure. Saying that is absolutely continuous with respect to means that the singular measure in the decomposition of is 0. From the above discussion, this is equivalent R to saying that for any measurable set E U , ..E// D E J dxdy. This reduces Rto showing thatR for any continuous function h with compact support contained in V , V hdxdy D U .h B /J dxdy. Using partitions of unity, it is sufficient to prove this when h has a small support: we will assume that the support of h is contained in the interior of a disc D V , and that 1 D is contained in a disc B U . Since 2 W 1;2 .U / one can construct, by taking the convolution of with a smooth kernel, a sequence .k / of smooth maps such that .k / converges to uniformly on k k / and . @ / converge a neighborhood of B and such that the partial derivatives . @ @x @y respectively to @ and @ in L2 .B/. @y @x The 2-form ! D h.z/dxdy is exact R R on C: let be a 1-form such that ! D d . By the Stokes formula, D hdxdy D @D . Also, since k is smooth Z Z Z Z .h B k /Jk dxdy D k .!/ D k D : B

B

@B

k .@B/

When k ! 1, the closed curve k .@B/ tends to .@B/. By the choice of B, .@B/ is homotopic to @D in the complement of the support of h. Therefore, for k sufficiently large, k .@B/ is homotopic to @D and is closed on the support of this homotopy. For those values of k, we have Z Z Z D D hdxdy: k .@B/

@D

D 1

Also, when k ! 1, R h B k tends to h B uniformly R Jk tends to J in L .B/ and .h B /J dxdy tends to over B. Therefore k R k B .h B /J dxdy. This gives the R B expected result: V hdxdy D U .h B /J dxdy.


47

One consequence is that @.z/ ¤ 0 for almost all z 2 U . If not, then J would vanish on a set E with .E/ > 0 and therefore E would have zero measure, by the above. But we know that the inverse 1 is also quasiconformal in the geometric sense. It is therefore absolutely continuous, contradicting that .E/ > 0. N @ Thus one can define a measurable fonction on U by setting D @ . Then K1 kk1 k where k D KC1 . This proves that is K-quasiconformal in the analytic sense. The reverse implication, any homeomorphism which is K-quasiconformal in the analytic sense is also K-quasiconformal in the geometric sense, could be proven also directly, but we won’t do it here. This can be seen as a consequence of the next paragraph on the Beltrami equation. Indeed it follows from the Existence Theorem of solutions to the Beltrami equation that any K-quasiconformal homeomorphism (for the analytical definition) is also a limit of K-quasiconformal homeomorphisms which are C 1 . Now when a C 1 diffeomorphism is K-quasiconformal, an easy application of the definition of the modulus by extremal length shows that is also K-quasiconformal in the geometric sense. By Theorem 2.5, the same is true for any uniform limit of such 0 s.

3 The Morrey–Bojarski–Ahlfors–Bers Theorem Theorem 3.1 (Solution of the Beltrami equation). Let U be equal to C or to the upper half-plane H D f.x; y/ j y > 0g. For any 2 L1 .U /, with kk1 < 1, there is a unique quasiconformal homeomorphism f W U ! U such that (i) f extends to Ux fixing 0, 1 and 1, and N D @f a.e. (ii) @f A classical lemma of Weyl implies that if W U ! V and W U ! W are two quasiconformal homeomorphisms with the same Beltrami coefficient, then there is a holomorphic homeomorphism h W V ! W such that D hB . This is the reason why the homeomorphism f provided by this theorem is unique: we call it the normalized solution of the Beltrami equation. The classical proof of Theorem 3.1 uses some difficult analysis, namely elements of the Calderon–Zygmund Theory (cf. [1], [17]). Another proof due to Adrien Douady, solves first the case where is C 1 with compact support on C. The case of an arbitrary 2 L1 .U / with kk1 < 1 follows from an approximation argument. This proof is done in detail in [8]. In the Douady approach, the case where is C 1 with compact support is solved using classical Fourier analysis in R2 . The difficulty amounts to showing that the solution is a homeomorphism. Here we present the proof by Alexey Glutsyuk [13]. One specificity of this proof is that is first supposed to be Z2 -periodic, i.e. invariant under the group ' Z2 of translations by vectors with

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integral coordinates. This amounts to solving the Beltrami equation on the torus T 2 D C=Z2 . There, Fourier analysis gets much simplified: Fourier series replace Fourier transform. We state now the theorem of Glutsyuk. Theorem 3.2 ([13], Theorem 1.10). Let W C ! C be a C 1 function which is Z2 periodic, and such that k k1 < 1. Then there exists a C 1 normalized diffeomorphism of C with Beltrami coefficient equal to . This theorem follows from the next result. Proposition 3.3. Let W C ! C be a C 1 function which is Z2 -periodic, and such that k k1 < 1. Then, there is a C 1 function f W C ! C which is Z2 -periodic and such that the differential 1-form !.z/ D f .z/.dz C .z/d z/ N is closed. Proof of Proposition 3.3. We look first for a solution f in the space L2 .T 2 /, idenspace of Z2 -periodic measurable functions h W C ! C such that Rtified with the 2 Œ0;1 Œ0;i jhj dxdy < 1. We will find a solution in that space, show that it is C 1 and then that it does not vanish. N D @. f /: In order The form ! D f .z/.dz C .z/d zN / is closed if and only if @f to solve this equation, one applies the homotopy method of Moser: one introduces a new parameter t 2 Œ0; 1 and one looks for a function f .t; z/ such that for all t , N .t; :/ D @.t f .t; ://: Assume that .t; z/ 7! f .t; z/ is differentiable. Then, after @f differentiating this equation with respect to t, one obtains @N fPt t@. fPt / D @. f /; where fPt .z/ denotes

(3.1)

@ f .t; z/. @t

Proposition 3.4. There is a unique unitary operator U in L2 .T 2 / such that (1) U.1/ D 1, (2) U leaves invariant the space of C 1 functions, and N (3) for any C 1 function h 2 L2 .T 2 /, @h D @.Uh/: Proof. Consider the standard Hilbert basis fep;q j .p; q/ 2 Z2 g of L2 .T 2 / where N p;q D i.p C i q/ep;q . ep;q .z/ D e 2i.pxCqy/ . Then @ep;q D i.p i q/ep;q and @e piq The operator U necessarily satisfies U.ep;q / D pCiq ep;q , for .p; q/ ¤ .0; 0/, and U.e0;0 / D e0;0 ; this defines clearly a unitary operator which satisfies (1). We now show that U preserves the space of C 1 functions. Recall that, for s 2 N, the Sobolev space W s;2 .T 2 / is the subspace of L2 .T 2 / consisting of the functions f whose s distributional derivatives @r x@@sr y belong to L2 .T 2 /, for all r s (equivalently, P W s;2 .T 2 / is the subspace of sequences .hp;q / such that jp C iqj2s jhp;q j2 < 1). This is a Hilbert space, for the Sobolev norm X jp C iqj2s jhp;q j2 C jh0;0 j2 : khk2s D Z2


49

It is clear from the definition that U leaves invariant each Sobolev space W s;2 .T 2 /. By the Sobolev Lemma, if h 2 W s;2 .T 2 / with s > k C 1, then h is C k .T 2 / (cf. [32], p. 272). More precisely, there is a constant Cs which only depends on s, such that any function h 2 W s;2 .T 2 / agrees a.e. with a C k function hQ whose C k norm Q k Cs khks : In particular a function h 2 L2 .T 2 / is C 1 if and only if satisfies khk C for any integer s, f 2 W s;2 .T 2 /. N It is clear that if h is C 1 , then @h D @.Uh/. If Equation (3.1) has a solution, then for all t , fPt t .U B /fPt D .U B /f C c.t /; where c.t/ W T 2 ! C is a constant function. We will construct a C 1 solution f .t; z/ of the equation fPt t .U B /fPt D .U B /f

(3.2)

such that f .0; :/ is the constant function equal to 1. Then, an integration with respect N .t; :/ D @.t f .t; ://: to t gives @f Since U is unitary and since k k1 < 1, the operator U B W f 7! U. f / has L2 norm strictly less than 1; therefore for all t 2 Œ0; 1 , Id t U B is an invertible operator of L2 .T 2 /. Thus Equation (3.2) can be written fPt D .Id tU B /1 .U B /f:

(3.3)

This is an ordinary differential equation in L2 .T 2 / which is Lipschitz in f for t 2 kk Œ0; 1 , since the norm of .Id t U B /1 B .U B / is smaller than 1kk . Therefore, 2 2 for any initial data f0 2 L .T /, the differential Equation (3.3) admits a unique solution f .t; :/, defined over Œ0; 1 and such that f .0; :/ D f0 . We will show that when the initial data is the constant function f0 D 1, the solution f .1; :/ satisfies the conclusions of Proposition 3.3. We will prove first that f .1; :/ is C 1 . To see this, it suffices to show that for any s, the second term of Equation (3.3) is Lipschitz in f for the Sobolev norm. Because then, the differential Equation (3.3) can be solved in W s;2 .T 2 / for any s. The uniqueness of the solution implies then that the solution f .t; :/ is contained in W s;2 .T 2 / for any integer s. Therefore f .1; :/ is C 1 . Claim 3.5. Let s be an integer 1. Then (1) for any 2 C s .T 2 / with k k1 < 1, U B leaves invariant W s;2 .T 2 /I (2) the W s;2 operator norm of U B tends to 0 when the C s norm of tends to 0I (3) .Id t U B /1 is a continuous operator of W s;2 .T 2 / whose norm is bounded independently of t 2 Œ0; 1 . Proof. By the Leibniz rule, if 2 C s .T 2 /, then f 2 W s;2 for all f 2 W s;2 .T 2 /. Furthermore k f ks ck kC s kf ks , where k kC s is the C s norm of and c only depends on s. We already noticed that U leaves invariant each Sobolev space and is a

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unitary operator on those spaces; therefore U B is a bounded operator of W s;2 with norm less than ck kC s . P n Let us prove (iii). Since .Id t U B /1 D 1 0 .t U B / , we only need to show s;2 n that the series of W operator norms k.UB / ks is convergent, once ı D k k1 < 1. We are going to prove that for any f 2 W s;2 .T 2 / k.U B /n f ks c 00 k ksC s .n C 1/s ı ns kf ks ; for a constant c only depending on s. It suffices to prove that the L2 norm of any s partial derivative @r x@@sr y .UB /n f is smaller than c.nC1/s ı ns kf ks for a constant c independant of n and f . Observe that U commutes with the operators so by the Leibniz rule,

.U B /n1 U B

@s @r x@sr y

@x

@y

m1 m01

and

@ ; @y

.U B /n f is a sum of terms

@m1 m01

@ @x

.U B /n2 ; : : : ; U B

@mk @x

m0k

@y

mk m0k

@l f @x l 0 @y ll 0

mj0

with mj and mj 1 for j D 1; ; k. Also m1 C C mk C l D s and n1 C C nk C k D n; in particular n1 C C nk n s. The operators .U B /nj mj contract the L2 norm by a factor less than ı ni . The operators U B m0 @ m m0 @x

j

@y

j

j

are continuous on L2 .T 2 / with norm less than ck kC s ; since the number k of such operators occuring in a given term is smaller than s, each term has an W s;2 norm less than c 0 k ksC s ı ns :kf ks , for a constant c 0 which only depends on s. As the number of terms in the sum is smaller than .s C 1/s .n C 1/s , the Ws;2 norm of .U B /n f is smaller than c 00 k ksC s .n C 1/s ı ns kf ks for a universal constant c 00 . When ı is < 1, this series is summable, proving (iii). Therefore, for any integer s the map t 7! f .t; :/ takes values in W s;2 .T 2 /. In particular, f .1; :/ is C 1 . To finish the proof of Proposition 3.3, we prove that f .1; :/ has no zeroes. We show first that for any t 2 Œ0; 1 , f .t; :/ does not vanish identically. Consider the scalar function u.t/ D kf .t; :/k2 , i.e. the square of the L2 norm of f .t; :/. From Equation (3.3), we deduce that u.t/ D kf .t; :/k2 satisfies ju0 .t/j 2

kU B k u.t /: 1 tkU B k

Since u.0/ D 1, we obtain that for t 2 Œ0; 1 kf .t; :/k 1 t kU B k: In particular for any t 2 Œ0; 1 , f .t; :/ does not vanish identically. The same differential inequation holds with the Sobolev norm. It follows that for t 2 Œ0; 1 the W s;2 norm of kf .t; :/ 1ks is bounded by C kU B ks for some universal constant C ; therefore from Claim 3.5, kf .t; :/ 1ks is bounded also by C 0 k kC s . The Sobolev Lemma [32] says that the Sobolev space W 3;2 .T 2 / injects continuously into C 1 .T 2 /. Therefore, there is a constant 0 > 0 such that for any t 2 Œ0; 1 the function f .t; z/ has no zeroes as soon as k kC 3 , the C 3 norm of , is smaller than 0 .


51

One corollary of this is the following: Claim 3.6. For any z0 2 C there exists a neighborhood V of z0 and a C 1 function N fz0 W Œ0; 1 V ! C such that for any t 2 Œ0; 1 the 1-form fz0 .z; t /.dz C t .z/d z/ is closed. Proof. Let us first suppose that .z0 / D 0. Then, by choosing > 0 sufficiently small the derivatives up to order 3 of the function z 7! .z0 C .z z0 // can be made arbitrarily small over the ball B.z0 ; 1=2/ of radius 1=2 centered at z0 . Now, let us choose a bump function h W C ! R which is equal to 1 on the ball B.z0 ; 1=4/ and which vanishes outside the ball B.z0 ; 1=2/. After making the function h.z/ .z0 C .z z0 // Z2 -periodic we obtain a Z2 -periodic function 0 whose C 3 norm is smaller than 0 if is chosen sufficiently small. From the above remark, there is a non-vanishing N is closed. Choosing C 1 function g.t; z/ such that the 1-form g.t; z/.dz C t 0 .z/d z/ V D B.z0 ; =4/ and fz0 .t; z/ D g.t; z0 C .z z0 /=/, the conclusions of Claim 3.6 are satisfied. Let us now suppose that .z0 / ¤ 0. Let us make an R-linear change of coordinates by defining D A.t; z/ D z C t .z0 /z. N In these new coordinates, the 1-form dz C N where 0 is C 1 , k 0 k1 < 1 and t .z/d zN can be written as ˛.t; /.d C 0 .t; /d /, 0 0

.t; A.t; z0 // D 0. After replacing .:; :/ by a Z2 -periodic function which is equal to 0 .:; :/ in a neighborhood of .t; A.t; z0 // we are reduced to a problem similar to the one studied above: finding a C 1 function g.t; / in a neighborhood of f.t; A.t; z0 //g N is closed. in Œ0; 1 T 2 which does not vanish and such that g.t; /.d C 0 .t; /d / g.t;A.t;z// Such a function exists by the first case. Then fz0 W .t; z/ 7! ˛.t;A.t;z// satisfies the conclusions of Claim 3.6. To finish the proof of Proposition 3.3 we now show that f .t; z/ has no zeroes on Œ0; 1 T 2 . Let us consider the set Z Œ0; 1 of t’s such that f .t; :/ has a zero. By definition, Z is closed. We show now that it is also open. Let t0 2 Z, and let z0 2 C such that f .t0 ; z0 / D 0. Since the function f .t0 ; :/ does not vanish identically, we may suppose that z0 is in the frontier of the set of zeroes of f .t0 ; :/. Let V be the neighborhood of z0 and g be the function provided by Claim 3.6. Then, up to reducing the neighborhood V , for each t 2 Œ0; 1 , the function Z z g.t; z/.dz C t d zN / z 7! .t; z/ D z0

is a quasiconformal diffeomorphism from V to an open subset R z in C. Integration of N the 1-form f .z/.dz C t d zN / leads to another map .t; z/ D z0 f .t; z/.dz C t d z/. 1 .t; z// is well-defined on Then, for each t 2 Œ0; 1 the function h.t; z/ D .t; N a neighborhood of 0 and @h.t; :/ D 0 on this neighborhood. Therefore h.t; z/ is holomorphic for any t 2 Œ0; 1 . By the Rouché Theorem the existence of a zero for the non-constant holomorphic map h.t; :/ is an open property in t . Therefore, Z is open. If Z ¤ ;, then Z D Œ0; 1 , contradicting f .0; :/ D 1. This concludes the proof of Proposition 3.3.

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Proof of Theorem 3.2. Theorem 3.2 follows simply from the proof of Proposition 3.3. N where f .t; z/ is the Define, for t 2 Œ0; 1 a 1-form on C by ! t D f .t; :/.dz R z C t d z/ solution of Equation (3.2). For all z 2 C, t .z/ D 0 ! t is well-defined (since ! t is closed) and for all t , t is a local diffeomorphism (since f vanishes nowhere). The N t . Then 1 will satisfy the conclusions map t W C ! C clearly satisfies @ t D t @ of Theorem 3.2 once we prove that it is a diffeomorphism of C. Because f is Z2 R1 Ri periodic, t .z C p C iq/ D t .z/ C p 0 ! t C q 0 ! t . Consider the set T Œ0; 1

R1 Ri of t ’s such that the periods 0 ! t and 0 ! t generate a lattice; for t 2 T , t is proper and therefore a diffeomorphism of C. Clearly T is open. If t 2 T , the determinant of R1 Ri the lattice generated by 0 ! t and 0 ! t is the integral of the Jacobian determinant of t over the square Œ0; 1 Œ0; i . This determinant is bounded from below uniformly on Œ0; 1 . Therefore T is closed. In particular 1 is a diffeomorphism of C with Beltrami coefficient . After rescaling 1 by homothety so that .1/ D 1, one obtains a normalized quasiconformal C 1 diffeomorphism with Beltrami coefficient . Proof of Theorem 3.1. By Proposition 3.3, for any C 1 function W C ! C with kk1 < 1 which is invariant under a rank-2 lattice there is a quasiconformal diffeomorphism of C with Beltrami coefficient . After post-composition with an affine map we may suppose that this diffeomorphism is normalized; we denote it by f . We now prove Theorem 3.1 when U D C. Let 2 L1 .C/ with kk1 D k < 1. Let us choose a sequence of C 1 functions n W C ! C, with k n k1 k such that each n has compact support and such that n ! a.e.; such a sequence n can be constructed by convolution of the restriction jD.0;n/ with an appropriate C 1 function R hn 0 satisfying hn .z/dxdy D 1. Then, we can transform each n to be invariant under a rank-2 lattice, by choosing such a lattice whose fundamental domain contains the support of n . In this way we obtain a sequence of C 1 functions .n / with norm less than k, which are invariant under a rank-2 lattice and such that n ! a.e. Since the homeomorphisms f n are normalized and are K-quasiconformal with the we may suppose, up to extracting a subsequence, that they same constant K D 1Ck 1k x to a K-quasiconformal homeomorphism converge uniformly on the Riemann sphere C f (cf. Theorem 2.5). The Beltrami coefficient of f is equal to . N n jD.0;R/ k of We observe that for any R > 0 the L2 norms k@f n jD.0;R/ k, k@f n N n the restrictions of @f , @f to the disc of radius R are bounded independently of n; as f n solves the Beltrami equation it suffices to see this for k@f n jD.0;R/ k. Recall N j2 : Therefore the L2 the Jacobian determinant of f is equal to Jf .z/ D j@f j2 j@f n norm k@f jD.0;R/ k is less than sZ p 1 1 Jf n .z/dxdy D p Area.f n .D.0; R///: p 1 k2 1 k2 D.0;R/ This area is bounded independently of n, because f n converges uniformly to f over compact sets. Therefore the L2 norm of @f n jD.0;R/ is bounded independantly


53

N n ) of n. We may suppose, up to possibly extracting a subsequence that @f n (resp. @f tends to an element g1 (resp. g2 ) weakly in L2loc .C/. N n Also, since f n converges uniformly to f the partial derivatives @f n and @f N converge to the distributional derivatives @f and @f as distributions. Therefore N . In order to prove that the Beltrami coefficient of f is , g1 D @f and g2 D @f N n D n @f n , it suffices to see we need only to check that g2 D g1 a.e. Since @f 2 that n @f n tends to g1 weakly in Lloc .C/. We write g1 n @f n D .g1 @f n / C . n /@f n : The first term on the right tends to 0 weakly in L2loc .C/ since @f n tends weakly to g1 and since kk1 k. For the other term we write, for any h 2 L2loc .C/, ˇ ˇZ ˇ ˇ h.n /@f n dxdy ˇ kh.n /jD.0;R/ kk@f n jD.0;R/ k: ˇ D.0;R/

By the Dominated Convergence Theorem, kh.n /jD.0;R/ k tends to 0 and, as we said before, k@f n jD.0;R/ k is bounded above independently of n. Therefore . n /@f n tends to 0 weakly in L2loc .C/. It follows that the Beltrami coefficient of f is . This proves Theorem 3.1 when U D C. Suppose now that U D H. Denote by W C ! C the map z ! zN . Extend to the lower half-plane by setting .z/ D ..z//. Let f be the normalized quasiconformal homeomorphism of C with Beltrami coefficient . An easy computation shows that the Beltrami coefficients of f B and B f are equal. Therefore f commutes with . Since f is orientation-preserving, this implies that f restricts to a quasiconformal homeomorphism of H.

4 Dependence of f on An important property of the solution of the Beltrami equation is the following regularity property of the map 7! f . It is one of the essential tools for the Bers construction of a complex structure on Teichmüller space. Theorem 4.1. Let ƒ be an open set of C k . Let W ƒ ! L1 .C/, 7! be a map such that (1) there exists k < 1 such that for 2 ƒ, k k1 k < 1, and (2) 7! is continuous, (resp. holomorphic), i.e. for almost every z 2 C, 7! .z/ is continuous (resp. holomorphic). x 7! f .z/ Then 7! f is continuous (resp. holomorphic), i.e. for any z 2 C, is continuous (resp. holomorphic). Recall for the proof the definition of “holomorphic” for a map taking values in a complex Banach space.

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Definition 4.2. Let ƒ be an open subset of C k and let H be a complex Banach space. A map h W ƒ ! H is holomorphic if it has a power series expansion in a neighborhood of any point; or alternatively if for 0 D .0i / 2 ƒ, there exists a polydisc D ƒ, D D fji 0i j i ; 8ig where h./ is represented by the Cauchy formula Z 1 k h./ d : h./ D 2i …. i i / @D Proof of Theorem 4.1. The statement about continuity in Theorem 4.1 was already established during the proof of Theorem 3.1. We saw indeed that if n ! a.e., then f n .z/ ! f .z/ for all z 2 C. For the proof of the rest of the theorem we first suppose that the coefficients are Z2 periodic, C 1 and that the C 3 norms of are bounded independently of . Denote by f the function constructed in Proposition 3.3 with D . Then 7! f is holomorphic as a function from ƒ to W 3;2 .T 2 /. Indeed, from the proof we know that this function can be expressed as an infinite sum f D 1 C U. / C .U B B U/. / C : By Claim 3.5 (1) this sum is convergent in W 3;2 .T 2 / uniformly over ƒ, since k k k < 1 and since the C 3 norm of is bounded independently of . This implies that 7! f is holomorphic as a map from ƒ to W 3;2 .T 2 / because each term is holomorphic. Since W 3;2 .T 2 / maps continuously to C 1 .T 2 /, 7! f is holomorphic also as a map from ƒ to C 1 .T 2 /. In particular theRfunction .; / 7! f ./ is bounded on compact sets. It follows z N is holomorphic; the same holds for the that for all z, 7! 0 f ./.d C ./d / normalized solution 7! f .z/. In the general case, one approximates by coefficients n like in the proof of Theorem 3.1, that is by taking the convolution of with a fixed function hn independant of . Then, for any z, 7! .z/ is a limit of bounded holomorphic maps and therefore is holomorphic (the functions .; / 7! f n .z/ are bounded on compact sets since they are K-quasiconformal with the same K and normalized).

4.1 The derivative of 7! f We compute here the derivative at 0 of the function t 7! f t .z/ when 2 L1 .C/ and t is in a neighborhood of 0 in C. This computation is an important point for showing the existence of a complex structure on Teichmüller space [10]. Its place is not really justified here but its proof fits in the framework of the Glutsyuk approach. Proposition 4.3. Let 2 L1 .C/. Then, for any z 2 C, Z 1 f t .z/ z ./ D z.z 1/ dxdy lim t!0 t . 1/. z/ C the convergence being uniform over compact sets.


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Proof. Suppose first that is C 1 and Z2 -invariant. Then the solution f .t; z/ constructed during the proof of Proposition 3.3 satisfies fP0 D U./. Consider the N One has differential form ! t .z/ D f .t; z/.dz C .z/d z/. d ! t j0 D U./dz C d z: N dt R z Since .t; z/ 7! f .t; z/ is C 1 on Œ0; 1 T 2 , f t .z/ D 0 ! t satisfies Z z f t .z/ z D .U./dz C d zN /; lim t!0 t 0 the convergence being uniform over compact sets of C. Since the diffeomorphism f t is normalized, it is a constant multiple of f t ; in particular f t .z/ z f t .z/ z cz D lim t!0 t !0 t t for some constant c. Therefore h W C ! C is a smooth function with the following N D and h grows at most linearly (since and properties: h.0/ D h.1/ D 0, @h U./ are bounded). In order to prove Proposition 4.3 it suffices to verify that there is only one function with these properties and that it is the function Z ./ 1 dxdy: g.z/ D z.z 1/ C . 1/. z/ h.z/ D lim

This function g is clearly continuous on C n f0; 1g. A change of variable near 0 or 1 shows that g extends continuously by 0 at these two points. Also, denoting by ı the Dirac mass at , one has for z ¤ 0 and ¤ 1:

z.z 1/ 1 D ı @N . 1/. z/ N D . in the sense of distributions. Therefore @g R z.z1/ A change of variables shows that z 7! C j .1/. jdxdy grows atmost like z/ jzj log jzj near 1; the same holds therefore for g.z/. N D @h, N gh is holomorphic on C. Since gh grows at most like jzj log jzj Since @g near 1, it is a polynomial of degree at most 1. Since this polynomial vanishes at 0 and 1, h D g. This proves the proposition when is C 1 and Z2 periodic. The case of a general follows from this one after approximating by a sequence .n / where each n is C 1 , has L1 norm kk and is invariant under a lattice. Since, for all z 2 C the functions t 7! f tn .z/ are holomorphic and tend uniformly d d f tn tends to dt f t uniformly over compact sets to t 7! f t .z/ as n tends to 1, dt over compact sets in C. This gives the expected formula for the derivative.

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5 The BMO-quasiconformal homeomorphisms 5.1 The David Theorem In this section we discuss the Beltrami equation when the coefficient satisfies kk1 D 1. Then the solutions are not necessarily in W 1;2 . We begin by reviewing the general setting of the Beltrami equation. Definition 5.1. Let U and V be open subsets of C and let W U ! V be an orientation-preserving homeomorphism which is ACL. Then, as it was said previously the Gehring–Lehto Theorem implies that is differentiable a.e. [12], [1]. One defines a measurable function W U ! C called the Beltrami coefficient of by setting N @.z/ D .z/@.z/ at any point z where is differentiable with @.z/ ¤ 0 and by setting .z/ D 0 otherwise. Notice that, since preserves orientation, kk1 1 a.e. One seeks for conditions on 2 L1 .U / which guarantee the existence of an orientation-preserving homeomorphism from U to an open subset of C which is ACL and whose Beltrami coefficient is equal to . Guy David made a decisive progress in 1988. He introduced a new class of Beltrami coefficients on C, those which satisfy a logarithmic condition. Definition 5.2. Let 2 L1 .C/ with jj < 1 a.e. One says that satisfies a logarithmic condition when there exist positive constants C and ˛ such that for any sufficiently small the Lebesgue measure .fz; j.z/j > 1 g/ is C e ˛= . Then he proved the following theorem. Theorem 5.3 ([7]). Suppose that 2 L1 .C/ satisfies a logarithmic condition. Then there is an orientation-preserving homeomorphism f W C ! C which is ACL and N D @f a.e. Furthermore there is only one such homeomorphism which satisfies @f up to post-composition with an affine map. In the same paper, David studied several important properties of f : the modulus of continuity, the area distortion by f and by f 1 . Since kk1 D 1, the dilatation function 1Cj.z/j is not in L1 and therefore the homeomorphism f provided by 1j.z/j the theorem of David is not quasiconformal in the classical sense. One possible generalization of the notion of quasiconformal homeomorphism is the notion of BMOquasiconformal homeomorphism (cf. [30]). We recall first the definition of BMO, the space of functions with bounded mean oscillation (cf. [31]). Definition 5.4. Let U be an open subset of C. Let Q W U ! R be a measurable function which Ris locally integrable. For any measurable set E U , we denote by mE .Q/ D jE1 j E Qd the average of Q over E. The space BMO.U / consists of


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those functions Q W U ! R which are locally integrable andRfor which there exists 1 a constant C < 1 such that for any round disc B U , jBj B jQ mB .Q/jd is C . The smallest constant C with this property is called the BMO norm of f . One says that Q is locally in BMO.U / if for any open set V U which is relatively compact in U , the restriction QjV belongs to BMO.V /. The space of those functions is denoted BMOloc .U /. In the Appendix we review the basic properties of BMO functions that we will use. One property is that if one replaces in the definition of BMOloc the Euclidean volume x the new space is identical to the d by the spherical volume of the Riemann sphere C former. Definition 5.5. Let U and V be open subsets of C and let W U ! V be an orientationpreserving homeomorphism which is ACL. One says that is BMO-quasiconformal if there is a function Q W U ! Œ1; 1Œ in BMO.U / such that the Beltrami coefficient of satisfies 1Cjj Q a.e. 1jj The class of BMO-quasiconformal homeomorphisms was introduced by Ryazanov, Srebro andYakubov in [30]. Proposition 2.1 in this reference shows the following relation between these homeomorphisms and those constructed in the David Theorem. If 2 L1 .U /, kk1 D 1, satisfies a logarithmic condition, then 1Cjj is bounded 1jj by a function in BMO.U /; furthermore, the converse is true when U is a quasi-disk. In the rest of this chapter we will prove the following theorem. Theorem 5.6. Let U be a connected open subset of C. Let 2 L1 .U /, kk1 D 1. Suppose that there exists a function Q W U ! R in BMOloc .U / such that 1Cjj Q 1jj a.e. Then there exists an open set V C and an orientation-preserving homeomorphism f W U ! V which is ACL and with Beltrami coefficient equal to . Furthermore if f W U ! V and g W U ! W are homeomorphisms with these two properties, then h D f B g 1 W W ! V is holomorphic. This theorem implies Theorem 5.3. What remains to be proven is that under the hypothesis of Theorem 5.3 the open set V in Theorem 5.6 equals C. This hypothesis clearly implies that satisfies the logarithmic condition for the spherical metric on x Therefore by Proposition 2.1 in [30], K D 1Cjj is bounded by a function in C. 1jj BMOloc .C/. Then it follows from the distortion estimate for the distortion of modulus (Lemma 5.8) that f .C/ D C.

5.2 Proof of Theorem 5.6: existence The homeomorphism f is constructed as a limit of a sequence of quasiconformal homeomorphisms fn W U ! Vn whose Beltrami coefficients have L1 norm < 1. Let n denote the Beltrami coefficient on U such that n .z/ D .z/ if j.z/j

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1 n1 and n .z/ D 0 if .z/ > 1 n1 . Choose two points z1 , z2 in U . Let fn be the quasiconformal homeomorphism of C whose Beltrami coefficient equals n on U and 0 on C n U and which fixes z1 , z2 and 1. Such a homeomorphism exists by Theorem 3.1. We will first prove that up to extracting a subsequence the homeomorphisms fn jU converge uniformly on compact sets in U to a homeomorphism f with the properties stated in Theorem 5.6. This proof is exactly like Steps (i) and (ii) in the proof of Theorem 2.5. However, the equicontinuity requires another argument. It will follow from an important distortion lemma due to Ryazanov, Srebro and Yakubov [30] which shows all the relevance of the property of Q being in BMO. First we recall how the classical distortion property of the modulus of annuli under K-quasiconformal homeomorphisms, m.fA/ Km.A/, can be made more precise by introducing the dilatation function instead of K. Claim 5.7 ([25]). Let f W U ! V be a quasiconformal homeomorphism with Beltrami . Let A U be an annulus which coefficient and dilatation function K D 1Cjj 1jj is relatively compact in U . Denote by F the family of curves W Œ0; 1 ! Ax whose x Then, for any admissible metric for endpoints areRin distinct components of @A. 1 2 F , m.fA/ .z/K .z/dxdy. Denote by G the family of closed Jordan curves contained in A which separate R the two boundary components. Then for any admissible metric for G , m.f A/ 2 .z/K .z/dxdy. Proof. To simplify denote the function K by K. Suppose first that f is a diffeomorphism. We of f F R 2observe that for any admissible metric for1F the modulus 1 is less than .z/K.z/dxdy. Indeed, define Q D B f .z/kD.f /z k where 1 is the norm of the tangent map of f 1 at z. Then Q is an kD.f 1 /z k D j@f j.1j.z/j/ admissible metric for the family of curves f F . Therefore Z Z m.f F / Q2 .z/dxdy D 2 .z/K.z/dxdy: (5.1) Since m.fA/ D m.f1 F / , the claim follows when f is a diffeomorphism. In the general case, one considers a sequence of kKk1 -quasiconformal diffeomorphisms .fn / which converges to f uniformly on compact sets of U , and such that Kn .z/ tends to K.z/ a.e., where Kn is the dilatation function of fn . Then f satisfies Inequality (5.1) by continuity. This implies the statement about the family F . The same proof holds for G . When the dilatation function z 7! K.z/ is bounded by a function in BMO. Claim 5.7 leads to useful modulus estimates. This is an important contribution from [30]. Let A.w; ; r/ U be the (round) annulus fz j jz wj rg; denote by D.w; t / the concentric (round) disc of radius t . The function .z/ D

log

log log

1 1 t

1 1 jz wj log jzwj


59

is admissible for F . The following result will be proven in the Appendix. Lemma 5.8 ([30], Lemma 3.2). Let 0 < t 1. Suppose that Q is in BMO.D.w; t // and that Q 0 a.e. Then for any 0 < e 2 t Z c 2 .z/Q.z/dxdy 1 log A.w;;t/ log 1 log

t

where c only depends on the BMO norm and on the L1 norm of the restriction QjD.w;t/ . With this modulus estimate, we prove now the existence part of Theorem 5.6, by adapting the arguments used for proving Theorems 2.5 and 3.1. (i) The sequence ffn g is equicontinuous on compact sets. Consider an annulus A.w; ; t / U . Lemma 5.8 says that the modulus of fn A.w; ; t / is bounded below by a positive constant independent of n and w (depending only on , t and Q). Arguing as in the step (i) of Theorem 2.5 we deduce that for any compact set K U the restrictions fn jK form an equicontinuous family for the spherical distance. Remark 5.9. More precisely, we see using Claim 6.1 that a modulus of continuity C of the maps fn jK is !.t / D ˇ , where the constant C only depends on K and .log 1t / where ˇ only depends on the BMO norm and the L1 norm of the restriction of Q to a compact neighborhood of K . Suppose after possibly extracting a subsequence that fn converges on compact x subsets of U to a map f W U ! C. (ii) f is a homeomorphism on its image and f U is contained in C. When A U is an annulus which is relatively compact in U , the modulus of m.fn A/ is bounded below by a positive constant independant of n (Claim 5.7). Then Step (ii) of the proof of Theorem 2.5 shows that f is locally injective; it is therefore a homeomorphism from U to f U as in Step (iii) of this proof. The same argument shows that f U C. To see this, it suffices to choose, for any point z 2 U , z ¤ z1 ; z2 an annulus A U which separates z and z1 from z2 and 1. (iii) f is ACL. The reasonning here is taken from the approach of Brakalova and Jenkins [4]. It is exactly along the same lines as in the classical situation. Consider a rectangle I J U . During the proof of Theorem 2.9 we showed that f .:; Y0 / is absolutely continuous for any derivability point of y 7! ƒ.y/ D .I Œ0; y /. Let Y0 J denote the set of those points. In the present case we need to consider a slightly smaller set. Recall that for any interval Œa; b I there is a full measure set YŒa;b J of points y such that Q.:; y/

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R is integrable over I and such that Œa;b fyg Q.x; y/dx is the derivative of the function T R y 7! Œa;b Œy0 ;y Q.x; y/dxdy. Let Y J be the set Y0 \ Œa;b YŒa;b where Œa; b describes the set of all intervals contained in I which have rational endpoints. The set Y has full measure. We prove now that when y0 2 Y then f .:; y0 / is absolutely continuous. Consider a finite union of disjoint intervals Œai ; bi contained in I with total length and with rational endpoints. Denote again by Riı the rectangle Œai ; bi Œy0 ; y0 C ı . Then by the Rengel inequality (2.3), X 2 X ƒ.y0 C ı/ ƒ.y0 / : di .ı/ ı m.f Riı / ı An upper bound for the moduli m.fRiı / can be deduced from Claim 5.7. Since the constant function D 1ı is admissible for the family of curves which join the horizontal sides of Riı , Z X 1 X ı m.fRiı / Q.x; y/dxdy : ı Œai ;bi Œy0 ;y0 Cı

Letting ı tend to 0, we obtain Z X 2 jf .bi / f .ai /j

tŒai ;bi

Q.x; y0 /dx ƒ0 .y0 /:

By continuity, this inequality remains true for an arbitrary union of disjoint intervals. Since Q.x; y0 / is integrable, it follows that x 7! f .x; y0 / is absolutely continuous. N D @f a.e. (iv) @f The argument is also taken from [4]. Like in the proof of Theorem 3.1, we will N n /) is bounded in Lq .U / for show that the sequence .@fn / (and therefore also .@f loc some q > 1, i.e. that for any compact set K U the Lq norms of the restrictions @fn jK are bounded. It will follow that a subsequence of .@fn / converges weakly in N D @f a.e. Lqloc ; the same reasoning as in the proof of Theorem 3.1 will give @f The derivative @fn is related to the Jacobian determinant Jn of fn by j@fn .z/j2 D Jn .z/ where Kn D

1Cjn j 1jn j

q 2

1 Jn .z/Kn .z/ 1 jn .z/j2

is the dilatation function of fn . In particular, for any q > 0 q

j@fn .z/jq .Jn .z// Kn .z/ 2 : Choose as compact set K a closed surface contained in U with smooth boundary. For any exponents p 0 > 1, q 0 > 1 with p10 C q10 D 1, the Hölder inequality gives Z 10 Z 10 Z p0 q q0 q p q q 2 2 j@fn j d Jn d Kn d : (5.2) K

K

K

R 1 Choose any q such that 1 < q < 2 and set p 0 D q2 . Then K Jn d p0 is bounded by a quantity independant of n since the integral is the algebraic area enclosed by


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q0 q R R q0 q fn .@K/ and since fn converges uniformly to f . Also K Kn 2 d K Q 2 d and this last quantity isRfinite since Q being in BMOloc is locally in Lp for all p > 0 (cf. Appendix). Thus K j@fn .z/jq d is bounded by a constant independent of n. Therefore .@fn / is bounded in Lqloc .U / for any q, 1 q < 2. This ends the proof of the existence in Theorem 5.6.

5.3 Proof of Theorem 5.6: uniqueness The general scheme for proving uniqueness is taken from [19], in particular the use of the harmonic extension, see [19], p. 254. Suppose that f W U ! V and g W U ! W are two orientation-preserving homeomorphisms which are ACL and which have the same Beltrami coefficient. Let D W x ! R which is the real part of the conbe a small round open disc. The function ˛ W @D 1 x still denoted ˛, which is tinuous function f Bg extends to a continuous function in D, harmonic on D. Let a be a holomorphic function on D whose real part equals ˛. Consider the Jordan domain 0 D g 1 .D/ contained in U ; then the map a B g W 0 ! C is ACL on 0 . By definition of a, the real part of h0 D a B g f W 0 ! C extends S0 (although h0 does not a priori extend contincontinuously to 0 at the boundary @ uously to the boundary). In particular, if z 2 C is such that 0 in , then for any point z in H./ degree deg.H; z/ is > 0. By an approximation argument, the same property holds for x ! C with H j locally in W 1;2 . any continuous map H W In our situation, hj is not locally in W 1;2 . However it is almost so thanks to the next claim. We first recall another definition. Definition 5.10. Say that a measurable function W ! C is locally in L2 = log L if R jj2 for any compact set K the integral K log.eCjj/ dxdy is finite. By analogy with 2

the Sobolev spaces W 1;p , the Orlicz–Sobolev space W 1;L = log L ./ is defined as the space of all measurable maps f W ! C which are locally integrable and whose distributional derivatives @f and @f are measurable functions which are locally in @y @x 2 L = log L. Claim 5.11. The map h D a B g f W ! C is ACL. Its Jacobian determinant is 2 0 a.e. and locally integrable. Furthermore, hj is in W 1;L = log L ./.

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Proof. Clearly h is ACL as a sum of ACL maps. Since a B g and f are two solutions N D @h a.e. In of the same Beltrami equation on , h D a B g f also satisfies @h 2 N 2 is 0 a.e. particular, since jj < 1 a.e., its Jacobian determinant Jh D j@hj j@hj Since a B g and f solve the same Beltrami equation, Jh 2 JaBg C Jf . Now, since g and f are homeomorphisms, their Jacobian determinants are integrable. Since a is C 1 on , the Jacobian determinant of a B g is also locally integrable. Therefore Jh is locally integrable. For the last statement of the proposition, it suffices to see that kDhk is locally in L2 = log L. We reproduce the proof from [18] and [19], p. 260. Let z 2 . Then N 2 is Jh .z/Q.z/. For some constant C , one has kDh.z/k2 D .j@hj C j@hj/ Q.z/Jh .z/ kDh.z/k2 C : log.e C kDh.z/k/ log .e C Jh .z// The following inequality holds for all positive real numbers x and y: xy x log.1 C x/ C .e y 1/: Let be the constant in the John–Nirenberg inequality (cf. Theorem 6.2). An appli Q.z/ kDh.z/k2 cation of the inequality with x D Jh .z/ and y D kQk gives that log.eCkDh.z/k/ is BMO bounded by Q.z/ C kQkBMO J.h; z/ log.1 C Jh .z// 1 C e kQkBMO log.e C Jh .z// Q.z/ C kQkBMO kQkBMO Jh .z/ C e : The claim follows now from the John–Nirenberg inequality (Theorem 6.2) The next theorem is the central result from [22] (proven in this reference for any 2 dimension). It means that maps in W 1;L = log L ./ whose Jacobian determinant is 0 almost everywhere are orientation-preserving. x ! C be a continuous map with F j Theorem 5.12 ([22], Theorem 2.4). Let F W 1;L2 = log L in W ./. Suppose that the Jacobian determinant JF is locally integrable and 0 almost everywhere. Then the degree of F at any point of F ./ F .@/ is > 0. This theorem will be discussed in the next section. We conclude now the proof of the uniqueness part in Theorem 5.6. By Claim 5.11, Theorem 5.12 can be applied with S0 / is contained in the y-axis. This implies that Jh D 0 a.e. F D h. It follows that h. N D 0 a.e. As h is ACL, h is constant. and therefore, since jj < 1 a.e., that @h D @h 1 This means that f D a B g on g D, where a is holomorphic on D. By analytic continuation a extends to a holomorphic map W ! V such that f D a B g. This concludes the proof of the uniqueness statement in Theorem 5.6.


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5.4 Topological degree of Sobolev maps Here we prove Theorem 5.12. An essential assumption is the L2 = log L integrability of the distributional derivatives of F . Indeed [22] contains an example of a continuous map F W Œ0; 1 2 ! C which has the following properties: (1) the distributional derivatives of F j 0;1Œ2 are locally in L2 = log2 L; (2) F restricts to the identity map on the boundary of Œ0; 1 2 (in particular the degree of F at any point of the open square is 1); (3) the Jacobian determinant is integrable and 0 almost everywhere. The core of the proof of Theorem 5.12 is the following change of variable formula. x ! C be a continuous map such Proposition 5.13 ([22], Theorem 2.3). Let F W 1;L2 = log L ./. Suppose that the Jacobian determinant JF is locally that F j 2 W integrable and 0 almost everywhere. Let u W C ! R be a C 1 function whose support is compact and disjoint from F .@/. Then Z Z u./deg.F; /d d D u B F .z/JF .z/dxdy: C

x ! C be a C 1 map. Let u W C ! R be a C 1 function whose support Proof. Let G W is compact and disjoint from G.@/. The classical change of variable formula says: Z Z u./deg.G; /d d D u B G.z/JG .z/dxdy: C

R

The term on the right is equal to G .ud ^ d/ where if is a differential form on C and G denotes the pull-back of under G: for any d -vector v at 2 , G .v/ D .DGx .v//: When G W ! C is only assumed to be differentiable a.e., the pull-back G is defined by the same formula. For instance when F satisfies the hypothesis of Proposition 5.13, then for any C 1 1-form , the pull-back F is a differential 1-form on whose coefficients are measurable functions which are the conclusion of Proposition 5.13 can be written as locally in L2 = log L. Therefore R R u./ deg.F; /d d D F .ud ^ d/. C 1 Choose a C 1-form such that d D ud ^d (for instance, can be taken equal to ?dv where v is a solution of v D u). We can write, still under the assumption that G is C 1 (C 2 would suffice) Z Z Z u./deg.G; /d d D G .d / D dG . /: C

1

Let W ! R be a C function with compact support and equal to 1 on a neighborhood of G 1 .supp.u//. Then Z Z Z dG . / D dG . / D G . / ^ d

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Jean-Pierre Otal

by Stokes formula. Combining these last two equations, we obtain: Z Z u./deg.G; /d d D G . / ^ d: C

(5.3)

x ! C is continuous with F j in W 1;1 ./. Let This formula holds also when F W 1 x x and Fi W ! C be a sequence of C maps which converges to F uniformly on 1;1 locally in W . Suppose in the same way as before, taking R R that u, and are defined F instead of G. Then Fi . /^d tends to F . /^d: Applying Equation (5.3) with G D Fi and observing that when i is sufficiently large, deg.Fi ; / D deg.F; / for any in the support of u, we obtain, letting i ! 1: Z Z u./deg.F; /d d D F . / ^ d: C

This holds assuming only that F is continuous and locally in W 1;1 ./. In order to prove Proposition 5.13, we need to show that under the stronger integrability assump2 tion F j 2 W 1;L = log L , then Z Z F . / ^ d D F .d /:

As is constant equal to 1 on the support of u B F , it suffices to prove that if F satisfies the hypothesis of Proposition 5.13, and if is a C 1 1-form with compact support on C, then, for any C 1 function W ! R with compact support, Z Z F . / ^ d D F .d /: (5.4)

The basic tool for proving this is the following formula of integration by parts of the Jacobian determinant. This formula in the case of Sobolev mappings is the subject of a vast literature (cf. Chapter 7 of [19]). The statement below, due to Luigi Greco, is proven in [14], using a result of Iwaniec–Sbordone ([21], Theorem 1). Theorem 5.14 ([14], Theorem 4.1). Let G D .G1 ; G2 / W ! C be a continuous 2 map which is in W 1;L = log L ./; suppose that the Jacobian determinant JG is locally integrable. Then for any C 1 function W ! R with compact support Z Z .z/JG .z/dxdy D G1 d ^ dG2 :

Equality (5.4) is a consequence of this theorem. Suppose first that is proportional @c to the 1-form dx, i.e. D c.x; y/dx. Denote d D @y dx ^ dy D u.x; y/dx ^ dy. Set G D .G1 ; G2 /, with G1 D c B F and G2 D F1 . Then G W ! C satisfies the hypothesis of Theorem 5.14 and its Jacobian determinant is JG .z/ D u.F .z//JF .z/ a.e. One has also G1 d ^ dG2 D F .d / ^ d. Therefore Equation (5.4) is a particular case of Theorem 5.14. The same can be done when is proportional to dy, D d.x; y/dy. This proves Equation (5.4) by linearity.

65


Proof of Theorem 5.12. Let B a ball F ./ F .@/ and let u be a non-constant function 0 with support contained in B. By Proposition 5.13 and since deg.F; z/ is constant over B and equal to deg.F; B/, Z Z u B F .z/JF .z/dxdy D deg.F; B/ u./d d:

C

Therefore, if deg.F; B/ D 0; then JF D 0 a.e. on the open set V D fu B F > 0g. Since jj < 1 a.e., DF D 0 a.e. on V . In particular, u B F is constant on V. But since u B F vanishes on the frontier of V , u B F vanishes identically, a contradiction.

6 Appendix 6.1 Estimates of moduli of annuli We give proofs of estimates for the modulus of annuli that were used in the text. x be an annulus and let be the infimum of the spherical diameters of the Let A C x n A. Define to be the function on C such that .z/jdzj is the two components of C x Then the spherical length of any curve in A which spherical Riemannian metric on C. x n A is 2 . Therefore the function D 1 is separates the two components of C 2 admissible for the family G ; in particular, one has (6.1) m.A/ 2 : A better upper bound for m.A/ is the following. x be an annulus; suppose that one component of the complement Claim 6.1. Let A C of A contains two points a and b, and the other component contains points c and d . Then when the spherical distance between c and d is sufficiently small, one has m.A/ C log

1

(6.2)

for a constant C which only depends on a lower bound for the mutual spherical distances between the points a, b and c. Proof. After applying a homography, we can suppose that a D 1, b D 1 and c D 0. The claim says that when jzj is sufficiently small, the modulus of an annulus A which separates the points 0 and z from 1 and 1 satisfies m.A/ 1 for some universal constant C . Consider the complete hyperbolic metric on C log jzj C n f0; 1; zg; for a homotopy class of closed curves on C n f0; 1; zg, denote by l the hyperbolic length of the geodesic in this homotopy class. Let a be a generator of the fundamental group of A. The annulus A lifts homeomorphically to the covering Aa of C n f0; 1; zg which has fundamental group the cyclic group generated by a:

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this covering is isomorphic to an annulus with modulus la . On the other hand, since A lifts isomorphically to an annulus contained in Aa the Schwarz lemma implies that m.A/ m.Aa / D la . To prove the claim, we now show that when jzj is sufficiently small the length of the shortest geodesic on C n f0; z; 1g is C 1 . Observe log

jzj

that thephyperbolic metric on C n f0; z; 1g is isometric to the hyperbolic metric on C n f0; z; p1z g (the one which is conformally equivalent to the Euclidean metric) by

the homothety of ratio p1z . Using the isometry ! 1 , one sees that the unit circle is a geodesic of the hyperbolic metric: denote by cpthe homotopy class it represents. The unit circle is the core of the round annulus jzj < jj < p1jzj embedded in p 2 2 C n f0; z; p1z g; therefore lc j log . In particular, when jzj is sufficiently small, lc jzjj is shorter than the Margulis constant. Recall that on any hyperbolic surface, the primitive geodesics which are shorter than the Margulis constant are simple curves and are p disjoint from each other. Since any simple closed curve on C n f0; p1z ; zg which is disjoint from the unit circle is homotopic to c or to one of the punctures, c is the only possible homotopy class containing a geodesic shorter than the Margulis constant. Therefore we are reduced to showing that lc is at least j logCjzjj for some C > 0. By the p p Schwarz lemma again the inclusion of C n f0; z; p1z g into C n f0; zg decreases the hyperbolic length; therefore lc ispat least equal to the length of the unit circle for the hyperbolic p metric on C n f0; zg. This length also equals the length of the circle of radius jzj for the hyperbolic metric on C n f0; 1g. To estimate this length, consider the covering of C n f0; 1g whose fundamental group is generated by the element running once around the puncture 0. The covering space is the punctured unit disc fw; jwj < 1; w ¤ 0g; the hyperbolic metric is jwjjjdwj . The covering map log jwjj w 7! .w/ is a local isometry between the hyperbolic metrics. As a holomorphic map, it also extends continuously at the puncture. The resulting map is holomorphic and its derivative at 0 is ¤ 0 for topological reasons. It follows that when jj is sufficiently 1 small, the hyperbolic metric is equivalent to jjj log ; thus when jzj is sufficiently j jj p 1 small, the length of the circle of radius jzj is C 1 for some C > 0. log

jzj

x be an annulus. Suppose that the spherical diameter of each component Let A C x of C n A is 2D and that the distance d between these two components is small compared to D. Then there is a round annulus A.d; D/ such that the circles of radii d x nA. Any element of the family of curves and D both intersect the two components of C G has to cross the annulus A.d; D/ at least twice. Let be the metric on A which is equal to the metric g D jdzj on the intersection A \ A.d; D/ and which is 0 outside jzj A.d; D/; then the -length of each curve in G is 2 log. D / and D .2 log. D //1 d d is an admissible metric. Therefore 1 : (6.3) m.A/ 4 log. D / d


67

6.2 BMO functions Let U C be an open set. Let f W U ! R be a measurable function which is R locally integrable. For any measurable set E U let mE .f / D jE1 j E f d denote the average of f over E. Then f 2 BMO.U / if and R only if there exists a constant 1 C < 1 such that for any round disc E U , jBj B jf mB .f /jd is C . The smallest constant C with this property is called the BMO norm of f and denoted kf kBMO . One first remark is that the BMO norm is not a norm on the space of BMO functions since it assigns 0 to any constant functions. It is indeed a norm on the quotient space of BMO by constant functions. From the definition of BMO.U / function it follows that one defines the same space (with an equivalent norm) by requiring in the definition the existence of a constant D R 1 jf d .f /jd is less and for any round ball B U of a constant dB such that jBj B B than D. Indeed if f satisfies this inequality then jdB .f / mB .f /j D; therefore f 2 BMO.U / and kf k 2D. One consequence is that if U is an open set in C, the property for a function of being in BMOloc .U / does not change if one replaces the Lebesgue measure on U by any Riemannian measure. This is true in particular if one chooses as measure on U x the restriction of the spherical measure of C. The next result is the John–Nirenberg inequality. Theorem 6.2 ([31], p. 144). There is a constant > 0 such that for any U C and any f 2 BMO.U / one has for any B U Z .f m .f // B 1 e kf kBMO d 2: (6.4) jBj B This theorem implies strong integrability properties for f 2 BMO.U /; in particular, f is locally in Lp for all p > 0. Also, if for 2 L1 .U / with jj < 1 a.e. the dilatation 1Cjj is bounded from above by a function Q 2 BMO.U /, then satisfies 1jj locally the logarithmic condition of David (Definition 5.2). The converse is also true. Proposition 6.3 ([30], Proposition 2.1). Let W U ! C be a measurable function with jj < 1 a.e. which satisfies locally the logarithmic condition. Then in a neighborhood is bounded from above by a function in BMO. of any point in U , 1Cjj 1jj We end this section with the proof of Lemma 5.8 which was a key point for the proof of the modulus estimate in Section 5.2. Proof of Lemma 5.8. We reproduce the proof of Lemma 3.2 from [30]. To simplify notation set w D 0; let t 1 and let Q 2 BMO.D.0; t // such that Q 0 a.e. Then we will prove that for all e 2 t , Z log 1 1 Qdxdy c log ; 2 2 log 1t A.;t/ r log r

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Jean-Pierre Otal

where c only depends on the BMO norm of the restriction QjD.0;t / and the L1 norm of QjD.0;t/ . Denote by An the ring A.e .nC1/ ; e n / and by Dn the disk D.0; e n /. Let p be the q qC1 . integer such that e p 1. Since x is a fixed point of S B T , by using the expression for S B T we obtain 0
0. Since ln1 1 1 Ejln D Ejl0 B .Ej1 l0 B Ejl1 / B B .Ejln2 B Ejln1 / B .Ejln1 B Ejln /;

E maps the geodesic ln to the geodesic connecting E.n/ D . 1e C e12 C C e1n / 1 to 1. Clearly, E.n/ converges to e1 as n goes to 1, and hence E maps the full hyperbolic plane H into the hyperbolic half plane to the right of the geodesic 1 to 1. Therefore E is not onto and hence it is not an earthquake connecting e1 map. A more interesting example was constructed in [9]. In that example, an earthquake measure .; L/ was inductively constructed such that the generalized earthquake map E corresponding to is not an earthquake map but the generalized earthquake map E2 corresponding to .2; L/ is actually an earthquake map. In [9], the work in Sections 2.1 and 2.2 was devoted to the construction of an earthquake map inducing a given earthquake measure with finite Thurston norm. It was also pointed out there that the procedure applies to the construction of a map similar to an earthquake map, but not satisfying the “onto” condition, for any Borel measure supported on a lamination. In our terminology, the work of Sections 2.1, 2.2 and 2.3 in [9] shows the following theorem.

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Jun Hu

Theorem 5.4. Given any earthquake measure .; L/, there exists a generalized earthquake map .E; L/ such that is the earthquake measure induced by E. Moreover, up to post-composition by a Möbius transformation, determines the isometries of E on all gaps, and for any leaf l 2 L, two possibly different isometries on l only differ by a hyperbolic isometry with axis l and translation length between 0 and the measure .l/ of l. Theorem 5.5 (Thurston). If an earthquake measure .; L/ is Thurston bounded, then there exists an earthquake map .E; L/ such that is the induced earthquake measure by E. Moreover, up to post-composition by a Möbius transformation, determines the isometries of E on all gaps, and for any leaf l 2 L, two possibly different isometries on l only differ by a hyperbolic isometry with axis l and translation length between 0 and the measure .l/ of l. In [21], a sufficient condition is introduced for a type of Thurston unbounded earthquake measures to be induced by earthquake maps, which is an analogue of David’s theorem ([2]) for solutions of the Beltrami differential equation ([1]) in earthquake theory. Given a lamination L, we use ˇ to denote an arbitrary geodesic arc transversal to L of hyperbolic length 1 and ı.ˇ/ to denote the Euclidean distance from the arc ˇ to the boundary of . Theorem 5.6. If an earthquake measure .; L/ satisfies .ˇ/

1 2 ln ln CC 3 ı.ˇ/

(5.1)

for a constant C > 0 and any geodesic arc ˇ transversal to L of hyperbolic length 1 and sufficiently close to the boundary in the Euclidean metric, then there exists an earthquake map .E; L/ such that is the earthquake measure induced by E. Moreover, up to post-composition by a Möbius transformation, determines the isometries of E on all gaps, and for any leaf l 2 L, two possibly different isometries on l only differ by a hyperbolic isometry with axis l and translation length between 0 and the measure .l/ of l. The main steps to prove Theorem 5.4 are: (1) Approximate the measure .; L/ by atomic measures .n ; Ln / supported on a nested sequence of laminations Ln L consisting of finitely many leaves; (2) Take a common leaf l0 in all laminations Ln and for each n 2 N construct a finite earthquake En determined by .n ; Ln / that induces the identity map on the leaf l0 ; (3) On each stratum A corresponding to the lamination L, the Möbius transformation EA on A is defined to be the limit of a convergent subsequence of fEn jA g1 nD1 ; (4) Show the collection fEjA W A is a stratum corresponding to Lg defines a generalized earthquake map E on the hyperbolic plane; (5) Check the earthquake measure induced by E is indeed equal to .; L/.

Chapter 2. Earthquakes on the hyperbolic plane

89

Then the main work to prove Theorem 5.5 or 5.6 is to show that the constructed generalized earthquake is an earthquake map; that is, it maps the hyperbolic plane onto itself. For interested readers, we refer to [9] for the detailed proofs of Theorems 5.4 and 5.5 and [21] of Theorem 5.6.

6 Equivalence of cross-ratio distortion norm and Thurston norm Given an orientation-preserving circle homeomorphism h, let .E; L/ be a left earthquake representation of h and the earthquake measure induced by .E; L/. Note that is uniquely determined by h. The goal of this section is to introduce a quantitative relationship derived in [12], saying that the Thurston norm of h is equivalent to the cross-ratio distortion norm of h. Definition 6.1. For any orientation-preserving homeomorphism h of the unit circle S1 , the cross-ratio distortion norm khkcr of h is defined as sup jln cr.h.Q//j ;

khkcr D

(6.1)

cr.Q/D1

where the supremum is taken over all quadruples Q D fa; b; c; d g of four points arranged in counter-clockwise order on the circle with cr.Q/ D 1, and where cr.Q/ D

.b a/.d c/ : .c b/.d a/

Furthermore, we say that h is quasisymmetric if khkcr is finite. Recalling Definition 4.5, the Thurston norm k kTh of is the supremum, over all hyperbolic geodesic segments ˇ of length one in the hyperbolic plane, of the total amount of shearing along the lines in the support of that intersect ˇ. Furthermore, we say that is Thurston bounded if k kTh is finite. Theorem 6.2 ([12]). There exists a universal constant C > 0 such that for any orientation-preserving homeomorphism h, 1 khkcr kkTh C khkcr ; C where is the earthquake measure induced by h. Theorem 6.2 immediately implies Corollary 6.3 (Thurston). Let h be an orientation-preserving circle homeomorphism and the earthquake measure induced by h. Then is Thurston bounded if and only if h is quasisymmetric.

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Jun Hu

The whole work of this section is devoted to the proof of Theorem 6.2, which is quite long. Before starting to do it, let us first recall briefly the progress towards a complete proof of the previous corollary due to Thurston. It was known to Thurston ([35]) that is Thurston bounded if and only if h is quasisymmetric. The “if” part was proved in [35] and the “only if” part was only given a sketch of proof there. A complete proof of the “only if” part was first presented in [9] by deriving first some regularity for the tangent vectors to an earthquake curve determined by and then obtaining the same regularity for the homeomorphism of S1 by integrating the vector field. Around ˘ c developed an alternative proof in his thesis ([27], and published in [28]) that time, Sari´ by showing that the corresponding earthquake curve is holomorphic on the parameter and then applying Slodkowski’s extension theorem ([32]) on holomorphic motions. ˘ c presented a short proof of the “only if” part in [29] by using the method Lately, Sari´ of proof by contradiction. All of these three proofs somehow miss an essence on how the Thurston norm of the earthquake measure explicitly controls the cross-ratio distortion of the boundary homeomorphism. The main work in [12] emphasizes that essence by using a quite elementary and direct approach (not long), and furthermore it leads to a universal quantitative dependence of the cross-ratio distortion norm of the homeomorphism on the Thurston norm of the earthquake measure. That is, there exists a universal constant C > 0 such that khkcr C k kTh

(6.2)

for any h. Note that it had already been a guide for the work in [9] to search for a quantitative relationship between the norms on the cross-ratio distortions and earthquake measures. In [9], we first obtained that there exists a constant C > 0 such that kkTh C khkcr

(6.3)

for any orientation-preserving homeomorphism h. In the converse direction, a partial result was derived there, that is, for any constant C0 > 0, there exists a constant C > 0 such that for any , if k kTh C0 then khkcr C . In this section, we present the proofs of Inequality (6.3) given in [9] and Inequality (6.2) in [12]. In the following, we first recall two important techniques used in the proofs, which we summarize into two lemmas and two corresponding corollaries. Let fa; b; c; d g be a quadruple of points on the real axis with a < b < c < d , and L, M , R and T stand for the left, middle, right and total intervals, i.e., L D Œa; b ; M D Œb; c ; R D Œc; d and T D Œa; d . We also identify L, M , R and T with the corresponding lengths b a, c b, d c and d a. Lemma 6.4. Assume a < b < c < x < y < d and hx and hy are simple left earthquake maps, each with multiplier > 1, and supported on the lines that join x to 1 and y to 1, respectively. Let Lx , Mx , Rx , Tx and Ly , My , Ry , Ty be the lengths of the images of the intervals L, M , R and T under the mappings hx and hy .


Then

Lx Rx Ly Ry : > My Ty Mx Tx

91

(6.4)

Moreover, if a < y < x < b < c < d , then Lx Rx Ly Ry < : Mx Tx My Ty

(6.5)

Proof. Both hx and hy are equal to the identity on L and M , so L D Lx D Ly and M D Mx D My . Also, Rx .d x/ C x c ; D Tx .d x/ C x a and

Ry .d y/ C y c : D Ty .d y/ C y a

Thus, Rx Ry > ; Tx Ty and Inequality (6.4) follows. The proof of (6.5) is similar. Lemma 6.5. With the same notation as in the previous lemma, suppose a < b < x < y < c < d . Then Ly Ry Lx Rx > : (6.6) My Ty Mx Tx Proof. Clearly, hx .z/ equals z for z < x and .z x/ C x for z x, and similarly, hy .z/ equals z for z < y and .z y/ C y for z y. We have hx .L/ D hy .L/ D L; hx .R/ D Rx D R D hy .R/ D Ry D R. And since x < y and > 1, hx .M / D Mx > hy .M / D My ; hx .T / D Tx > hy .T / D Ty , which implies the lemma. Given a quadruple Q D fa; b; c; d g with a < b < c < d , we denote, as before, c/ LR by cr.Q/ the cross ratio M D .ba/.d . We have the following corollaries of the T .cb/.d a/ previous lemmas respectively. Corollary 6.6. Let Q D fa; b; c; d g be a quadruple on the real line with a < b < c < d , and c s d and d < t. Suppose that A.s;t / is the hyperbolic Möbius transformation with repelling fixed point at s and attracting fixed point at t and whose derivative at the repelling fixed point is equal to > 1. Let f.s;t / W R ! R be equal to A.s;t/ on the interval Œs; t and equal to the identity on the complement of Œs; t . Let a, b, c, d and be temporarily fixed. Then the cross-ratio of the image quadruple f.s;t / .Q/ considered as a function of two variables s 2 Œy; z and t 2 .z; C1/ decreases in s for each fixed t and increases in t for each fixed s.

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Corollary 6.7. With the same notation as in the previous corollary, suppose b s c and d t. Then the cross-ratio of the image quadruple f.s;t / .Q/ is increasing in s for each fixed t and also increasing in t for each fixed s. In the course of deriving Inequality (6.3), we show first that there exists a positive constant C1 such that for any h, if khkcr < 1, then .ˇ/ C1 khkcr :

(6.7)

Then we show that there exists another positive constant C2 such that .ˇ/ khkcr C C2

(6.8)

for any h. Thus if khkcr 1, then .ˇ/ .1 C C2 /khkcr : Consequently, for any h, we have .ˇ/ maxfC1 ; .1 C C2 /gkhkcr : To prove Inequalities (6.7) and (6.8) we use Corollary 6.6 and Lemma 6.7. Let l1 and l2 denote the lines in L that meet ˇ at points that are the maximal distance apart (the distance being measured on ˇ). There are three situations of relative positions of l1 and l2 we need to consider: l1 and l2 share no endpoints, or one endpoint or two endpoints. Let us first assume that l1 and l2 do not have a common endpoint. In this case we replace ˇ by the geodesic segment perpendicular to both l1 and l2 , which we continue to denote by ˇ. By a normalization we may assume ˇ is an arc on the imaginary axis between i and ib with 1 b e. Clearly, .ˇ/ D .Œb; 1 Œ1; b / and the length of ˇ is log b. We also assume that l1 is the geodesic connecting 1 to 1 and l2 is the one connecting b to b. By post-composition by a Möbius transformation, we can assume that h fixes the geodesic l2 . Consider the sublamination L0 of L consisting of those lines that intersect ˇ and the measure 0 which is the measure restricted to the closed subset L0 . Then 0 induces a homeomorphism h0 . We assume that h0 also fixes the geodesic l2 . To prove Inequalities (6.7) and (6.8), we will compare the cross ratio distortions by h and h0 on certain quadruples Q and give bounds for the cross ratio distortions by h0 on the quadruples. To prove (6.7), we take Q D fb; 1; b; 0g, and to prove (6.8), we take Q D f1; b; 1C.b/ ; 1g. Note that in each case cr.Q/ D 1. 2 In the situation where l1 and l2 share two endpoints, i.e., l1 D l2 , we substitute the above b by 1 and b by 1 everywhere. In the remaining situation where l1 and l2 share one endpoint, we replace ˇ by a geodesic segment of small hyperbolic length which is perpendicular to l1 and transversal to l2 (still call it ˇ); through a conjugation by a Möbius transformation, we normalize ˇ to be an arc on the imaginary axis between i and i b 0 with b 0 > 1, and assume that l1 is the geodesic connecting 1 to 1 and l2 is the geodesic connecting 1 to b

93


with b > 1; we can further assume that b e by replacing ˇ by a segment of small hyperbolic length; finally we substitute the above b by 1 everywhere. The rest of the proofs for the three situations are the same and are as follows. Let us begin with the proof of (6.7). First note that the cross ratio distortion by h on Q is bounded below by the cross ratio distortion by h0 on Q. To estimate the amount that h0 distorts the cross ratio of Q, we approximate h0 by compositions of finitely many hyperbolic isometries. For any positive integer n, by the definition of .ˇ/, there exist finitely many hyperbolic isometries A1 ; A2 ; : : : ; Ak whose axes Li are non-intersecting lines with one endpoint on Œb; 1 and the other on Œ1; b , such that the composition h0n D A1 B A2 B B Ak coincides with h0 on the intervals .1; b and Œb; 1/, .ˇ/ n .ˇ/ D

k X

.Ai / < .ˇ/ C

iD1

1 ; n

and jcr.h0n .Q// cr.h0 .Q//j
cr.h0n .Q//

b C an 1 1 : n b an n

Let n go to infinity, e khkcr where a D

1 C1

bCa ; ba

and log D .ˇ/. Since 1 b e and 0 a < 1, e khkcr

eCa bCa : ba ea

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Jun Hu

Let v./ D

eCa .e C 1/ C .e 1/ D : ea .e 1/ C .e C 1/

Elementary analysis of v./ yields that v./ e khkcr and e khkcr < 1

v .e to v 1 ,

khkcr

/. But v

1

.e/ D

e 2 C1

2eC1e 2

eC1 e1

imply that

< 5, so by the mean value theorem applied

v 1 .e khkcr / D v 1 .1/ C .v 1 /0 .u/.e khkcr 1/ 1 C

1 v 0 .5/

.e khkcr 1/;

where 1 u v 1 .e khkcr / < v 1 .e/ < 5 since khkcr < 1. Therefore log 1 when khkcr < 1. Let C1 D

e v 0 .5/

D

1 v 0 .5/

.e khkcr 1/

.6e4/2 4

e v 0 .5/

khkcr

D .3e 2/2 . Then

.ˇ/ D log C1 khkcr for any h with khkcr 1. This proves Inequality (6.7). Also note that when the cross-ratio norm khkcr decreases to 0, the constant C1 decreases to 1. In the following, we prove Inequality (6.8) by considering the cross-ratio distortion ; 1g. Just as in the previous case, we first of h on the quadruple Q D f1; b; 1b 2 notice that the cross-ratio distortion of h on Q is less than or equal to the cross-ratio distortion of h0 on Q. By Lemma 6.5, the amount by which h0n distorts the cross-ratio of Q is the greatest if we assume all of the mass of n in ˇ is concentrated on one geodesic line joining 1 to b and the map h0n is the identity outside the interval Œ1; b

and inside this interval it is given by the Möbius transformation z 7! w D h0n .z/, where wC1 zC1 D n : wb zb Taking the limit when n goes to infinity, we see that the amount by which h0 distorts the cross-ratio of Q is the greatest if we assume all of the mass of in ˇ is concentrated on one geodesic line joining 1 to b and the map h0 is the identity outside the interval Œ1; b and inside this interval it is given by the Möbius transformation z 7! w D h0 .z/ D A.z/, where wC1 zC1 D : wb zb to A. 1b / D w1 , and 1 to In that case, h maps 1 to 1, b to b, 1b 2 2 1 1 A.1/ D w2 . If B.s/ D bC1 .s C 1/, then B B A B B has multiplier , fixed points at 0 and 1, and t B B A B B 1 .t/ D : (6.9) . 1/t C 1


95

3b 2 D B. 1b / D 2.bC1/ ; z D B.1/ D bC1 . We must 2 calculate the distortion of the map hQ that fixes 1 and b and maps y to B BABB 1 .y/ and z to B B A B B 1 .z/. From (6.9), the distortion is

Let x D B.b/ D

Q cr.h.Q// D D

1b ;y 1Cb

y z .1/yC1 .1/zC1 y x .1/yC1

2 .y

.z y/ : yx/ C ..y yx/.1 z/ C .yx x/z/ C .yx x/.1 z/

Thus, the reciprocal of this distortion is

z.y yx/ yz.2z 1/ C y x.y C z/ x.y 1/.1 z/ C C : zy zy .z y/

(6.10)

Since the last two terms in (6.10) are positive, we obtain log

1 zy.1 x/ log C log : Q zy cr.h.Q//

Q Since cr.h.Q// cr.h0 .Q// cr.h.Q// 1, Q khkcr j log cr.h.Q/j j log cr.h.Q//j D log and hence khkcr log C log

1 ; Q cr.h.Q//

zy.1 x/ : zy

By substituting in the values of x; y and z, the term log zy.1x/ is a function of b for zy 3

.eC1/ 1 b e which has a lower bound log 4e.3e/ . Therefore,

log khkcr C log

.e C 1/3 : 4e.3 e/

We have completed the proof of Inequality (6.8). Notice that by a careful study of the function of defined by Formula (6.10) when is near 1, one can also prove Inequality (6.7). The method used in the proof of Inequality (6.7) has more applications in Section 5.4. In summary, we have proved the following theorem. Theorem 6.8 ([9]). If h is a quasisymmetric homeomorphism of the circle and is the induced earthquake measure by h, then kkTh C1 khkcr and k kTh khkcr C C2 ;

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.eC1/ where C1 D maxf.3e 2/2 ; 1 C C2 g and C2 D log 4e.3e/ .

In the rest of this section, we show Inequality (6.2). Before presenting its proof, we summarize two more techniques into two lemmas. As before we use D to denote the open unit disk centered at the origin of the complex plane C and H to denote the upper-half plane. Lemma 6.9. A quadruple Q D fa; b; c; d g has cr.Q/ D 1 if and only if the geodesic ac from a to c is perpendicular to the geodesic bd from b to d , and ifpand only if the hyperbolic distance from ab to cd (or bc to da) is equal to ln.3 C 2 2/. Proof. Let A be the Möbius transformation mapping D onto H such that A.a/ D 1; A.b/ D 0 and A.c/ D 1. Since cr.A.Q// D cr.Q/ D 1, A.d / D 1. Clearly, the geodesic from 1 to 1 is perpendicular to the geodesic from 0 to 1, and then the geodesic ac is perpendicular to bd . Let ˇ denote the geodesic in D which is perpendicular to both the geodesics ab and cd . There exists a Möbius transformation B from D to H that maps ˇ to the imaginary axis, and a to 1 and b to 1. Assume D x p that B.d / D x and B.c/ jdzj with x > 1. Since cr.B.Q// D 1, x D 3 C 2 2. And therefore, taking y as the p hyperbolic metric in H, the distance from ab to cd is ln.3 C 2 2/. By a symmetry argument, the hyperbolic distance from bc to da is equal to the same value. In thispsection, we let C0 D 2, which is the smallest positive integer greater than ln.3 C 2 2/ D 1:7627471 : : : . Lemma 6.10. Consider the hyperbolic plane H. Let ln denote the geodesic connecting e n to e n for each n 2 f0g [ N, L the lamination consisting of the ln ’s, GnC1 the gap between ln and lnC1 and G0 the remaining gap. Suppose that an earthquake map E is defined as follows: EjG0 is the identity map; for each n 2 f0g [ N, the comparison map Ej1 Gn B EjGnC1 is the hyperbolic Möbius transformation with axis ln and hyperbolic translation length ln n , and Ejln D EjGnC1 . Let h denote the extension of E to the boundary of H, and Q the quadruple f1; 1; 1; 0g. If there exists 1 such that n for each n 2 f0g [ N, then there exists a constant C1 > 0, independent of , such that 0 ln cr.h.Q// C1 ln : Proof. Set A0 D EjG0 , and An D Ej1 Gn1 B EjGn for each n 2 N. Clearly h.Q/ D f1; 1; 1; h.0/g, and h.0/ D lim A1 B A2 B B An .0/: n!1

n

Let xn denote the point e and yn the point e n on the real axis for each n 2 f0g[N. Since the derivative of An decreases from n to 1n on the interval Œxn ; yn , by the

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mean value theorem, n 1 : yn An .0/ D An .yn / An .0/ yn 1 n e

Then yn1 An1 B An .0/ D An1 .yn1 / An1 B An .0/ .yn1 An .0//1 n1 .yn1 yn C yn An .0//1 .e nC1 e n /1 C e n 2 : Inductively, y0 A0 B A2 B B An .0/ .1 e 1 /1 C .e 1 e 2 /2 C C .e nC1 e n /n C e n n1 D .e 1/Œe 1 1 C e 2 2 C C e n n C e n .nC1/ : As n ! 1, y0 h.0/ Now we estimate cr.h.Q//. Since EjG0

e1 : e 1 D id ,

cr.h.Q// D cr.h.f1; 1; 1; 0g// D cr.h.f1; 1; 0; 1g//1 D Therefore 1 cr.h.Q//

e1 e1 e1 e1

2

D

h.0/ C 1 : 1 h.0/

2e e 1 : e1

Let D ln , then 0 ln cr.h.Q// ln Let . / D ln 2e

1C e1

e1

2e 1C e 1 : e1

. Clearly 0 ./ D

1

1

eC1 2e 1C

. Then 1 0 . /

2e e1

for

any 2 Œ0; C1/. By the mean value theorem again, ./ .0/

2e : e1

Since .0/ D 0, 0 ln cr.h.Q// . / C1 ln ; where C1 D

2e . e1

Let h denote an orientation-preserving circle homeomorphism, .E; L/ an earthquake representation of h, and the induced earthquake measure by .E; L/.

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Theorem 6.11 ([12]). There exists a universal constant C > 0, independent of h and , such that khkcr C k kTh : In fact, one can take C D C0 C p 2C1 C2 , where C0 is the smallest positive integer 2e greater than or equal to ln.3 C 2 2/, C1 D e1 , and C2 is the smallest positive p 2 integer greater than or equal to ln.e C e 1/. We divide the proof into several cases. Given a quadruple Q D fa; b; c; d g with cr.Q/ D 1, we first assume that three points a, b and c belong to the same stratum A, and estimate cr.h.Q// in this case. By a Möbius change of coordinates, we may assume that a D 1; b D 1; c D 1; d D 0, and by post-composing E with another Möbius transformation, we may also assume that the earthquake E is the identity map on the stratum A. We will show that there is a constant C such that 0 ln cr.h.Q// C k kTh : n

Let xn denote the point e and yn the point e n on the real axis for each n 2 f0g[N. Let L0 denote the collection of lines in L that connect points of the interval Œ1; 0/ to 0 points of .0; 1 . We denote by L 0 the collection of lines in L connecting points of C Œx0 ; x1 / to points of .0; y0 , and L0 the collection of lines in L0 connecting points of C 0 Œx0 ; 0/ to points of .y1 ; y0 . Let L0 D L 0 [L0 . Then any line in L nL0 must connect a point in Œx1 ; 0/ to a point in .0; y1 . Inductively, for each n 2 N, let L n denote the collection of lines in L0 n .L0 [ L1 [ [ Ln1 / that connect points of Œxn ; xnC1 / 0 to points of .0; yn , and LC n be the collection of lines in L n .L0 [ L1 [ [ Ln1 / C which connect points of Œxn ; 0/ to points of .ynC1 ; yn , and Ln D L n [ Ln . We have the following three lemmas. Lemma 6.12. For each n 2 f0g [ N, any line in Ln must connect a point in Œxn ; 0/ to a point in .0; yn . Proof. This can be easily proved by an induction on n. Lemma 6.13. There exists a constant C2 > 0, independent of h and , such that .Ln / C2 k kTh for any n 2 f0g [ N. Proof. For each n 2 f0g[N, let ln denote the geodesic line connecting the point xn to the point yn . For each n 2 N, let ln denote the geodesic connecting the point xn to 0, and lnC the geodesic connecting 0 to the point yn . The hyperbolic distance p from ln to C lnC1 (or lnC1 ), n 2 f0g [ N, is equal to a constant, which is equal to ln.e C e 2 1/. Note that the measure of each line in Ln may be as big as k pkT h . Let C2 be the smallest positive integer that is greater than or equal to ln.e C e 2 1/. Then .Ln / C2 kkTh for each n 2 f0g [ N.


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Let Ez be the same earthquake map as the one defined in Lemma 6.10, with ln n D .Ln /. Lemma 6.14. For any quadruple Q D fa; b; c; d g with cr.Q/ D 1, if a; b and c belong to the same stratum, then z cr.h.Q// cr.E.Q//: Proof. For each n 2 f0g [ N, let ln0 denote the geodesic line in Ln whose endpoints have the maximal distance on the real line, and let En be the earthquake that induces the earthquake measure . jLn ; Ln / and that is the identity on the stratum above the geodesic line ln0 . In the case where Ln is an empty collection, we let En be the identity map. Denote hn D E0 B E1 B B En . Clearly h.d / D h.0/ D lim hn .0/: n!1

We only need to show that for each n 2 f0g [ N, cr.hn .Q// cr.A0 B A1 B B An .Q//; where the Ai ’s are the maps defined in the proof of Lemma 6.10. Now we compare cr.An .Q// with cr.En .Q//. Let the geodesic lines li be the ones defined in Lemma 6.10. By Lemmas 6.6 and 6.12, if we move the weights of the geodesic lines in Ln to the geodesic line ln , we only increase the cross ratio of the image of Q, that is, cr.En .0// cr.An .Q//. Therefore En .0/ An .0/. Since An1 is monotone increasing on the interval Œe n ; e n , An1 .En .0// An1 .An .0//: By Lemmas 6.6 and 6.12 again, if we move the weights of the geodesic lines in Ln1 to the geodesic line ln1 , we only increase the cross ratio of the image of the quadruple f1; 1; 1; En .0/g, that is, cr.En1 .f1; 1; 1; En .0/g// cr.An1 .f1; 1; 1; En .0/g//: Hence En1 B En .0/ An1 B En .0/ An1 B An .0/: Inductively, for each 0 i n, we see that EiC1 B B En1 B En .0/ AiC1 B B An1 B An .0/: By the monotonicity of Ai on the interval Œe i ; e i , one has Ai .EiC1 B B En1 B En .0// Ai .AiC1 B B An1 B An .0//: And by Lemmas 6.6 and 6.12 and moving the weights of the geodesic lines in Li to the geodesic line li , we have cr.Ei .f1; 1; 1; EiC1 B B En1 B En .0/g// cr.Ai .f1; 1; 1; EiC1 B B En1 B En .0/g//;

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and therefore Ei B EiC1 B B En1 B En .0/ Ai B EiC1 B B En1 B En .0/ Ai B AiC1 B B An1 B An .0/: When i D 0, the above inequality implies cr.hn .Q// cr.A0 B A1 B B An .Q//: By taking the limit as n ! 1, we have z cr.h.Q// cr.E.Q//: Lemmas 6.14, 6.10 and 6.13 imply the following proposition. Proposition 6.15. Let Q D fa; b; c; d g be a quadruple of four points on the unit circle arranged in the counter-clockwise order. If cr.Q/ D 1 and a, b, c belong to the same stratum of the earthquake representation .E; L/ of h, then 0 ln cr.h.Q// C1 C2 kkTh : Proposition 6.16. Let Q D fa; b; c; d g be a quadruple of four points on the unit circle arranged in the counter-clockwise order. If cr.Q/ D 1 and a, c belong to the same stratum of the earthquake representation .E; L/ of h, then 0 ln cr.h.Q// 2C1 C2 kkTh : Proof. Recall that we may assume that a D 1, b D 1, c D 0, d D 1, and the earthquake E is the identity map on the stratum A containing the points a and c. Let E1 and E2 be the maps defined by partially changing E to the identity map on one side of A respectively. Then E D E2 B E1 . By applying the previous proposition to E1 and E2 respectively, we obtain 0 ln cr.fa; b; c; h.d /g C1 C2 kkTh and 0 ln cr.fa; h.b/; c; d g C1 C2 kkTh : Thus 1 h.d / e C1 C2 k kTh

and

1 h.b/ e C1 C2 kkTh :

Therefore 1 cr.a; h.b/; c; h.d // D

h.d / e 2C1 C2 kkTh ; h.b/

which implies 0 ln cr.h.Q// 2C1 C2 kkTh :

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Proposition 6.17. Suppose cr.Q/ D 1, and assume that there exists at least one geodesic line in the lamination L which separates the vertices a, b from the vertices c, d , then j ln.cr.Q//j .C0 C 2C1 C2 /k kTh :

a

d

LII LV

LI

LIII

b

LIV c

Figure 3. Five subcollections of L used in the proof of Proposition 6.17.

Proof. Given two points x and y on the unit circle, we use Œx; y to denote the arc on S1 from x to y in the counter-clockwise order. Let LI denote the collection of geodesic lines in L that connect points of the arc Œd; a to points of the arc Œb; c . By Lemma 6.9, .LI / C0 k kTh . Let LII denote the collection of lines in L that connect points of the arc .d; a/ to points of the arc .a; b/, LIII the collection of lines in L that connect points of the arc .a; b/ to points of the arc .b; c/, LIV the collection of lines in L that connect points of the arc .b; c/ to points of the arc .c; d /, and finally LV the collection of lines in L that connect points of the arc .c; d / to points of the arc .d; a/. First notice that the motion of a (resp. c) under the earthquake map E along the lines in LII (resp. LIV ) only decreases the cross ratio of Q. Therefore the cross ratio cr.h.Q// is less than or equal to the cross ratio cr.E 0 .Q//, where E 0 is the new earthquake obtained by omitting the earthquake motion along the geodesic lines in L n .LI [ LIII [ LV /. By a Möbius change of coordinates, we may assume a D 1, b D 1, c D 0, d D 1. Then the geodesic lines in LI connect the points of Œ1; 0 to the points of Œ1; C1 . By Lemma 6.7, if one moves all the lines in LI to the geodesic line

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from 0 to 1 without changing the total amount of shearing along the lines in LI to obtain a new earthquake E 00 , then the cross ratio of the image quadruple of Q under the earthquake is possibly increased, that is, cr.E 0 .Q// cr.E 00 .Q//. As usual, by post-composing with a Möbius transformation, we may assume that E 00 is the identity map on the geodesic line from 0 to 1. Let EIII (resp. EV ) denote the earthquake obtained by omitting the earthquake motion along all lines in L n LIII (resp. L n LV ). Then by Proposition 6.15 and the same argument as in the proof of Proposition 6.16, we have 1 EV .d / e C1 C2 k kTh

and

1 EIII .b/ e C1 C2 kkTh :

Then 1 E 00 .d / e .C1 C2 CC0 /k kTh

and

1 E 00 .b/ e C1 C2 kkTh :

Therefore cr.E.Q// cr.E 0 .Q// cr.E 00 .Q// e .2C1 C2 CC0 /kkTh : Now we omit the motion of E along the geodesic lines in LIII and LV , and move all geodesic lines in LI to the geodesic line bd . By a similar reasoning, we obtain cr.E.Q// e .2C1 C2 CC0 /kkTh : This completes the proof of the proposition. Now the proof of Theorem 6.11 can be organized as follows. Proof. Let C D C0 C 2C1 C2 . We show that for any quadruple Q with cr.Q/ D 1, j ln cr.h.Q//j C kkTh : We divide the proof into three cases. Case 1: The quadruple Q has three vertices belonging to the same stratum. Then either a, b, c or b, c, d or c, d , a or d , a, b belong to the same stratum. By Proposition 6.15, either 0 ln cr.h.Q// C1 C2 kkTh or 0 ln cr.h.fb; c; d; ag// C1 C2 kkTh : Clearly cr.h.fb; c; d; ag// D

1 , cr.h.Q//

and hence

j ln cr.h.Q//j C1 C2 kkTh < C kkTh : Case 2: The quadruple Q has two opposite vertices belonging to the same stratum. Then either a and c or b and d belong to the same stratum. By Proposition 6.16 and the same reasoning as in Case 1, we have j ln cr.h.Q//j 2C1 C2 kkTh < C kkTh :


103

Case 3: The quadruple Q has no opposite vertices belonging to the same stratum. Then either there exists a geodesic line in L which separates a and b from c and d or there exists a geodesic line in L which separates b and c from d and a. By Proposition 6.17 and the same reasoning as in Case 1, we have j ln cr.h.Q//j .C0 C 2C1 C2 /k kTh D C kkTh : This completes the proof. Our Theorems 6.8 and 6.11 imply Theorem 6.2. Remark 6.18. From Theorems 3.2, 5.2, 5.5 and 6.2, we conclude that associating each quasisymmetric circle homeomorphism with its corresponding earthquake measure defines a bijection † from the universal Teichmüller space T to the space M of Thurston bounded earthquake measures. Our Theorem 6.2 implies that with respect to a base point in T , † and †1 are continuous at the base point and the trivial point with respect to the Teichmüller metric on T and the Thurston norm on M. It is an interesting problem to introduce a meaningful topology on M such that † improves to be a homeomorphism between T and M with respect to the Teichmüller metric ˘ c recently provide such a topology on T . Although quite implicit, Miyachi and Sari´ on M (see [26]). It is considerably difficult to make such a topology transparent or geometric. The reason is that for earthquake measures, there is nothing like a chain rule for the Beltrami coefficient of the composition of two quasiconformal maps. A more subtle and very important open problem is to study if there is a rule for the earthquake measure of the composition of two circle homeomorphisms that can be viewed as a replacement of the chain rule for the Beltrami coefficient.

7 Earthquake curves Let .; L/ be a Thurston bounded earthquake measure. For each t 0, there exists an earthquake map E t inducing t . Take a stratum A and normalize E t by requiring E t to be the identity map on A. Then the restriction h t of E t on the boundary circle gives a continuous curve of homeomorphisms of the unit circle S1 , which is called an earthquake curve determined by or t . The curve h t .x/; t 0, is differentiable in t for each point x on the boundary circle and satisfies a non-autonomous ordinary differential equation ([9]). The holomorphic dependence of h t .x/ on t for each fixed x is developed in [27] or [28] by pairing earthquake for real parameter with bending for purely imaginary parameter. Applications of earthquake curves to Teichmüller theory and to the deformation theory of the unit disk can be found in [25], [30] and [5]. In this chapter we focus on the real theory of earthquakes, and in this section we introduce the proof of the following theorem in [9]. Before giving the statement of the theorem, we recall that each earthquake measure .; L/ is a Borel measure on X supported on the set of pairs consisting of endpoints of geodesics in L, where X is the space

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S1 S1 n fthe diagonalg factorized by the equivalence relation .a; b/ .b; a/. Then the double integrals involved in the statement of the following theorem are integrations with respect to on the space X. More explicitly, given any two disjoint arcs I and J on S1 , .I J / is the measure of the geodesics in L connecting points of I to points of J . For example, if a and b are two distinct points on S1 , then .a; b/ is equal to the weight of the geodesic connecting a and b if it belongs to L and is equal to 0 if not. Theorem 7.1 ([9]). Let .; L/ be a Thurston bounded earthquake measure and h t , t 0, be an earthquake curve determined by . Then for each x 2 S1 , h t .x/ is differentiable in t for all t 0. Moreover, let us take the upper-half plane model of the hyperbolic plane and assume that the upper-half imaginary axis is contained in a stratum determined by L, then d h t .x/ D V t .h t .x//; dt where V t is the vector field given by “ V t .x/ D Eh t .a/h t .b/ .x/d.a; b/ C a quadratic polynomial; where Eab .x/ D ular,

.xa/.xb/ ab

V0 .x/ D

(7.1)

(7.2)

if x 2 Œa; b , and otherwise Eab .x/ D 0; and in partic-

“ Eab .x/d.a; b/ C a quadratic polynomial.

(7.3)

Furthermore, if the upper-half imaginary axis happens to be contained in the stratum which normalizes the earthquake curve h t , then “ V t .x/ D Eh t .a/h t .b/ .x/d .a; b/I (7.4) and in particular,

“ V0 .x/ D

Eab .x/d .a; b/:

(7.5)

An earthquake curve h t , t 0, is a solution of the non-autonomous ordinary differential equation (7.1), and it is a one-parameter Lie group if we consider h t , t 0, as the boundary maps of the right earthquakes determined by t with the same normalization. Specifically, if t and s are nonnegative, we have the following equation on h tCs . Without loss of generality, we assume that there is at least one leaf in the lamination L. Let L t denote the target lamination of E t (which is the image of L under E t ), and let t D ht ./ be the push-forward of the measure .; L/ under h t . Then the support of t is L t . Now let hQ s , s 0, be the curve of the boundary maps of the earthquake maps Ezs determined by .s t ; L t / which are normalized to be


105

the identity map on E t .A/. Then for any point x 2 S1 , h tCs .x/ D hQ s .h t .x//:

(7.6)

Hence the tangent vector V t of h t at time t is the tangent vector Vzs of hQ s at s D 0, and then by (7.3), “ Eab .x/d t .a; b/ C a quadratic polynomial. (7.7) V t .x/ D This is how we obtain V t in (7.2) in [9]. Note that in the special case where the support L of an earthquake measure consists of only finitely many geodesics, one can quickly derive the properties in the previous theorem by using the chain rule in computing derivatives. For a Thurston bounded earthquake measure , the proof of Theorem 7.1 in [9] is divided into two steps. In the first step, we show that an earthquake curve corresponding to any locally finite earthquake measure .; L/ (not necessarily Thurston bounded) satisfies the ordinary differential equation (7.1) at any point x on the boundary of a stratum. Notice that the union of the boundary points of all strata does not necessarily cover the whole circle, and the earthquake map E t , determined by .t ; L/, is not necessarily extendable continuously. In the second step, we show that if kkTh < 1, then the ordinary differential equation extends to any point on the boundary circle and the earthquake curve is the unique solution to the ordinary differential equation (7.1). Step I. Let be a locally finite earthquake measure supported on a lamination L, and l0 be a line in L. Let E t D E t be the earthquake map determined by .t ; L/ with the normalization of fixing the line l0 for t 0. Let S denote the union of the boundaries of all strata on the circle. Then E t extends to a continuous injective map h t D h t from S into the circle. In the following we show that for each x 2 S , h t .x/ is differentiable in t and satisfies the ordinary differential equation (7.1). We first show that h t .x/ is differentiable at t D 0. Lemma 7.2. For each x 2 S, lim

t!0C

where Eab .x/ D

h t .x/ x D t

.xa/.xb/ ab

“ Eab .x/d .a; b/;

if x 2 Œa; b , and otherwise Eab .x/ D 0.

Proof. Let T denote the stratum whose boundary on the circle contains the point x, l the geodesic boundary line of T which is the closest one to l0 in the hyperbolic metric, and r the geodesic arc perpendicular to both l0 and l. (In the case where l0 and l1 share one endpoint, we may choose any geodesic arc which is transversal to both of them.) Denote by L0 the set of all leaves of L in the strip bounded by l0 and l that intersect the geodesic arc r. Let 0 denote the restriction of to L0 and E t0 D E t 0 be the earthquake map determined by .L0 ; t 0 / normalized to fix the line l0 . Then E t0 extends to a homeomorphism h0t of the circle S1 . Clearly h t .x/ D h0t .x/ for any

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t 0. Let V .x/ D

’

Eab .x/d.a; b/, then “ V .x/ D Eab .x/d 0 .a; b/:

Let P D fzi gniD0 be a partition of the transversal arc r, and d be the maximum of the hyperbolic lengths of the small arcs in the partition. Assume that z0 belongs to l0 and zn belongs to l. Denote by Ti the stratum that contains zi , 0 i n. Let ri denote the arc .zi1 ; zi on r, and li a line in L0 intersecting ri , 1 i n. Let 10 D 0 .Œz0 ; z1 /, and i0 D 0 .ri /; 2 i n. Let Ln denote the collection of li ; 1 i n, n the measure supported on Ln with the weight i0 on li , 1 i n. Let F tn D F1t B F2t B B Fnt denote the curve of the finite earthquake maps that fix the line l0 and induce earthquake measures (Ln , t n ), where t 0. Let Eit D E t0 jTi . Then h t .x/ D h0t .x/ D E t0 .x/ D E1t B E2t B B Ent .x/: Clearly each Fit is a hyperbolic Möbius transformation with axis li and translation length i0 . Each Eit is also a hyperbolic Möbius transformation with axis intersecting the arc Œzi 1 ; zi , and by Proposition 4.4 its translation length differs from i0 by an amount O.i0 l.ri //, where l.ri // denotes the hyperbolic length of the arc ri . Sublemma. jh0t .x/ F tn .x/j D O.td.n/ 0 .r/e t mum of the hyperbolic lengths of the ri ’s.

0 .r/

/; where d.n/ denote the maxi-

Let us first use this sublemma to prove the differentiability of h t .x/ at t D 0. Define “ Vn .x/ D

Eab .x/dn .a; b/:

One can easily show that Vn .x/ approaches V .x/ as n approaches infinity. Now we write h t .x/ x h0 .x/ x V .x/ D t V .x/ t t F n .x/ x E 0 .x/ F tn .x/ CŒ t Vn .x/ C ŒVn .x/ V .x/ : D t t t Let denote an arbitrary small positive number. Because of the sublemma, for evE 0 .x/F n .x/ j < 3 . It is easy to see that ery > 0, if n is sufficiently large, j t t t jVn .x/ V .x/j < 3 when n is sufficiently large. Because of the differentiability of the earthquake curves induced by atomic earthquake measures, for a fixed large F n .x/x value of n, if t is small enough then j t t Vn .x/j < 3 . Therefore there exists ı > 0 such that for any 0 t < ı, j This means that

d h .x/j tD0 dt t

h t .x/ x V .x/j < : t

D V .x/.

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t t Now we prove the previous sublemma. Let xit D EniC1 B EniC2 B B Ent .x/ t t B FniC2 B B Fnt .x/, 1 i n. By Proposition 4.4, x1t D and yit D FniC1 t t t En .x/ differs from y1 D Fn .x/ by at most O.l.rn /t 0 .rn // D O.d.n/t n0 /. By Proposition 4.4 again, 0 t t x2t D En1 .x1t / C O.d.n/t n1 /: .x1t / D Fn1

By the mean value theorem, 0

t t jFn1 .x1t / Fn1 .y1t /j e t n1 jx1t y1t /j;

and therefore 0

0 x2t D y2t C O.d.n/tn1 / C O.d.n/e t n1 t n0 /:

Inductively using Proposition 4.4 and the mean value theorem, we obtain 0

0

0

xnt D ynt C O.d.n/t10 / C O.d.n/e t1 t 20 / C O.d.n/e t 1 Ct 2 t 30 / C 0

0

0

C O.d.n/e t1 Ct2 CCtn1 t n0 / D ynt C O.d.n/e t

0 .r/

t 0 .r//;

which proves the sublemma. Suppose that t and s are positive. Let h t denote the curve of earthquake maps that fix l0 , have source lamination L and measure t . Let L t denote the target lamination of h t and ht ./ be the push-forward of the measure .; L/ under h t , whose support is L t . Let hQ s be the curve of earthquake maps that fix the line l0 and have lamination L t and measure sht ./, s 0. Lemma 7.3. For each x 2 S , h tCs .x/ D hQ s .h t .x//: Proof. We use the notation in the proof of the previous lemma. Let Ezs0 , s 0, be the earthquake map determined by .E t0 .L0 /; s.h0t / . 0 // normalized to fix the line l0 D h0t .l0 /, and hQ0 s be the extension of Ezs0 to the boundary circle. We only need to show h0t Cs .x/ D hQ0 s .h0t .x//: Let hnt denote the extension of the finite earthquake map F tn to the boundary circle. The method used to show the sublemma in the proof of Lemma 7.2 implies that hnt n converges to h0t on the boundary circle. For each t 0 and s 0, F tCs B.F tn /1 is also a left earthquake with source lamination F tn .Ln / and measure s.hnt / .n /. Let Ezsn denote the finite earthquake curve determined by .F tn .Ln /; s.hnt / .n // normalized to fix l0 , and hQ ns the extension of Ezsn to S1 . Then for each y 2 S1 , hntCs .y/ D hQ ns .hnt .y//: In particular,

hntCs .x/ D hQ ns .hnt .x//:

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As n approaches infinity, hntCs .x/ approaches h0t Cs .x/. In fact, for any t 0, hnt converges to h0t uniformly on S1 . It remains to show that the right-hand side of the above equation converges to hQ0 s .h0t .x//. We first show that hQ ns converges to hQ0 s in the C 0 topology. Since hnt converges to h0t uniformly on S1 , the images of the leaves in Ln under F tn are uniformly close to their images under E t0 . Now we write the map hQ0 s as a long composition of the comparison maps with respect to the finitely many lines in the lamination E t0 .Ln /. Because we require Ezs0 to fix the line l0 , the comparison maps are hyperbolic Möbius transformations. And of course the finite earthquake map Ezsn is also a long composition of hyperbolic Möbius transformations. Consider respectively the Möbius transformations in these two long compositions. They have nearby axes and nearly equal translation lengths. Again using the method to show the sublemma in the proof of the previous lemma, we conclude that maxy2S1 jhQ ns .y/ hQ 0s .y/j converges to zero as n approaches infinity. Now we write hQ ns .hnt .x// hQ0 s .h0t .x// D ŒhQ n .hn .x// hQ0 s .hn .x// C ŒhQ0 s .hn .x// hQ0 s .h0 .x// ; s

t

t

t

t

and deduce the convergence of hQ ns .hnt .x// to hQ0 s .h0t .x//. Theorem 7.4. For each x 2 S, h t .x/ is differentiable for any t 0, and d h t .x/ D V t .h t .x//; dt ’ where V t .y/ D Eh t .a/h t .b/ .y/d.a; b/. Proof. By definition, h t Cs .x/ h t .x/ d h t .x/ D lim : s!0 dt s When s > 0, by Lemma 7.3, hQ s .h t .x// h t .x/ hQ s .y/ y h tCs .x/ h t .x/ D D ; s s s where y D h t .x/. By Lemma 7.2, lim

s!0C

’

hQ s .y/ y D Vz .y/; s

where Vz .y/ D Eab .y/dht .a; b/. And clearly “ “ Eh t .a/h t .b/ .y/d .a; b/ D V t .y/: Vz .y/ D Eab .y/dht .a; b/ D Therefore lim

s!0C

h tCs .x/ h t .x/ D V t .h t .x//: s


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t .x/ Now we consider the limit lims!0C h t s .x/h . Following arguments similar s to the arguments in Lemma 7.3, one can show that h t s .x/ D hN s .h t .x//, where hN s , s 0, is the right earthquake map determined by .h t .L/; sht . // normalized to fix l0 . Just as in Lemma 7.2, one can show that “ “ hN s .y/ y D Eba .y/dh t .b; a/ D Eab .y/dht .a; b/ D V t .y/: lim s s!0C

Therefore lim

s!0C

h ts .x/ h t .x/ D V t .h t .x//: s

This completes the proof. Step II. If the measure has bounded Thurston norm then each earthquake map E t , t 0, extends to a homeomorphism of the boundary circle. In the following we will first show that for a Thurston bounded measure , h t .x/ is differentiable in t 0 for any point x on the circle. We will also prove that under that condition, the normalized earthquake curve is the unique solution to the ordinary differential equation (7.1) with the initial condition that h0 is the identity map. Given a lamination L, a point x on the boundary circle is called an accumulation point with respect to L if there exists an infinite sequence fln g1 nD0 of distinct, nonintersecting lines in L such that both endpoints of ln converge to x in the Euclidean metric. Let S a denote the set of all accumulation points with respect to L. Let h t .x/ be the curve of earthquake maps determined by .L; t / normalized to fix l0 in L. In order to conclude that the normalized curve h t .x/ is differentiable on t 0 for any x on the circle, by Theorem 7.4 we only need to prove that h t .x/ is differentiable on t for any accumulation point x of L. Lemma 7.5. For any x 2 S c , h t .x/ x D lim C t t!0 where Eab .x/ D

.xa/.xb/ ab

“ Eab .x/d .a; b/;

if x 2 Œa; b , and otherwise Eab .x/ D 0.

Proof. Let l0? denote the geodesic that passes through x and is perpendicular to l0 . Conjugating by a Möbius transformation, we may assume that l0? is the imaginary axis of the upper-half plane, x is at the origin, and l0 is the geodesic connecting 1 to 1. Let L0 denote the collection of those lines in L that connect points in Œ1; 0/ to points in .0; 1 , and 0 the restriction of to L0 . Let an D e n and bn D e n , where n 0. The following classification of the lines in L0 is the same one given before Lemma 6.12, which is a key step to derive Proposition 6.15. We denote by 0 L 0 the collection of lines in L connecting points of Œa0 ; a1 / to points of .0; b0 , and C L0 the collection of lines in L0 connecting points of Œa0 ; 0/ to points of .b1 ; b0 . Let C 0 L0 D L 0 [ L0 . Then any line in L n L0 must connect a point in Œa1 ; 0/ to a

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point in .0; b1 . Inductively, for each n 2 N, let L n denote the collection of lines in L0 n .L0 [ L1 [ [ Ln1 / that connect points of Œan ; anC1 / to points of .0; bn , and 0 LC n be the collection of lines in L n .L0 [ L1 [ [ Ln1 / which connect points C of Œan ; 0/ to points of .bnC1 ; bn , and Ln D L n [ Ln . It is clear that for each n 2 f0g [ N, any line in Ln must connect a point in Œan ; 0/ to a point in .0; bn . By Lemma 6.13, there exists a constant C > 0 such that .Ln / C k kTh for any n 2 f0g [ N, p where C denotes the smallest positive integer that is greater than or equal to ln.e C e 2 1/. If some Ln is empty, then we add a non-intersecting line with zero weight to the lamination L0 such that Ln is no longer empty. Now for each n 2 f0g [ N, let an0 be the supremum of the left endpoints of lines in Ln and bn0 the infimum of the right endpoints of lines in L. Then jan0 bn0 j jan1 bn1 j D 2e .n1/ : For each n 2 f0g [ N, let .L.n/ ; .n/ / denote the restriction of .L0 ; 0 / on Œa0 ; an0 / .bn0 ; b0 , and E t.n/ be the curve of the earthquake maps determined by .L.n/ ; t .n/ / with the normalization of being the identity on l0 . Sublemma 1. If t is small enough such that C t k kTh < 1, then max jE t.nC1/ .y/ E t.n/ .y/j D O.t kke n.1C t kkTh / /:

y2S1

In the remaining part, we briefly denote k kTh by k k. Proof. By Lemma 6.13, the measure 0 0 0 0 0 .Œan1 ; an0 / .bn1 ; bn0 / D .Œan1 ; an0 / .bn1 ; bn0 /

is bounded above by C k k for each n 2 N. Clearly y an0 .E t.n/ /1 B E t.nC1/ .y/ an0 e C t kk .y an0 / for each y 2 Œan0 ; bn0 . Note that for each y 2 Œan0 ; bn0 , y an0 0. By subtracting y an0 from the previous double inequality, we obtain 0 .E t.n/ /1 B E t.nC1/ .y/ y .e C tk k 1/jy an0 j .e C t kk 1/jbn0 an0 j: This implies that max jE t.nC1/ .y/ E t.n/ .y/j .e C t kk /n .e C t kk 1/2e .n1/ :

y2S1

Thus max jE t.nC1/ .y/ E t.n/ .y/j D O.t k ke n.1C t kk/ /:

y2S1

111


From this sublemma we see that if t < C k1kTh , then E t.n/ converges uniformly to E t0 on the circle. In the remaining part of the proof, we assume x D 0. Then E t .x/ E t.n/ .x/ D E t0 .x/ E t.n/ .x/ D O.t kke n.1C t kk/ /: ’ ’ If Vn .x/ D Eab .x/d .n/ .a; b/, and V .x/ D Eab .x/d .a; b/ then clearly “ V .x/ D Eab .x/d 0 .a; b/: Sublemma 2. limn!1 Vn .x/ D V .x/. Proof. Since

“ V .x/ Vn .x/ D

D O.

1 X

0 ;0/.0;b 0 Œan n

Eab .x/d 0 .a; b/

jak0 bk0 jk 0 k/ D O.e .n1/ kk/;

kDn

Vn .x/ converges to V .x/ as n goes to infinity. We now write

E 0 .x/ E t.n/ .x/ E .n/ .x/ x h t .x/ x V .x/ D t C t Vn .x/ CŒVn .x/ V .x/ : t t t Given > 0, because of the above sublemmas, there exists an n big enough such E 0 .x/E

.n/

.x/

t j < 3 and jVn .x/ V .x/j < 3 . Since x is a boundary point with that j t t respect to the lamination L.n/ , by Theorem 7.4, for large enough n, there exists ı > 0 such that for any 0 t < ı, ˇ ˇ ˇ ˇ E .n/ .x/ x ˇ ˇ t V .x/ ˇ< : ˇ n ˇ 3 ˇ t

Therefore lim

t!0C

h t .x/ x D V .x/: t

We need the following lemma, which is analogous to Lemma 7.3. Lemma 7.6. For each x 2 S c and t; s 0, h tCs .x/ D hQ s .h t .x//: Proof. We follow the notation of Lemma 7.5. It suffices to show that h0t Cs .x/ D hQ0 s .h0t .x//;

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where h0t , t 0, denotes the boundary homeomorphism of the normalized earthquake map (by fixing l0 ) determined by .L0 ; t 0 / and hQ0 s , s 0, denotes the boundary homeomorphism of the normalized earthquake map (by fixing l0 ) determined by .E t0 .L0 /; s.h0t / . 0 //. Let h.n/ denote the extension of E t.n/ to the boundary circle, t .n/ and Ezsn be the curve of earthquake maps determined by .E t.n/ .L.n/ /; s.h.n/ // t / . .n/ n Q z with the normalization of fixing l0 , and hs the extension of Es to the boundary circle. Since x is a boundary point of some stratum of L.n/ , by Lemma 7.3, Q .n/ .n/ h.n/ t Cs .x/ D hs .h t .x//: We need to show the equality continues to hold after taking limits. First, we know 0 that h.n/ t .x/ converges to h t .x/ as n goes to infinity. Secondly, by the same argument used in Sublemma 1 in the proof of Lemma 7.5, if s 0 is small enough, then hQ .n/ s converges to hQ0 s uniformly on the boundary circle, where the size of s depends on the Thurston norm of .h0t / . 0 /, and hence depends on t and . Now we rewrite Q .n/ .n/ Q0 .n/ Q0 .n/ h.n/ t Cs .x/ D Œhs .h t .x// h s .h t .x// C h s .h t .x//: Finally, by taking the limits, we complete the proof. Theorem 7.4 and Lemmas 7.5 and 7.6 imply the following theorem. Theorem 7.7. Let .; L/ be a Thurston bounded measure and h t .x/, t 0, be the curve of the earthquake maps determined by .t ; L/ normalized to fix l0 in L. Then d h t .x/ D V t .h t .x//; dt ’ where V t .y/ D Eh t .a/h t .b/ .y/d.a; b/. In the remainder of this section, we show that the solution to the ordinary differential equation (7.1) is uniquely determined by the initial vector V0 given by (7.3). Theorem 7.8. Let kkTh < 1. Then any earthquake curve h t is the unique solution to the ordinary differential equation (7.1) with initial vector V0 given by (7.3). Proof. Note that the vector field V t is a Zygmund bounded function in x (see Definition 8.1) and hence V t .x/ has an j log j-modulus of continuity in x that is uniform for 0 t t0 . Let !.s/ D s log.1=s/ for 0 < s < 1=2. Then Z 1=2 ds D 1; !.s/ 0 which means that the modulus of continuity of V t .x/ in x satisfies the so-called Osgood condition for the uniqueness of solution to an ordinary differential equation. Instead of applying Osgood’s Theorem to conclude in Theorem 7.8, we recapitulate the proof.


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Let us use the upper-half plane as the hyperbolic plane. Let x.t / D h t .x/ be a normalized solution to (7.1). Then x.0/ is the identity map on the real line. Suppose y.t/ is another solution with y.0/ D x.0/. Then xP D W .t; x/

and

yP D W .t; y/;

where W .t; z/ D V t .z/: Thus the difference z.t / D y.t / x.t / satisfies zP D W .t; y/ W .t; x/

and

jzj P C !.jzj/;

where !./ is the modulus of continuity. If jz.t /j is not equal to 0 constantly, then it stays positive or negative on an interval. The proof for these two cases are similar. We provide the proof for one case in which we assume that there is a value t0 where z0 D z.t0 / D 0 and another value t1 > t0 where z1 D z.t1 / > 0 and z.t / > 0 for t0 < t < t1 . Then Z t1 Z z1 dz z.t P / D dt; z0 !.z/ t0 !.z.t // and

Z t1 zP .t / dt C dt D C.t1 t0 /: t0 !.z.t // t0 R z dz D 1. This completes the proof. But this is a contradiction because z01 !.z/ Z

t1

Remark 7.9. In fact, if is a Thurston bounded earthquake measure then the dependence of an earthquake curve h t on t 0 is real analytic ([24]) and is complex analytic ([27] or [28]).

8 Infinitesimal Earthquake Theorem Let be a Thurston bounded earthquake measure and h t , t 0, be an earthquake curve determined by t . From the previous section, we know that h t satisfies the ordinary differential equation (7.1) and the initial derivative of h t at t D 0 is given by the formula (7.3); that is, “ V0 .x/ D Eab .x/d.a; b/ C a quadratic polynomial. Definition 8.1. A continuous function V W S1 ! C is said to be Zygmund bounded if jV .e 2 i.Ct/ / C V .e 2 i.t / / 2V .e 2 i /j M jt j for a constant M > 0, where and t are real and jt j
0 such that for any Thurston bounded earthquake measure 2 M, if V is the initial tangent vector to an earthquake curve h t , t 0, determined by t , then 1 kkTh kV kcr C kkTh : C Proof. Let .; L/ be a measure in M with finite Thurston norm, and h t , t 0, be an earthquake curve determined by t . Let C be the universal constant in Theorem 6.2. Then 1 kt kTh kh t kcr C kt kTh : C Hence for any t > 0, 1 1 kkTh kh t kcr C kkTh : C t By Theorem 7.1, the earthquake curve h t is differentiable from the right at t D 0. Let V denote the right derivative of h t at t D 0. Then lim

t!0C

1 kh t kcr D kV kcr : t

Therefore 1 kkTh kV kcr C kkTh : C


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Let M0 denote the collection of all Thurston bounded earthquake measures defined on X. Then the initial tangent vector given by (7.3) defines a functional operator from the space M0 into the space of Zygmund bounded continuous vector fields on S1 . The Zygmund boundedness of V0 was first proved by Gardiner in [6] and then V t was proved to be Zygmund bounded ([9]) by showing that the pushforward ht . / of under h t is Thurston bounded. Without applying the differentiability of earthquake curves, a direct proof of Theorem 8.3 is constructed in [15] by using strategies similar to the proof of Theorem 6.2. An interesting problem arises: Can every Zygmund bounded continuous vector field on S1 be expressed by the integration (7.3) over a Thurston bounded earthquake measure? The answer is the so-called infinitesimal earthquake theorem developed by Gardiner in [6]. Theorem 8.4 ([6]). For any Zygmund bounded continuous vector field V on S1 , there exists a Thurston bounded earthquake measure such that “ (8.5) V .x/ D Eab .x/d.a; b/ modulo a quadratic polynomial; furthermore, if two vector fields V differ by a quadratic polynomial then the corresponding measures are the same. The strategy in [6] to prove the previous theorem is to develop first a finite version of that theorem and then take the limit of a sequence of finite approximations. We refer the readers to [6] for a full proof. In the following we present the essence of the proof by recapitulating the construction of an example given in [8], which follows the procedure in [6] to work out the integral expression in (8.5) for a vector field V defined on a finite subset A of S1 . For convenience, we work with the real line R and the upper-half plane H instead of the circle S1 and the unit open disk D. Assume that A is a subset of R and V takes values only at the points in A. This procedure yields a finite lamination L consisting of non-intersecting hyperbolic geodesics l in the upper-half plane H together with nonnegative weights l associated to each geodesic l in L. To describe this procedure we refer to the example illustrated in Figure 4, which treats a case where A [ f1g consists of 9 points. We will need the formulae for the parabolic bump function Eab .x/ given by ´ .xa/.xb/ for a x b, ab Eab .x/ D 0 otherwise, and the special cases ´ Ea1 .x/ D

xa 0

for a x 1, otherwise

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and

´ E1b .x/ D

.x b/ 0

for 1 x b, otherwise.

l1

l3 l2 x1

x2

x3 x4

x5

x6

x7

x8

R

Figure 4

The procedure consists of the following three steps. Step 1. Draw a line l1 which meets the graph of V over A at the point farthest to the left, which meets the graph of V in at least one more point p and which lies on or below the graph. In Figure 4, this is the line l1 and p D x4 . Step 2. If p is the farthest right point on the graph of V on A, then stop. Otherwise draw a line l2 which meets the graph of V at the point .p; V .p//, meets the graph of V at at least one more point q and which lies on or below the graph of V . Replace p by q, continue this step inductively until p is the farthest right point on the graph of V on A. Then we obtain a finite sequence of line segments l1 ; l2 ; : : : ; lk . In Figure 4 there are three such line segments, l1 , l2 and l3 . Step 3. Let R.x/ be the piecewise linear function whose graph consists of these line segments. Over each linear piece of the graph of R.x/ add a function ab Eab .x/ where a and b are the left and right endpoints of the linear piece, and ab 0 and is as large as possible so that the graph of R.x/ C ab Eab .x/ lies on or below the graph of V . Continue inductively to add parabolic bump functions aj bj Eaj bj .x/ in the above way until the graph of the resulting function passes through all points on the graph of V . Now we obtain a function Vz .x/ of the form X aj bj Eaj bj .x/ C B.x/ (8.6) Vz .x/ D aj bj


117

where the .aj ; bj /’s are pairs of points in A [ f1g, the hyperbolic geodesic lj with endpoints aj and bj do not intersect, and B.x/ is affine. Moreover, it satisfies (i) Vz .xj / D V .xj / for each xj in A; (ii) the graph of each parabolic or linear segment of Vz , if extended, lies entirely on or below the graph of V . In the example depicted in Figure 4, the weight assigned to the line connecting x4 to 1 is equal to the slope of l2 minus the slope of l1 and the weight assigned to the line connecting x7 to 1 is equal to the slope of l3 minus the slope of l2 . All weights are given by the numbers aj bj in the sum (8.6). For example, the weight assigned to the geodesic connecting x4 to x7 is the coefficient of Ex4 x7 in the sum (8.6). Figure 5 is the translation of Figure 4 from the upper-half plane to the unit disk. 1 x1 x8

x2 l1 x3

l3 l2

x4

x7 x5

x6

Figure 5. Translation of Figure 4 to the unit disk.

Remark 8.5. There is a necessary and sufficient condition on the Fourier series approximations of a continuous function V for V to be Zygmund bounded. An analogue of this result in infinitesimal earthquake theory is given in [8].

9 Smooth circle homeomorphisms and earthquake measures vanishing near infinity Let be a Thurston bounded earthquake measure, h be the boundary homeomorphism of an earthquake map E inducing , and V be the initial tangent vector of an earthquake curve h t ; t 0, determined by t . Techniques have been developed in [9] and

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[12] to derive one-to-one correspondences among vanishing conditions on , smooth conditions on h and vanishing conditions on V . In this section, we give a brief summary of some results in [9] and [12]. For some open questions in this direction, see [13]. Now we consider the open unit disk D centered at the origin in the complex plane C as the hyperbolic plane. Definition 9.1. Let D be a disk in D, and let ı.D/ denote the Euclidean distance from D to the boundary S1 of D and mass .D/ the total amount of shearing along the lines of the lamination of that intersect D. Given any ˛ 0, we say that an earthquake measure is vanishing (resp. strongly vanishing) to order ˛ if mass .D/ O.ı.D/˛ /

.resp. mass .D/ o.ı.D/˛ //

for all disks D of hyperbolic diameter 1. Definition 9.2. Assume that four points a, b, c, d on the unit circle S1 are labeled in counter-clockwise order. Define the minimum scale of a quadruple Q D fa; b; c; d g to be smin .Q/ D minfja bj; jb cj; jc d j; jd ajg: An orientation-preserving homeomorphism h of the unit circle S1 is defined to be smooth (resp. strongly smooth) of order ˛ for some ˛ 0 if j log cr.h.Q//j D O.smin .Q/˛ / .resp: j log cr.h.Q//j D o.smin .Q/˛ // for all quadruples Q with cr.Q/ D 1. Note that h is smooth of order 0 means h is quasisymmetric; and h is strongly smooth of order 0 means h is symmetric. A 4-tuple of points a < b < c < d on the real line is called a standard quadruple if b a D c b D d c, and is denoted by Q0 . In [33], Sullivan classified the smoothness of one-dimensional homeomorphisms h according to the comparison between the cross-ratio distortion of h on Q0 and the scale of Q0 . (For applications of this type of classification in dynamical systems, see [33] and [22].) Definition 9.3. We say that a homeomorphism h W R ! R is Sullivan smooth (resp. strongly Sullivan smooth) of order ˛ for ˛ 0 if the cross-ratio distortion of h on any standard quadruple Q0 is O.s.Q0 /˛ / (resp. o.s.Q0 /˛ /). Let S1 D fe i W 0 < 2g and U1 D S1 n f1g, U2 D S1 n fi g, U3 D S1 n f1g and U4 D S1 n fi g. For each 1 n 4, let xn D S1 n Un and define z C xn ˆn W Un ! R W z 7! : i.z xn / Consider f.Un ; ˆn /g4nD1 as a system of coordinate charts for the unit circle S1 . Then a circle homeomorphism h is Sullivan smooth (resp. strongly Sullivan smooth) of order


119

˛ if h is Sullivan smooth (resp. strongly Sullivan smooth) of order ˛ in each chart .Un ; ˆn /. Definition 9.4. A continuous function V W S1 ! C is said to be vanishing (resp. strongly vanishing) to order ˛ if jV ŒQ j D O.smin .Q/˛ / .resp: jV ŒQ j D o.smin .Q/˛ / for all quadruples Q with cr.Q/ D 1. Note that V is Zygmund bounded if and only if V is vanishing to order 0. Similarly, we give the following. Definition 9.5. We say that a continuous function V W R ! C is Sullivan vanishing (resp. strongly Sullivan vanishing) to order ˛ if jV ŒQ0 j D O.s.Q0 /˛ / .resp: jV ŒQ0 j D O.s.Q0 /˛ // for all standard quadruples Q0 . A continuous function V W S1 ! C is said to be Sullivan vanishing (resp. strongly Sullivan vanishing) to order ˛ if V is Sullivan vanishing (resp. strongly Sullivan vanishing) to order ˛ in each chart .Un ; ˆn /, where f.Un ; ˆn /g4nD1 is the system of coordinate charts for the unit circle S1 given in Definition 9.3. The following theorem is a comprehensive conclusion of some theorems in [9] and [12]. Theorem 9.6. For each 0 < ˛ < 1, the following statements are equivalent: (1) h is smooth (resp. strongly smooth) of order ˛; (2) is vanishing (resp. strongly vanishing) to order ˛; (3) V is vanishing (resp. strongly vanishing) to order ˛; (4) h is Sullivan smooth (resp. strongly Sullivan smooth) of order ˛; (5) V is Sullivan vanishing (resp. strongly Sullivan vanishing) to order ˛. Let h denote an orientation-preserving circle homeomorphism and h denote the earthquake measure induced by an earthquake map whose restriction on the boundary circle is h. Assume that Vh is formally defined to be the integration in (8.5). Then the following two theorems correspond to the situations where ˛ D 0, which are the comprehensive conclusions of some theorems in [9], [12] and [6]. Theorem 9.7. The following statements are equivalent: (1) h is quasisymmetric; (2) h is smooth of order 0; (3) h is Thurston bounded;

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(4) Vh is Zygmund bounded; (5) h is Sullivan smooth of order 0; (6) Vh is Sullivan vanishing to order 0. Theorem 9.8. The following statements are equivalent: (1) h is symmetric; (2) h is strongly smooth of order 0; (3) h .D/ vanishes uniformly as the disk D approaches the boundary; (4) Vh is strongly vanishing to order 0; (5) h is strongly Sullivan smooth of order 0; (6) Vh is strongly Sullivan vanishing to order 0. Remark 9.9. There are nontrivial circle homeomorphisms h that are Sullivan smooth of order ˛ with ˛ 2 Œ1; 2 (resp. strongly Sullivan smooth of order ˛ with ˛ 2 Œ1; 2/) (see [33]). In the meantime, it is observed in [31] that any earthquake measure vanishing to order ˛ with ˛ > 1 is trivial. Open problems are to characterize those circle homeomorphisms in terms of geometric conditions on their corresponding earthquake measures (see [13]). Remark 9.10. The space of symmetric circle homeomorphisms inherits a Banach manifold structure from the universal Teichmüller space. Since its particular introduction in [11], there have been several studies on asymptotically conformal Teichmüller spaces ([10]). Remark 9.11. Recently, techniques have been developed in [18] to apply the crossratio distortion norm of h to estimate, directly, the quasiconformality of the Douady– Earle extension ˆ ([3]) of h to the unit disk; and then the techniques of [18] are generalized in [19] to show that the quasiconfomality ([3]) and asymptotic conformality ([4]) of ˆ are local properties. A similar theory of Douady–Earle extensions of circle endomorphisms is also being developed in [20]. Motivated by the results in this section, it seems to be possible to investigate how the asymptotic conformality of the Douady–Earle extension ˆ of h depends on the smoothness of h.

References [1] [2]

L. Ahlfors and L. Bers, Riemann mapping’s theorem for variable metrics. Ann. of Math. 72 (1960), 385–404. 88 G. David, Solutions de l’équation de Beltrami avec kk1 D 1. Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), 25–70. 88

Chapter 2. Earthquakes on the hyperbolic plane [3] [4]

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A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle. Acta Math. 157 (1986), 23–48. 120 ˘ c, Barycentric extension and the Bers embedding for C. J. Earle, V. Markovic, and D. Sari´ asymptotic Teichmüller space. In Complex manifolds and hyperbolic geometry, Contemp. Math. 311, Amer. Math. Soc., Providence, R.I., 2002, 87–106. 120

[5]

D. B. A. Epstein, A. Marden, and V. Markovic, Complex earthquakes and deformations of the unit disk. J. Differential Geom. 73 (2006), 119–166. 103

[6]

F. P. Gardiner, Infinitesimal bending and twisting in one-dimensional dynamics. Trans. Amer. Math. Soc. (3) 347 (1995), 915–937. 72, 115, 119

[7]

F. P. Gardiner and J. Hu, Finite earthquakes and the associahedron. In Teichmüller theory and moduli problems, Ramanujan Math. Soc. Lect. Notes Ser. 10, Ramanujan Math. Soc., Mysore 2010, 179–194, 71, 72, 75, 77, 78

[8]

F. P. Gardiner and J. Hu, An earthquake version of the Jackson-Zygmund theorem. Ann. Acad. Sci. Fenn. Math. 30 (2005), 237–260. 115, 117

[9]

F. P. Gardiner, J. Hu, and N. Lakic, Earthquake curves. Stony Brook IMS Preprint No. 7, 2001; in Complex manifolds and hyperbolic geometry, Contemp. Math. 311, Amer. Math. Soc., Providence, R.I., 2002, 141–195. 71, 87, 89, 90, 95, 103, 104, 105, 115, 117, 118, 119

[10] F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller theory. Math. Surveys Monogr. 76, Amer. Math. Soc., Providence, R.I., 2000. 71, 72, 74, 75, 77, 79, 120 [11] F. P. Gardiner and D. P. Sullivan, Symmetric structures on a closed curve. Amer. J. Math. 114 (1992), 683–736. 120 [12] J. Hu, Earthquake measure and cross-ratio distortion. Stony Brook IMS Preprint No. 8, 2001; in In the tradition of Ahlfors and Bers, III, Contemp. Math. 355, Amer. Math. Soc., Providence, R.I., 2004, 285–308. 71, 72, 89, 90, 98, 118, 119 [13] J. Hu, Thurston’s earthquake measures of Sullivan’s circle diffeomorphisms. In Complex dynamics and related topics (Y. Jiang and Y. Wang, eds.), New Stud. Adv. Math. 5, International Press, Somerville, Mass., 2004, 198–217. 118, 120 [14] J. Hu, On a norm of tangent vectors to earthquake curves. Adv. Math. (China) 33 (2004), no. 4, 401–414. 114 [15] J. Hu, Norms on earthquake measures and Zygmund functions. Proc. Amer. Math. Soc. 133 (2005), 193–202. 115 [16] J. Hu, From left earthquakes to right. In In the tradition of Ahlfors-Bers, IV, Contemp. Math. 432, Amer. Math. Soc., Providence, R.I., 2007, 75–92. 81 [17] J. Hu, Finite earthquakes, the associahedron, and forgetful maps. Preprint in preparation. 77 [18] J. Hu and O. Muzician, Cross-ratio distortion and Douady–Earle extension: I. A new upper bound on quasiconformality. Preprint 2010; J. London Math. Soc., to appear. 120 [19] J. Hu and O. Muzician, Cross-ratio distortion and Douady–Earle extension: II. Quasiconformality and asymptotic conformality are local. Preprint 2010; J. Analyse Math., to appear. 120 [20] J. Hu and O. Muzician, Conformally natural extensions of continuous circle maps: I. The case when the pushforward measure has no atom. Preprint, Graduate Center of CUNY,

122

Jun Hu March 2012; in Quasiconformal mappings, Riemann surfaces, and Teichmüller spaces, Contemp. Math., Amer. Math. Soc., Providence, R.I., to appear. 120

[21] J. Hu and M. Su, Thurston unbounded earthquake maps. Ann. Acad. Sci. Fenn. 32 (2007), 125–139. 88, 89 [22] J. Hu and D. Sullivan, Topological conjugacy of circle diffeomorphisms. Ergodic Theory Dynam. Systems 17 (1997), no. 1, 173–186. 118 [23] S. Kerckhoff, The Nielsen realization problem. Ann. of Math. 117 (1983), 235–265. 71 [24] S. Kerckhoff, Earthquakes are analytic. Comment. Math. Helv. 60 (1985), 17–30. 113 [25] C. T. McMullen, Complex earthquakes and Teichmüller theory. J. Amer. Math. Soc. 11 (1998), 283–320. 103 ˘ c, A Fréchet topology on measured laminations and earthquakes [26] H. Miyachi and D. Sari´ in the hyperbolic plane. Preprint, arXiv:1006.0941v1 [math.GT]. 103 ˘ c, Complex earthquake curves are holomorphic. Ph.D. thesis, Graduate School & [27] D. Sari´ University Center of CUNY, New York, N.Y., June 2001. 90, 103, 113 ˘ c, Real and complex earthquakes. Trans. Amer. Math. Soc. 358 (2006), 233–249. [28] D. Sari´ 90, 103, 113 ˘ c, Bounded earthquakes. Proc. Amer. Math. Soc. 136 (2008), 889–897. 90 [29] D. Sari´ ˘ c, Geodesic currents and Teichmüller space. Topology 44 (2005), no. 1, 99–130. [30] D. Sari´ 103 ˘ c, Some remarks on bounded earthquakes. Proc. Amer. Math. Soc. 138 (2010), [31] D. Sari´ 871–879. 120 [32] Z. Slodkowski, Holomorphic motions and polynomial hulls. Proc. Amer. Math. Soc. 111 (1991), 347–355. 90 [33] D. P. Sullivan, Bounds, quadratic differentials, and renormalization conjectures. In American Mathematical Society centennial publications, Vol. II, Amer. Math. Soc., Providence, R.I., 1992, 417–466. 118, 120 [34] W. P. Thurston, Geometry and topology of three-manifolds. Princeton University Lecture Notes, 1979. http://library.msri.org/books/gt3m/ 71 [35] W. P. Thurston, Earthquakes in two-dimensional hyperbolic geometry. In Low-dimensional topology and Kleinian groups, London Math. Soc. Lecture Note Ser. 112, University Press, Cambridge 1986, 91–112. 71, 72, 75, 77, 78, 79, 81, 82, 85, 87, 90

Chapter 3

Kerckhoff’s lines of minima in Teichmüller space Caroline Series Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kerckhoff’s original paper . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Filling up a surface . . . . . . . . . . . . . . . . . . . . 2.2 Kerckhoff’s main results . . . . . . . . . . . . . . . . . . . . . 2.3 Lines of minima and earthquakes . . . . . . . . . . . . . . . . . 2.4 Action of the mapping class group . . . . . . . . . . . . . . . . 2.5 The analogy with Teichmüller geodesics . . . . . . . . . . . . . 3 Further straightforward properties . . . . . . . . . . . . . . . . . . . 3.1 Lines of minima and fixed length horoplanes . . . . . . . . . . 3.2 The simplex of minima . . . . . . . . . . . . . . . . . . . . . . 3.3 Concrete examples . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Limiting behaviour at infinity . . . . . . . . . . . . . . . . . . . 4 Lines of minima and quasifuchsian groups which are almost Fuchsian 4.1 Applications and examples . . . . . . . . . . . . . . . . . . . . 4.2 Quakebends and the Fuchsian limit . . . . . . . . . . . . . . . . 4.3 The bending measure conjecture . . . . . . . . . . . . . . . . . 5 Relationship to Teichmüller geodesics . . . . . . . . . . . . . . . . . 5.1 Comparison in the thick part of Teichmüller space . . . . . . . . 5.2 Comparison in the thin part of Teichmüller space . . . . . . . . 6 Short curves on Teichmüller geodesics . . . . . . . . . . . . . . . . 6.1 Flat and expanding annuli . . . . . . . . . . . . . . . . . . . . 6.2 Moduli of annuli and hyperbolically short curves . . . . . . . . 6.3 Rafi’s thick-thin decomposition for the q-metric . . . . . . . . . 6.4 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . 6.5 Rafi’s combinatorial formula for Teichmüller distance . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The usual geometries of Teichmüller space relate largely to analytic properties of Riemann surfaces such as quasiconformal maps and quadratic differentials. Lines of minima were introduced by Kerckhoff [19] as a means of endowing Teichmüller space with a geometry directly related to the hyperbolic structure uniformising the surface.

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The fundamental observation is the following. Suppose that and are measured laminations on a hyperbolic surface S, which fill up S in the sense that all the complementary components are disks or once punctured disks. Then the sum of the hyperbolic lengths l C l has a unique minimum M.; / on the Teichmüller space T .S / of S . This result is a consequence of Thurston’s earthquake theorem and the convexity of lamination length along earthquake paths. The proof is sketched in Section 2. Now consider convex combinations of the form .1 t /l C t l . The set of minima M..1 t /; t / for t 2 .0; 1/ is a 1-manifold L; embedded in T .S /, called the Kerckhoff line of minima of and . Kerckhoff showed that the lines L; mimic many properties of geodesics in the Poincaré disc model of hyperbolic space H2 . For example, two projective laminations in PML.S /, the Thurston boundary of T .S /, determine a unique line of minima. The lines L; for a fixed measured lamination foliate T .S/. Any two points in T .S/ lie on at least one line, although whether or not this line is unique, is unknown. Lines of minima are intimately connected with earthquakes, in particular M.; / is the unique point in T .S / at which @=@t D @=@t , where @=@t is the tangent to the earthquake path along . More such results are detailed in Section 2. In Section 3 we discuss various extensions and examples of Kerckhoff’s results, most of which are extracted from [8], [9], [36]. Thus far, the main application of lines of minima has been to problems about small deformations of Fuchsian groups, via a link discovered by Series [36], [37]. One can deform a Fuchsian group by bending along a lamination , a bend or quakebend being the complex analogue of an earthquake, see Section 4.2. For small values of the bending parameter this gives a quasifuchsian group the boundary of whose convex core on one side is bent along the lamination . It turns out that the bending lamination on the other side of the convex core is , if and only if the initial Fuchsian group lies on L; . Subsequently Bonahon [3] used these ideas partially to prove a conjecture of Thurston about the uniqueness of groups with given bending data. These results are explained in Section 4. It is natural to make a comparison between lines of minima and Teichmüller geodesics. The minimisation property of the length functions is analogous to an important minimisation property along Teichmüller geodesics. A Teichmüller geodesic is also determined by a pair of laminations , which fill up S , namely the horizontal and vertical foliations of the defining quadratic differential, see Section 2.5. Gardiner and Masur [12] showed that this Teichmüller geodesic G; can be characterised as the line along which the product of the extremal lengths of , is minimised. Unlike the sum of lengths along L; , this product remains constant along G; . The limiting behaviours of lines of minima and Teichmüller geodesics are at least partly comparable, see Section 3.4 and [9]. More detailed questions about the relationship between lines of minima and Teichmüller geodesics have been explored by Choi, Rafi and Series. In [6] they gave a combinatorial estimate of the distance between the two paths and in [7], they proved that a line of minima is a Teichmüller quasi-geodesic. These results are discussed in Section 5.

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The methods of Section 5 are heavily dependent on some techniques originally due to Minsky [25] and developed further by Rafi [32], [33], [34], on curves which are short on surfaces along a Teichmüller geodesic. Since this work contains some powerful and interesting techniques, we take the opportunity to summarize it briefly in Section 6. I would like to thank Athanase Papadopoulos for giving me the opportunity to present this relatively recent body of work in this volume.

2 Kerckhoff’s original paper 2.1 Basic definitions Let S be a surface of hyperbolic type with genus g and b punctures, and denote its Teichmüller space by T .S/. For simplicity, Kerckhoff restricted his statements in [19] to surfaces without punctures, but remarked that the work easily generalises to the finite area case. Here we state the results for surfaces with punctures, noting some of the points where the original proofs need a little attention. We denote the spaces of measured and projective measured laminations on S, by ML.S / and PML.S/ respectively. Then T .S / and ML.S / are open balls of real dimension 2d with d D 3g 3 C b. The Thurston compactification of T .S / adjoins PML.S /, homeomorphic to the 2d 1-sphere, as a boundary ([11], Exposé 8). For 2 ML.S / we denote by Œ its projective class and by jj its underlying support. Let denote the set of homotopy classes of simple non-peripheral curves on S . We call a lamination 2 ML.S/ rational Pif jj is a disjoint union of closed geodesics ˛i 2 . We write such laminations i ai ˛i , where ai 2 RC and ˛i (more properly ı˛i ) represents the lamination with support ˛i which assigns unit mass to each intersection with ˛i . The hyperbolic length l of a lamination 2 ML.S / is the function on T .S / which associates to each p 2 T .S/ the total mass of the measure which is the product of hyperbolic distance along the leaves of with the transverse measure . (Here and in what follows, the hyperbolic structure on S is the one P which uniformises the conformal structure on p 2 T .S/.) In particular, if D i ai ˛i is rational, then P l D a l , where l .p/ is the hyperbolic length of the geodesic ˛i on the i ˛ ˛ i i i surface p 2 T .S/. The length is a continuous function ML.S / T .S / ! R>0 . Likewise the geometric intersection number i.; / of ; 2 ML may be defined as a continuous function ML.S/ ML.S / ! R0 extending the usual geometric intersection number i.˛; ˛ 0 / of two geodesics ˛; ˛ 0 2 , see for example [18]. 2.1.1 Filling up a surface Definition 2.1. Laminations ; 2 ML.S / fill up S if i.; / C i.; / > 0 for all 2 ML.

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This condition clearly only depends on the projective classes of , in PML. An equivalent condition is that every component of S jj [ jj contains at most one puncture, and whose closure, after filling in the puncture if needed, is compact and simply connected. For the equivalence of these two conditions see [19] Proposition 1.1 and Lemma 4.4, or [29] Proposition 2. These authors treat the case of closed surfaces, however there is no essential difference for surfaces with punctures provided we allow complementary components which are punctured disks. For rational laminations it is enough to require that the components of the complement be disks or punctured disks. For general laminations this is not enough, since the complement of a generic lamination is a union of ideal polygons. This is why one needs in addition that the complementary components be compact. If laminations are replaced by foliations, the condition of filling up a surface is also equivalent to a pair of foliations in the measure equivalence classes being realised transversally, see Section 2.5. Earthquakes. The time-t left earthquake [18] along a lamination 2 ML is a real analytic map E .t/ W T .S/ ! T .S/ which generalises the classical Fenchel–Nielsen twist. The earthquake shifts complementary components of the lamination jj on the surface p 2 T .S/ a hyperbolic distance t./ relative to one another, where ./ is the -measure of a transversal joining the two components. If p 2 T .S/, we denote by E D E .p/ the earthquake path E .t /.p/; t 2 R. This path induces a flow and hence a tangent vector field @=@t on T .S /. In [18], Kerckhoff shows that if 2 ML, then the length l is a real analytic function of t along E .p/. One has ([18], Lemma 3.2, [43], Theorem 3.3): “ cos d d; (2.1) @l =@t D where the integral is over all intersection points of a leaf of jj with one of jj, and for each such intersection point, is the ’ anticlockwise angle from to . Following [19], we sometimes write Cos.; / D cos d d. It follows that l is strictly convex along E .p/ if i.; / > 0 and constant otherwise, in particular, l has a unique minimum on E .p/. Equation (2.1) also immediately gives Wolpert’s antisymmetry formula ([42], Theorem 2.11) @l =@t D @l =@t ;

(2.2)

from which one deduces easily that the minimum points for l along E and l along E coincide. Wolpert also showed ([43], Theorem 3.4) that if i.; / > 0, then at the unique minimum, @2 l =@t2 > 0 (see [10], 3.10, for the same result for laminations). This observation becomes crucial in Bonahon’s Theorem 4.7 below. Thurston’s earthquake theorem [18], [41], see also [13], states that for any pair of points p; p 0 2 T .S/, there is a unique 2 ML.S / such that E .1/.p/ D p 0 . (The proof of the earthquake theorem in [18] requires a bit of attention when S has punctures. The main point is that in the approximation arguments, one has to use


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the well known fact that every leaf in the support of a lamination 2 ML avoids a definite horocycle neighbourhood of each cusp.)

2.2 Kerckhoff’s main results The foundational result on the existence of minima of length functions is a classic application of the earthquake theorem. In fact the proof gives a rather stronger statement: Theorem 2.2 ([19], Theorem 1.2). Let ; 2 ML be laminations which fill up S . Then l C l has a unique critical point on T .S / which is necessarily a minimum. Proof. We first show f; D l C l W T .S / ! RC is proper. We have to show that f; .pn / ! 1 whenever pn ! 1 in T .S /. Pick a subsequence of .pn / which converges to 2 PML. Then there exist cn ! 1 such that l =cn ! i.; / and l =cn ! i.; /, see [11], [28]. Since , fill up S, we have i.; / C i.; / > 0 so f; ! 1 as claimed. It follows that f; has at least one minimum on T .S /. Now we show there is a unique critical point, which from the above must be a minimum. Suppose that p; p 0 2 T .S/ are both critical points. By Thurston’s earthquake theorem there is a unique 2 ML such that E .1/.p/ D p 0 . Both l and l are convex along this path. Moreover since , fill up S, we have i.; /Ci.; / > 0 so that at least one of l , l must be strictly convex. Thus so is f; , hence the critical point along this path is unique. The next result shows that the assumption that , fill up S is necessary. Proposition 2.3 ([19], Theorem 2.1, Part II). Suppose that ; 2 ML do not fill up S . Then l C l has no critical point on T .S /. Proof. It suffices to show that if and fail to fill up S, then l C l can always be decreased. Precisely this statement is proved in Theorem 2.1 (Part II) in [19]. The argument needs minor changes if S has cusps to allow for the possibility that the complementary components of various laminations may be punctured disks. Definition 2.4. Suppose that ; 2 ML fill up S . The line of minima L; of , is the image of the path .0; 1/ ! T .S/; t 7! M..1 t /; t /. Note that this definition depends only on the projective classes of , , and that equivalently, L; is the image of the path .0; 1/ ! T .S/; k 7! M.; k/. Since the notation is slightly lighter, we shall from now on frequently parameterise the line of minima in this way. At the critical point p D M.; k/, we have @=@t .l C kl / D 0, equivalently using (2.1): Cos.; / C k Cos.; / D 0 (2.3) for all 2 ML.

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Proposition 2.5 ([19], Theorem 2.1, Parts I, III). The map k ! M.; k/ is a continuous injection .0; 1/ ! T .S/. Proof. Injectivity is a special (easy) case of [19], Theorem 2.1, Part III. To see this, use (2.3) at the point M.; k/, together with the easy observation that Cos.; / certainly does not vanish simultaneously for all , to show that M.; k/ D M.; k 0 / implies k D k 0 . Continuity is proved in [19], Theorem 2.1, Part I. One of Kerckhoff’s main results is that the lines L; for varying foliate T .S / in a manner analogous to the foliation of H2 by geodesics emanating from one point on the boundary @H2 . Fixing Œ 2 PML.S /, we therefore need to understand PML , the set of laminations which together with fill up S . Clearly, PML PML fŒg R6g7C2b D R2d 1 . If is uniquely ergodic and maximal, then by definition i.; / D 0 implies 2 Œ. It is easy to see in this case that PML D PML fŒg. In general, PML can be smaller, nevertheless we have: Theorem 2.6 ([19], Theorem 4.7, see also [12], Theorem 8). Suppose 2 ML n f0g. Then PML is homeomorphic to R2d 1 . Proof. If is rational, one can prove this as follows. Extend the support curves of to a pants decomposition A D f˛1 ; : : : ; ˛d g of S . It is not hard to see that , fill up S if and only if i.; ˛i / > 0 for all ˛i 2 A. Setting qi ./ D i.; ˛i /, let f.pi ./; qi .//diD1 g denote the Dehn–Thurston coordinates of 2 ML relative to A, see for example [31]. Then PML is the image Q in PML of the set f 2 ML W the point in Rd †d 1 whose qi ./ > 0 for all i g. We can map any point in Q to P th d d 1 are pi =qi and qi = qj P respectively, where †d 1 i coordinates in R and † denotes the d 1 dimensional simplex f.x1 ; : : : ; xd / W xi D 1; xi > 0g. This shows that Q is homeomorphic to Rd †d 1 . For the general case, see [19]. Fix a continuous section j W PML ! ML, for example by choosing a fixed point . p0 2 T .S / and defining j.Œ/ D l .p 0/ Theorem 2.7 ([19], Theorem 2.1). The map PML .0; 1/ ! T .S / which sends .Œ; k/ to M.; kj.Œ//, is a homeomorphism. Proof. By Theorem 2.6, the domain and range are both balls of dimension 2d . The proof consists in showing that the map is continuous, proper and injective. If the map is not proper, one shows that there exists .; k/ 2 PML .0; 1/ such that l C kl has a critical point at p 2 T .S/, contradicting Proposition 2.3. To prove injectivity, the question is reduced using (2.3) to showing that if Cos.; / D Cos.; / for all 2 ML at some point p 2 T .S/, then D . This is resolved using the more technical but useful Proposition 2.8 which follows.


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Proposition 2.8 ([19], Theorem 4.8). Suppose that ; 2 ML and that i.; / > 0. Let p 2 T .S/. Then there exists 2 ML such that at p: Cos.; / > Cos.; /;

Cos.; / > Cos.; / and

Cos.; / > Cos.; /:

It follows from Theorem 2.7, that for each p 2 T .S /; 2 ML, there is a unique D .p/ such that l C l is minimised at p. Let

1

PML D PML Œ1; 1/=PML f1g be the cone over PML with the cone point as PML f1g. Then:

1

Theorem 2.9 ([19], Theorem 3.6). Fix p 2 T .S /. Then the map PML ! T .S / which sends .; k/ to M.; k .p// is surjective. Corollary 2.10 ([19], Corollary 3.7). There is a line of minima between every pair of points in T .S/. It is not known whether or not this line is unique, see [19], p. 188.

2.3 Lines of minima and earthquakes There is a close connection between lines of minima and earthquakes. As remarked above, the earthquake flow E .t/ generates a field of tangent vectors @=@t on T .S /. Likewise, the length function l defines a cotangent vector field d l . Theorem 2.11 ([19], Theorem 3.5). For all p 2 T .S /, the maps 7! @=@t .p/ and 7! d l .p/ are homeomorphisms from ML to the tangent space Tp .T .S // and the cotangent space Tp .T .S// respectively. Proof. To prove that 7! @=@t .p/ is a homeomorphism, one uses invariance of domain. The key step is to prove that the map 7! @=@t .p/ is injective, which follows from Proposition 2.8. The second statement can be proved using (2.2). Remark 2.12. The fact that fd l˛ ; ˛ 2 g span Tp .T .S // is classical and goes back to Fricke–Klein, see [43], p. 229. Corollary 2.13 ([19], Theorem 3.4). Given ; 2 ML which fill up S, and k 2 .0; 1/, there is a unique p 2 T .S/ such that d l D kd l , or equivalently @=@t D k@=@t , at p. Moreover p D M.; k/. Proof. By Theorem 2.2, d l D kd l at p 2 T .S / if and only if p D M.; k/. At M.; k/, we have @=@t .l C kl / D 0 for every 2 ML. Using (2.2) this @l @l gives @t D k @t for all . We conclude by Theorem 2.11 that @=@t D k@=@t . Reversing the argument concludes the proof.

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In fact, the conditions on k, , in Corollary 2.13 are automatic: Corollary 2.14 ([36], Proposition 4.7). Suppose that ; 2 ML and that … Œ. Suppose also that @=@t D k@=@t for some k 2 R. Then and fill up S and k > 0. In particular, p D M.; k/. Proof. The main part of the proof is the case i.; / > 0. By Proposition 2.8, for any point p 2 T .S/, there exists a lamination 2 ML such that at p, Cos.; / > Cos.; /

and

Cos.; / > Cos.; /:

From @=@t D k@=@t we deduce k Cos.; / D Cos.; / D 0 and hence Cos.; / > 0 > Cos.; / D k Cos.; / which forces k > 0. By the same argument as Corollary 2.13, we deduce that d l D kd l at p and the conclusion follows from Proposition 2.3.

2.4 Action of the mapping class group Lines of minima are natural for the action of the mapping class group: Theorem 2.15 ([19], Theorems 3.1, 3.2). A pseudo-Anosov mapping class fixes the unique line of minima defined by its stable and unstable laminations. More generally, the action of the mapping class group carries lines of minima to lines of minima. Proof. Let ˙ be the two fixed laminations of , so that .˙ / D ˙1 .˙ / for some 2 RC , see [11], [28]. From this we deduce immediately that L.C /;. / D LC ; . The second statement follows in a similar way.

2.5 The analogy with Teichmüller geodesics Let q be a quadratic differential on a Riemann surface R. Its horizontal trajectories p equipped with the vertical measure j= q.z/dzj form a measured foliation Hq on p R, similarly the vertical trajectories with the horizontal measure j< q.z/dzj form a measured foliation Vq , see [40]. Hubbard and Masur [14] showed that on every compact Riemann surface, every equivalence class of measured foliations is realised as the horizontal foliation of a unique quadratic differential q. Suppose K 2 .0; 1/. Then q also determines a Teichmüller map from R to another surface R0 with dilatation K. If q 0 is the terminal quadratic differential on R0 with the same norm as q, then the horizontal and vertical foliations of q 0 are K 1=2 Hq and K 1=2 Vq respectively. The foliations Hq , Vq are clearly transverse. Gardiner and Masur, see [12], Theorem 5.1, showed that conversely, for any pair of transversally realisable measured foliations F , F 0 , there is a unique Teichmüller


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geodesic determined by a quadratic differential q on a Riemann surface R for which F , F 0 are measure equivalent to Hq and Vq respectively. We note that the condition of being transversally realisable is equivalent to the two corresponding pairs of measured laminations filling up the surface, see [12], Lemmas 3.4 and 5.3. The Gardiner–Masur Teichmüller line is found by the following minimisation property. Generalising the usual definition of extremal length of a curve, define the extremal length of a measured foliation ŒF as Z jq.z/jdxdy; Ext.ŒF / D kqk D R

where q is the unique quadratic differential on R for which F is equivalent to the horizontal foliation Hq . It is not hard to see that the product Ext.ŒHq / Ext.ŒVq / is constant along the Teichmüller geodesic G determined by q. Conversely, it is proved in Theorem 5.1 of [12] that Ext.ŒHq / Ext.ŒVq / achieves its infimum along G , and that at any point not on G the product is strictly larger. In the course of the proof, it is shown that if and are not transversally realisable, then the product can always be decreased, in analogy with Proposition 2.3.

3 Further straightforward properties In this section, we describe some relatively straightforward extensions of Kerckhoff’s results.

3.1 Lines of minima and fixed length horoplanes In the very special case 2d D dimR T .S/ D 2, the analogy between H2 and T .S / suggests viewing lines of minima as geodesics and earthquake paths as horocycles. It is not hard to see, [15], Lemma 6, that in this case a line of minima meets an earthquake path in exactly one point. One way to generalise this is as follows. Note that along the earthquake path E , the hyperbolic length of the lamination is constant. For 2 ML; c > 0, define the horoplane HIc D fp 2 T .S / W l .p/ D cg, so that E HIc , with equality in the case d D 1. As above, let ML D f 2 ML W ; fill up Sg, and let j be a fixed section PML ! ML. The following result is an improvement on Theorem 7.2 in [36]. Theorem 3.1. For each 2 ML , the restriction of l to HIc has a unique minimum. The minimum point p is also the unique point in which L; meets HIc . The map PML ! H;c sending to Œ to pj.Œ/ is a homeomorphism. Proof. The proof in [36] can be substantially simplified by using Lagrange multipliers to minimise l subject to the constraint that l D c. This shows that at a critical point, d l D d l for some 2 R, which is exactly the condition in Corollary 2.14.

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By contrast, if we prescribe the lengths of all the curves in a pants decomposition A D f˛1 ; : : : ; ˛d g, the situation is more subtle. For c D .c1 ; : : : ; cd /; ci > 0, define the shearing plane EA;c D fp 2 P T .S/ W l˛i .p/ D ci ; i D 1; : : : ; d g. Let MLA be the set of laminations such that i ˛i and together fill up S . We have: Theorem 3.2 ([36], Theorem 7.3). Let A be a pants decomposition of S and let open set U MLA such that c D .c1 ; : : : ; cd /, ci > 0. Then there is a non-empty P for all 2 U , there are no laminations D i ai ˛i for which L; meets EA;c . However [8] gives an example of a pants decomposition A and a curve 2 MLA for which LA; meets EA;c for any choice of c.

3.2 The simplex of minima Series [36] and Díaz and Series [8] studied the special case of lines of minima for families of disjoint curves A D f˛1 ; : : : ; ˛N g and B D fˇ1 ; : : : ; ˇM g which fill up S . The simplex of minima †A;B associated to A and B is the union of lines of minima L; , where ; 2 ML.S/ are strictly positive linear combinations of f˛i g and fˇi g, respectively. We can regard †A;B as the image of the affine simplex in y A;B RN CM 1 spanned by independent points A1 ; : : : ; AN ; B1 ; : : : ; BM , under † P P Ai / C s. P the map ˆ P D ˆ.A; B/ P which sends the point .1 s/. i ai P j bj Bj / to M..1 s/. i ai l˛i /; s. j bj lˇj //, where 0 < s; ai ; bj < 1; ai D 1; bj D 1. The methods of [19] show that ˆ is continuous and proper, extending continuously to the faces of †A;B corresponding to those subsets of A B which still fill up S . The map ˆ may or may not be injective. This may be studied by examining the case in which A and B are pants decompositions, so that M D N D d . Let M.A; B/ @l P P be the d d matrix @t˛ˇji . If D i ai ˛i , we have @=@t D i ai @=@t˛i from P which @lbj =@t D i ai @lˇj =@t˛i for each i and hence .@lˇ1 =@t ; : : : ; @lˇd =@t / D M.A; B/.a1 ; : : : ; ad /T :

P ˇj . At the point M.; k/ we have @=@t D k@=@t . Hence Set D j bjP @lˇj =@t D k i bi @lˇj =@tˇi D 0 for all j so that det M.A; B/ D 0 at M.; k/. The converse statement, that det M.A; B/ D 0 at p 2 T .S / implies p D M.; k/, is false. This is because although Pthe conditionPimplies that @=@t D k@=@t for some formal combinations D i ai ˛i ; D j bj ˇj , the coefficients ai ; bj may not all have the same sign. Nevertheless, elaborating these observations, we have: Proposition 3.3 ([8], Proposition 4.6). Let A; B be two pants decompositions which fill up S . Then rank M < d on Im ˆ. Moreover ˆ is a homeomorphism onto its image if and only if rank M D d 1 on Im ˆ. If rank M D d 1, then p 2 Im ˆ if and only if the adjoint matrix of M has all its entries of the same sign (with possibly some entries vanishing).


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It can be shown by example, see Section 3.3, that both cases rank M D d 1 and rank M < d 1 occur.

3.3 Concrete examples Not many explicit computations of lines of minima have been made. In the case of the once punctured torus S1;1 , there are two special cases, corresponding to the laminations , being supported on curves ˛, ˇ which intersect once and twice respectively. It turns out, see Section 11 in [16], that the lines L; are the lines in T .S / which correspond to the rectangular and the rhombic tori respectively. One can see this as follows. In the first case, let .V; W / be a generator pair for 1 .S1;1 /, so that we may take V D ˛; W D ˇ. Then any point on L˛;ˇ is fixed by the rectangular symmetry R W .V; W / 7! .V; W 1 /, and it follows that L˛;ˇ must be the fixed line of R. In the second case, again taking .V; W / to be a generator pair of 1 .S1;1 /, the curves ˛ D V W and ˇ D V W 1 intersect twice. In this case, any point on L˛;ˇ is fixed by the rhombic involution V ! W , W ! V , so that L˛;ˇ must be the fixed line of the rhombic symmetry which is given by the equation Tr W D Tr V . If ˛, ˇ intersect once, one can also determine L; as follows. The condition @=@t˛ D k@=@tˇ on L; implies by (2.1) that ˛ and ˇ are orthogonal. Let t˛;ˇ be the Fenchel–Nielsen twist coordinate of ˛ about ˇ, normalised to be 0 when ˛ and ˇ are orthogonal. Then relative to the Fenchel–Nielsen coordinates .l˛ ; t˛;ˇ / for T .S1;1 /, L˛;ˇ is the line .l; 0/, 0 < l < 1. If ˛, ˇ are curves on S1;1 which meet exactly once, we have ([30], Theorem 2.1): cos.t˛;ˇ =2/ D cosh.l˛ =2/ tanh.lˇ =2/:

(3.1)

It follows that L˛;ˇ is also defined by the equation sinh.l˛ =2/ sinh.lˇ =2/ D 1 obtained by setting t˛;ˇ D 0 in (3.1). More interesting examples were computed in [8] for the twice punctured torus S1;2 . Let A D f˛1 ; ˛2 g and B D fˇ1 ; ˇ2 g be pairs of non-separating, disjoint, simple closed curves on S1;2 such that each ˇi intersects each ˛j exactly once. Also let D D fˇ1 ; ı1 g, where ı1 is a separating curve, disjoint from ˇ1 and ˛2 and intersecting ˛1 twice. Using Fenchel–Nielsen coordinates relative to the pants systems A, extensive computations in [8] located the simplices of minima †A;B and †A;D in T .S1;2 / D R4 . These two examples serve to illustrate the distinction made in Proposition 3.3. We find rank M.A; B/ D d 1 D 3 on †A;B whereas rank M.A; D/ D d 2 D 2 on y A;B and that †A;D . It follows that ˆ.A; B/ is an embedding of the affine simplex † y A;D under ˆ.A; D/ is not †A;B is a 3-submanifold of T .S1;2 /, while the image of † embedded and †A;D is a 2-submanifold.

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3.4 Limiting behaviour at infinity It might be natural to suppose that the ends of L; would always converge to the points Œ, Œ in Thurston’s compactification of T .S /, namely PML. However Kerckhoff ([19], p. 192) indicates that this may not always be the case. In fact: Theorem 3.4 ([9], Theorems 1.1, 1.2). Suppose that ; 2 ML fill up S. For P t 2 .0; 1/, let M t D M..1 t /; t /. If D N iD1 ai ˛i 2 ML is rational with ai > 0 for all i then lim M t D Œ˛1 C C ˛N 2 PML:

t!0

If is uniquely ergodic and maximal, then lim M t D Œ 2 PML:

t!0

This should be compared with the analogous results of Masur [23] for Teichmüller geodesics. A geodesic ray is determined by a base surface p 2 T .S / and a quadratic differential q on p. Masur shows that if q is a Jenkins–Strebel differential, that is, if the horizontal foliation H is supported (apart from saddle connections) on closed leaves, then the associated ray converges in PML to the barycentre of the leaves (the foliation with the same closed leaves all of whose cylinders have unit height). At the other extreme, if H is uniquely ergodic and every leaf apart from saddle connections is dense in S, the ray converges to the boundary point defined by H . Recently Anna Lenzhen [20] has shown the existence of Teichmüller geodesics which do not converge to any point in PML. One can presumably apply [6] to show the same is true of lines of minima, although to the author’s knowledge this has not been written down. Proof of Theorem 3.4. The condition of being uniquely ergodic and maximal is equivalent to the condition that i.; / D 0 for all 2 ML implies that 2 Œ ([9], Lemma 2.1). The second statement of Theorem 3.4 is thus easily proved by showing that as t ! 0, any subsequence of .M t / necessarily has a convergent subsequence which limits on Œ 2 PML. The proof of the first part of Theorem 3.4 is easiest when f˛i g form a pants decomposition A of S. Let .l˛i ; t˛i / be the Fenchel–Nielsen coordinates of S relative to A, where the twist t˛i is the signed hyperbolic distance twisted round ˛i measured from some suitable base point. It is not hard to show by constructing surfaces in which the lengths l˛i are specified, that l˛i .M t /=t is uniformly bounded away from 0; 1 as t ! 0 for each i. An earthquake about a curve ˛ fixes l˛ , while the length l of any transverse curve is a proper and convex function of the earthquake parameter. It is therefore reasonable to suppose that for fixed lengths l˛i , the sum l C l attains its minimum within bounded distance of the point where t˛i D 0. To make this more precise, we


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replace by a homotopic piecewise geodesic curve O which runs alternately along the pants curves ˛i , and across pairs of pants in such a way that it meets the boundary components orthogonally, so that adjacent segments of O always meet orthogonally. Now the hypotenuse of a right angled hyperbolic triangle is, up to bounded additive constant, equal to the sum of the lengths of the other two sides. It follows that if the lengths l˛i are bounded above (so that each curve ˛i is surrounded by a collar of definite width), the length l is coarsely approximated by the length lO , with error comparable in magnitude to the intersection number of with A. By the hyperbolic collar lemma, the width of the collar round a short curve of length l is approximately log 1= l, see [22], [27]. Using the observation that the twist parameter remains bounded at M t gives the estimate l .M t / D 2

N X j D1

i.˛j ; / log

1 C O.1/ l˛j .M t /

(3.2)

for a curve transverse to the pants curves A. Since we have already observed that l˛j .M t / t for all i , the result follows from the definition of convergence to a point in PML, see [11]. The argument in the case that A is not a full pants decomposition is considerably more subtle, see [9], Section 6.

4 Lines of minima and quasifuchsian groups which are almost Fuchsian As discovered by Series [36], [37], lines of minima are closely related to small deformations of Fuchsian into quasifuchsian groups. To understand the connection, recall first some basic facts about quasifuchsian groups and their convex cores; for more details see [21]. A Kleinian group (that is, a discrete subgroup of SL.2; C/) is called quasifuchsian if it is isomorphic to 1 .S/ for some surface S of hyperbolic type and if its limit set is a Jordan curve. Equivalently, a Kleinian group G is quasifuchsian if the associated hyperbolic 3-manifold H3 =G is homeomorphic to S .0; 1/. Let C be the convex hull of the limit set of G in H3 , see [10]. Then C =G is the smallest closed convex set containing all closed geodesics in H3 =G and is homeomorphic to S Œ0; 1. Moreover C has two simply connected boundary components @˙ C whose quotients @˙ C=G are pleated surfaces [10], each homeomorphic to S. Each component is bent along a geodesic lamination on S , the amount of bending being measured by the bending (or pleating) measures pl˙ D pl˙ .G/ 2 ML.S /. The following is a special case of a fundamental result of Bonahon and Otal: Theorem 4.1 ([4], Théorème 1). Let S be a hyperbolisable surface and suppose that ; 2 ML.S/. Suppose also that every closed leaf of and of has weight strictly less than . Then there is a quasifuchsian group G.; / for which plC .G/ D and pl .G/ D , if and only if ; fill up S . If , are rational, then G.; / is unique.

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The following conjecture, known as the bending measure conjecture, was originally made by Thurston. Conjecture 4.2. For any ; 2 ML satisfying the conditions of Theorem 4.1, the quasifuchsian group G.; / for which plC .G/ D and pl .G/ D is unique. Moreover groups for which both bending laminations are rational are dense in the space of all quasifuchsian groups. This was proved by Series [38] for the very special case in which S is a once punctured torus. We shall return to this conjecture in Section 4.3. From now on, we denote by G.; / any quasifuchsian group for which plC .G/ D and pl .G/ D . It seems reasonable to assume that as ; ! 0, the convex cores of the groups G.; / flatten out approaching a Fuchsian limit. Topologising the space Q.S / of quasifuchsian groups with the algebraic topology inherited from the space of representations 1 .S/ ! SL.2; C/, we have: Theorem 4.3 ([37], Theorems 1.4, 1.6). Let , be two measured laminations which fill up S , and suppose that n ! , n ! in ML.S / and that n ! 0. Then the sequence of groups G.n n ; n n / converges to M.; / as n ! 1. For a partial proof, see Proposition 4.6 below. It is however essential that the bending measures stay in bounded proportion: Theorem 4.4 ([37], Theorem 1.5). Let , be two measured laminations which fill up S . Then any sequence of groups G.n ; n / with n ; n ! 0 diverges (that is, no subsequence has an algebraic limit) unless n = n is uniformly bounded away from 0 and 1. It is well known that Q.S/ is a 2d -dimensional complex manifold. (This may be seen using complex Fenchel–Nielsen coordinates [39].) Via the uniformisation theorem, T .S/ may be identified with the space of Fuchsian groups F .S /, naturally embedded as a 2d -dimensional totally real submanifold in Q.S /. The following close connection between lines of minima and bending measures of groups in Q.S / was originally noted by Keen and Series [16] in the context of the once punctured torus. Given two non-zero measured laminations and , the pleating variety P; consists of all quasifuchsian groups G. ; /, ; 2 RC . Theorem 4.5 ([37], Theorem 1.7). Let , be two measured laminations which fill up S . Then the closure of P; meets F .S/ precisely in L; . To prove Theorem 4.5 one needs to invoke Theorem 4.1 for the existence of the groups G.n ; n /. Since for rational laminations this in turn is based on the Hodgson–Kerckhoff theory of deformations of cone manifolds, Theorem 4.5 rests ultimately on the same thing.


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We deduce from Theorem 4.5 that given any point p 2 F .S / and lamination 2 ML.S /, it is possible to move away from p into Q.S / along a path in P; if and only if p 2 L; . Using Theorem 3.1, we deduce that for any 2 ML, c > 0 and any 2 ML one can bend (see Section 4.2) along to produce groups in P; for which l D c. On the other hand, Theorem 3.2 implies the rather surprising negative result that if A D f˛1 ; : : : ; ˛d g is a pants decomposition of S, then for any choice of fixed lengths l˛i D ci , i D 1; : : : ; d , there are laminations 2 MLA with the property that it is impossible to bend on the lines ˛i obtaining groups with pl 2 Œ.

4.1 Applications and examples The examples of simplices of minima described in Section 3.3 were expanded in [8] into direct computations of pleating varieties in Q.S1;2 /. With notation as in that section, complex Fenchel–Nielsen coordinates were used to locate the pleating varieties PA;B and PA;D (those groups G 2 Q.S / for which pl˙ .G/ are supported on the curves A, B or A, D respectively) in QF .S1;2 / D C 4 . The computations confirm Theorem 4.5, showing by explicit computation that the closures of PA;B and PA;D meet F .S/ in the simplices of minima †A;B and †A;D . As expected (see Proposition 4.9) both pleating varieties are smooth submanifolds of real dimension 4.

4.2 Quakebends and the Fuchsian limit The main part of the proof of Theorem 4.3 involves geometry in H3 and is beyond the scope of this chapter. However the proof that if the limit of the groups G.n n ; n n / exists, it must be M.; /, is a nice illustration of the results in Section 2.2. The proof uses quakebends ([10], Section 3.5), an extension of earthquakes into the complex domain. Starting from a Fuchsian group G0 , representing a point p 2 F .S /, the quakebend construction produces a quasifuchsian group G by bending, Q denote the more generally quakebending, G0 along a lamination 2 ML. Let jj union of the lifts of the (geodesic) leaves supporting in the hyperbolic structure on p to H2 . This gives a decomposition of H2 into plaques, each plaque being a connected component of H2 n jj. Q Let D t C i 2 C. The time- left quakebend Q . / reglues these plaques, in such a way that if two plaques are connected by a transversal to jj, Q they are reglued at an angle ./ after being shifted a relative distance t./, resulting in a pleated surface Q . /.H2 / in H3 . This surface is invariant under the action of a group G D Q . /.G0 / SL.2; C/. Keen and Series [15] showed that for sufficiently small j j, Q . /.H2 / is embedded in H3 and hence that G D Q . /.G0 / is quasifuchsian. Moreover Q . /.H2 / is one of the two boundary components of the hyperbolic convex hull C of the limit set of G in H3 , and plC .G/ D .

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Proposition 4.6 ([37], Proposition 3.1, see also [3], Proposition, 6). Let , be measured laminations which fill up S . Suppose that as n ! 0, the groups G.n ; n / converge to p 2 F .S/. Then p D M.; /. Proof. By Theorem 2.11 and Corollary 2.13 it is enough to show that @l @l .p/ C .p/ D 0 @t @t

for all 2 ML:

Now for any 2 ML and p 2 F .S/, the quakebend construction with D i gives a one parameter family of quasifuchsian groups G./ for which plC D for all small > 0. Throughout this deformation, the Fuchsian structure p C . / on @C C .G. // (see [10]) remains fixed. Thus we can reach q./ D G.; / 2 Q.S / either by starting at p C ./ and making the pure bend Q .i /, or by starting at p . / and making the pure bend Q .i/. One can extend the definition of the complex length of a curve 2 to that of an arbitrary lamination 2 ML, either by taking limits of real lengths and extending along a suitable branch from F .S/ into QF .S / ([16], Theorem 3), or using Bonahon’s shearing coordinates [2]. The resulting complex length function .Q . // is a holomorphic function of . Hence we can expand as a Taylor series about p ˙ . / (checking the second derivatives are uniformly bounded) and compare the results:

.q.// D l .p C .// C i

@l C .p . // C O. 2 / @t

.q.// D l .p .// i

@l .p . // C O. 2 /: @t

and

Equating imaginary parts gives @l @l C .p .n // C .p .n // D O.n /: @t @t

(4.1)

It is shown in [37], Proposition 1.8, that up to subsequences, the groups G.n ; n / necessarily converge in such a way that the real analytic structures on p ˙ .n / are close. Viewing l as a real analytic function on F .S /, this gives @l C @l .p .n // C .p .n // D O.n /: @t @t Combining (4.1) and (4.2) and taking limits completes the proof.

(4.2)

4.3 The bending measure conjecture We return to Conjecture 4.2. Although the conjecture in general remains open, Bonahon proved:


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Theorem 4.7 ([3], Theorem 1). There exists an open neighbourhood U of the Fuchsian submanifold F .S/ of QF .S/ such that the bending map ˇ W Q.S / ! ML.S / ML.S /, ˇ.q/ D .plC .q/; pl .q//, is a homeomorphism from U onto its image. This provides an alternative proof of Theorem 4.5. One can modify the proof to show that the length map ` W Q.S/ ! RC RC which sends q 2 Q.S / to .lj.ŒplC .q// ; lj.Œpl .q/ / (where j is a section PML ! ML), is also a homeomorphism from U onto its image (Series, unpublished). The key steps in the proof of Theorem 4.7 are summarized below. The essential idea is to translate the non-degeneracy of the Hessian of the function l C l at the minimum M.; / into a suitable transversality statement about the pleating varieties P ˙ ./ D fq 2 Q.S/ W pl˙ .q/ 2 Œ [ f0gg. Proposition 4.8 ([3], Proposition 3). Let ; 2 ML fill up S. Then the sections p 7! @=@t .p/, p 7! @=@t .p/ from F .S / to T .F .S // are transverse. Proof. By [43], see also [44], the Weil–Petersson symplectic form induces the isomorphism @=@t ! d l between the tangent bundle T .F .S // and the cotangent bundle T .F .S//, see also Theorem 2.11. Thus the statement can be converted into the transversality of the sections d l ; d l W F .S / ! T .F .S //. The intersection of the tangent spaces to these sections at p D M.; / consists of all vectors which can be written in the form Tp .d l /.w/ D Tp .d l /.w/, where w 2 Tp .F .S// and Tp .d l /; Tp .d l / W Tp .F .S // ! Tp .T .F .S /// are the tangent maps of the two sections. One shows by calculation in local coordinates that Tp .d l /.w/ Tp .d l /.w/ is the image of w under the natural isomorphism Tp .F .S // ! Tp .F .S// induced by the Hessian of l C l . Since by [42] (see also Section 2.1.1) the Hessian is non-degenerate, the result follows. Proposition 4.9 ([3], Lemma 7). Let 2 ML. The pleating varieties P ˙ ./ are submanifolds of Q.S/ of dimension 2d C 1 and with boundary F .S /. P Proof. If D i ai ˛i is a rational lamination with weight ai > 0 on all the curves in a pants decomposition A, this follows using complex Fenchel–Nielsen coordinates . ˛i ; ˛i / relative to A ([39], see also [15]). Here ˛i is the complex length given by 2 cosh ˛i =2 D Tr ˛i and ˛i is the complex twist as in the quakebend construction 4.2. A necessary condition for a curve ˛ to be contained in the bending locus is that its complex length be real ([16] Proposition 22), giving d conditions ˛i 2 R. The bending angle on ˛i is the imaginary part of ˛i , giving a further d 1 conditions to ensure the bending angles are in the proportion Œa1 W a2 W : : : W ad specified by Œ. To prove the result in general is substantially harder. One needs to use Bonahon’s shear-bend coordinates [2] which provide a substitute for Fenchel–Nielsen coordinates for laminations which are not rational. { / to be the The final key step is to blow up Q.S/ n F .S / along F .S /. Define Q.S union of Q.S/ n F .S/ with the unit normal bundle N 1 .F .S // of F .S / Q.S / with

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{ suitable topology. Then Q.S/ is a manifold with boundary N 1 .F .S //. The complex structure gives a natural identification of N .F .S // with i T .F .S //. The inclusion P C ./ ! Q.S/ extends uniquely to an embedding P C ./ [ { F .S / ! Q.S/ by sending p 2 F .S/ to n .p/, where n .p/ is the unit normal vector in the direction i@=@t at p. We define P{ C ./ to be the image of P C ./ [ F .S / under this map. Likewise we embed P ./ by the map p 7! n .p/, and define P{ ./ in a similar way. The following result is then a translation of Proposition 4.8. Proposition 4.10 ([3], Proposition 9). The boundaries of P{ C ./ and P{ ./ in N 1 .F .S // have non-empty intersection if and only if , fill up S . If they fill up, the intersection is transverse and is equal to the image of L; under the map { p 7! n .p/ D n .p/ 2 Q.S/. The injectivity of the map ˇ in Theorem 4.7 follows from the transversality of { /. Using invariance of P{ C ./; P{ ./ in a neighbourhood of N 1 .F .S // in Q.S domain, one then shows that ˇ is a homeomorphism in a neighbourhood U of F .S / in Q.S /nF .S/. The density of groups with rational pleating loci (see Conjecture 4.2) is immediate from the density of rational laminations in ML ML.

5 Relationship to Teichmüller geodesics In two papers [6], [7], Choi, Rafi and Series investigated the relationship between Teichmüller geodesics and lines of minima. The first paper derives a combinatorial formula for the Teichmüller distance between the time-t surfaces on these two paths, and the second proves that a line of minima is a Teichmüller quasi-geodesic. The first step is to parameterise both paths in a comparable way. Given laminations ; 2 ML which fill up S , define t D e t ;

t D e t :

We define the line of minima t 7! L t to be the path which takes t to the minimising surface M. t ; t /. As explained in Section 2.5, for each t , there is a unique Riemann surface G t and quadratic differential q t such that t ; t are the horizontal and vertical foliations of q t respectively. The path t 7! G t is a Teichmüller geodesic. For ˛ 2 let l˛ .G t /; l˛ .L t / denote the lengths of ˛ in the hyperbolic metrics on G t ; L t respectively. We say a curve is extremely short if its hyperbolic length is less than some fixed constant 0 > 0 determined in the course of the proofs. For functions f , g we write f g and f g to mean respectively that there are constants C; c > 1, depending only on the topology of S, such that 1 g.x/ C f .x/ cg.x/ C C c

and

1 g.x/ f .x/ cg.x/: c


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Theorem 5.1 ([6], Theorems A and D). The extremely short curves in the hyperbolic metrics on L t and G t coincide. The Teichmüller distance dT between L t and G t is given by 1 l˛ .G t / dT .L t ; G t / D log max C O.1/; ˛ l˛ .L t / 2 where the maximum is taken over all simple closed curves ˛ that are extremely short in G t . In particular, the distance between corresponding thick parts of L t and G t is bounded. Theorem 5.2 ([7], TheoremA). The line of minima t 7! L t , t 2 R, is a quasi-geodesic with respect to the Teichmüller metric dT . In other words, dT .La ; Lb / jb aj for any a; b 2 R. The constants involved in these two theorems depend only on the genus and number of punctures of S . The method is to compare the curves which are short on the time-t surfaces G t ; L t . Perhaps surprisingly, although the same curves are short on the two surfaces, their lengths are not necessarily in bounded proportion. Thus one can construct examples of surfaces and pairs of laminations for which the two paths t 7! G t and t 7! L t are unboundedly far apart. Theorem 5.1 is proved using estimates for short curves in L t and G t which involve two quantities D t .˛/; K t .˛/ defined in Section 5 below. Both D t .˛/ and K t .˛/ have a combinatorial interpretation in terms of the topological relationship between ˛, and : D t .˛/ is large if and only if the relative twisting of and about ˛ is large, while K t .˛/ is large if and only if and have large relative complexity in S n ˛ (the completion of S minus ˛), in the sense that every essential arc or closed curve in S n ˛ must have large intersection with both and . Theorem 5.3 ([32], see also [6], Theorem B). Suppose that ˛ 2 is extremely short on G t . Then 1 maxfD t .˛/; log K t .˛/g: l˛ .G t / Theorem 5.4 ([6], Theorem C). Suppose that ˛ 2 is extremely short on L t . Then p 1 maxfD t .˛/; K t .˛/g: l˛ .L t / The motivation for this approach stems in part from a central ingredient of the proof of the ending lamination theorem. Suppose that N is a hyperbolic 3-manifold homeomorphic to S R. The ending lamination theorem states that N is completely determined by the asymptotic invariants of its two ends. A key step is to show that if

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these end invariants are given by the laminations , , then the curves on S which have short geodesic representatives in N can be characterized in terms of their combinatorial relationship to and . (The relationship is expressed using the complex of curves of S . Roughly speaking, a curve is short in N if and only if the distance between the projections of and to some subsurface Y S is large in the curve complex of Y , see [5].) Since the proofs of Theorems 5.1–5.4 involve some interesting techniques, the remainder of Section 5 will be spent outlining their proofs.

Definitions of the combinatorial parameters. With the terminology explained in Section 6, D t .˛/ is approximately the modulus of the maximal flat annulus round ˛ while log K t .˛/ is approximately the modulus of the maximal expanding annulus. The following definitions are independent of these notions. The term K t .˛/ depends on the (possibly coincident) hyperbolic thick components Y1 , Y2 of the surface G t adjacent to ˛. Let q t be the area 1 quadratic differential on G t whose horizontal and vertical foliations are respectively t and t . Associated to q t is a singular Euclidean metric; we denote the geodesic length of a curve in this metric by l .q t /, see Section 5.1. By definition ²

Y2

Y1 ; K t .˛/ D max l .q t / l .q t /

³

where Yi is the length of the shortest non-trivial non-peripheral simple closed curve on Yi with respect to the q t -metric. If Yi is a pair of pants, there is a slightly different definition, see [6]. To define D t .˛/, we need the notion of the relative twist d˛ .; / of and around ˛. Following [27], for p 2 T .S/ and 2 ML, define tw˛ .; p/ D inf s=l˛ .p/; where l˛ .p/ is the hyperbolic length of ˛ in the surface p, s is the signed hyperbolic distance between the perpendicular projections of the endpoints of a lift of a geodesic in jj at infinity onto a lift of ˛, and the infimum is over all lifts of leaves of jj which intersect ˛. Now define d˛ .; / D inf jtw˛ .; p/ tw˛ .; p/j; p

where the infimum is over all points p 2 T .S /. Then tw˛ .; p/ tw˛ .; p/ is independent of p up to a universal additive constant, see [27]. Note that d˛ .; / agrees up to an additive constant with the definition of subsurface distance between the projections of jj and jj to the annular cover of S with core ˛, as defined in [24] Section 2.4 and used throughout [32], [33]. The definition of D t .˛/ also involves the balance time of ˛, namely the unique time t D t˛ for which i.˛; t˛ / D i.˛; t˛ /. (If i.˛; / D 0 define t˛ D 1, if

143


i.˛; / D 0 define t˛ D 1.) Finally, define D t .˛/ D e 2jt t˛ j d˛ .; /: By [33], Theorem 3.1, the length l˛ .G t / is approximately convex along G and is close to its minimum at t˛ . For later use we note that the twist is closely related to the normalised Fenchel– Nielsen twist coordinate s˛ .p/ defined in Section 5.2: Lemma 5.5 (Minsky [27], Lemma 3.5). For any lamination 2 ML and any two hyperbolic metrics p; p 0 2 T .S/, j.tw˛ .; p/ tw˛ .; p 0 // .s˛ .p/ s˛ .p 0 //j 4:

5.1 Comparison in the thick part of Teichmüller space It is relatively easy to compare the two surfaces G t and L t when both are contained in the thick part of Tthick. / of T .S/, consisting of all surfaces on which the hyperbolic injectivity radius has some fixed lower bound > 0. We have: Theorem 5.6 ([6], Theorem 3.8). If G t ; L t 2 Tthick. / then dT .G t ; L t / D O.1/: To gain some insight into how the two surfaces may be compared, we outline the proof. Quadratic differential metrics. A finite area holomorphic quadratic differential q on a Riemann surface p 2 T .S/ defines a singular Euclidean metric, which away from the singularities is just the Euclidean metric defined by the horizontal and vertical foliations Hq ; Vq , see Section 2.5 and [40], [25]. On the surface G t , by definition Hq and Vq are equivalent to the laminations t ; t respectively. Every simple closed curve in .S; q/ either has a unique q-geodesic representative, or is contained in a family of closed Euclidean geodesics foliating an annulus whose interior contains no singularities, see [40] and Section 6.1. Thus on G t , the horizontal and vertical lengths of are i.; t / and i.; t / respectively, from which it follows that its q-geodesic length l .q t / satisfies

l .q t / i. t ; / C i. t ; /:

(5.1)

Short markings. Define a marking of a surface S to be a collection of pants curves A, together with a dual set of curves which intersect each curve in A either once or twice, depending on the topology (see Section 3 in [6]). The marking is short with respect to a hyperbolic metric on S, if there is a uniform upper bound to the lengths of all curves in A, and if in addition the dual curves are as short as possible among all choices which have the same intersections with A. By a result of Bers [1], one can

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P always choose A so that ˛2A l˛ is less than some universal upper bound depending only on the genus of S. Since curves in A may be arbitrarily short, there is in general no upper bound to the length of the dual curves. However in Tthick , the lengths of the dual curves are uniformly bounded above. It follows, see e.g. Lemma 4.7 in [26], that for any hyperbolic metric h 2 Tthick and short marking Mh , we have

l .h/ i.; Mh / and lMh .h/ 1

(5.2)

for any 2 , where lMh .h/; i.; Mh / are the sums over curves in Mh of the lengths and intersections numbers with respectively. Proof of Theorem 5.6. Let MG t and ML t be short markings for G t , L t respectively. In Tthick , any two metrics, in particular the hyperbolic metric and the quadratic differential metric, are comparable. Hence using (5.1) and (5.2):

lMG t .G t / lMG t .q t / i.MG t ; t / C i.MG t ; t / l t .G t / C l t .G t /:

(5.3)

In a similar way

lML t .G t / lML t .q t / i.ML t ; t / C i.ML t ; t / l t .L t / C l t .L t /: Since L t minimizes l t .h/ C l t .h/ over all hyperbolic metrics h 2 T .S /: l t .G t / C l t .G t / l t .L t / C l t .L t /: Putting together the preceding three equations, we have

lMG t .G t / lML t .G t /: We deduce from (5.2) that

lML t .G t / 1:

(5.4)

It is not hard to see that the set of surfaces for which a given marking M has a diameter bounded by B > 0 has bounded diameter with respect to the Teichmüller distance, with a bound which depends only on B. The result follows.

5.2 Comparison in the thin part of Teichmüller space Let us assume for a moment the results of Theorems 5.3 and 5.4. If ˛ 2 is short in the hyperbolic metrics on two surfaces p; p 0 2 T .S /, then Kerckhoff’s formula [17] dT .p; p 0 / D

1 Ext .p/ log sup 0 2 2 Ext .p /

(5.5)

(where Ext./ is the extremal length of ) together with Theorem 6.3 below, shows that the expression 1 l˛ .G t / log max ˛ l˛ .L t / 2


145

in Theorem 5.1 is an approximate lower bound for dT .G t ; L t /. To turn this into a precise estimate, one uses Minsky’s product region theorem. This is worth explaining in its own right. Minsky’s product region theorem. This theorem, proved in [27], is an approximate formula for the Teichmüller metric dT in the part of T .S / in which a collection A D f˛1 ; : : : ; ˛n g of disjoint, homotopically distinct, simple closed curves on S are y if necessary, we fix Fenchel–Nielsen short. Extending A to a pants decomposition A y coordinates .l˛i ; s˛i / on A, where s˛ is the normalised twist, that is the hyperbolic signed distance twisted away form some base point divided by the length l˛i . Let Tthin .A; 0 / T .S/ be the subset on which all curves ˛i 2 A have hyperbolic length at most 0 . Let SA denote the surface obtained from S by removing all the curves in A and replacing the resulting boundary components by punctures. For ˛ 2 A, let H˛ be the hyperbolic plane and let dH˛ be half the usual hyperbolic metric on H˛ . Define …˛ W T .S/ ! H˛ by …˛ .p/ D s˛ .p/ C i= l˛ .p/ 2 H˛ : Also define …0 W T .S/ ! T .SA / by forgetting the coordinates of the curves in A and keeping the same Fenchel–Nielsen coordinates for the remaining surface. Theorem 5.7 (Minsky [27]). Let p; p 0 2 Tthin .A; 0 /. Then dT .p; p 0 / D maxfdT .SA / .…0 .p/; …0 .p 0 //; dH˛i .…i .p/; …i .p 0 /g ˙ O.1/: i

A consequence of this formula is that unless the difference between the twist coordinates s˛ .p/; s˛ .p 0 / is extremely large in comparison to l˛ .p/; l˛ .p 0 /, their contribution to dT .p; p 0 / can be neglected. Lemma 5.5 shows that we can replace js˛ .p/ s˛ .p 0 /j with jtw˛ .; p/ tw˛ .; ; p 0 /j, for any 2 ML. In fact: Corollary 5.8 ([6], Corollary 4.7). Suppose that p; p 0 2 Tthin .˛; 0 / and that there exists 2 ML such that jtw˛ .; r/j lr .˛/ D O.1/ for r D p; p 0 . Then ˇ ˇ ˇ l˛ .p/ ˇˇ 0 ˇ ˙ O.1/: dH˛ .…˛ .p/; …˛ .p // D ˇlog l .p 0 / ˇ ˛

Proof of Theorem 5.1. Theorems 5.3 and 5.4 show that, with suitable choices of constants, the extremely short curves on L t and G t coincide. To use Minsky’s product theorem to deduce Theorem 5.1 from Theorems 5.3 and 5.4, we need to estimate the Teichmüller distance between the hyperbolic thick components of G t and L t , and also the difference between the Fenchel–Nielsen twist coordinates corresponding to the short curves in the two surfaces. The first part is done by an elaboration of the method in Theorem 5.6. One needs a substitute for the fact that the surface L t minimises l t C l t over T .S /. Let Q be a thick part of the hyperbolic surface L t . One shows ([6], Proposition 7.4)

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that the contribution to l t C l t coming from the intersection with Q is less, up to multiplicative constants, than the similar sum for the surface G t . For the second part, by Corollary 5.8, it is enough to show that jtw˛ .; p/jl˛ .p/ D O.1/ for p D L t ; G t and some 2 ML, where as usual l˛ .p/ means the hyperbolic length of ˛ in the hyperbolic metric uniformising p. On the surface L t , the result follows from equation (2.3) with D if t > t˛ and D if t < t˛ , see [6], Theorems 6.2 and 6.9. For the surface G t , we use the analogous notion of the twist tw˛ .; q/ of about ˛ with respect to a quadratic differential metric q, introduced by Rafi [33]. We have: Proposition 5.9 ([33], Theorem 4.3, [6], Proposition 5.7). Suppose that p 2 T .S / and that q is a compatible quadratic differential. For any geodesic lamination intersecting ˛, we have l˛ .p/jtw˛ .; p/ tw˛ .; q/j D O.1/: As explained in Section 6, a hyperbolically short curve ˛ on G t is surrounded by an annulus of large modulus which in the associated quadratic differential metric q t is either flat (that is, isometric to a Euclidean cylinder) or expanding. It follows immediately from the Gauss–Bonnet theorem (6.1) that the contribution to tw˛ .; q/ and tw˛ .; q/ from an expanding annulus is bounded. In a flat annulus F , at the balance time t˛ the leaves of the horizontal and vertical foliations make angles of ˙ =4 with @F . It is an exercise in Euclidean geometry to determine the angle with the boundary at time t t˛ , from which one deduces the required bounds on twists with the aid of Proposition 5.9. Proof of Theorem 5.2. It follows from Theorem 5.1, that along intervals on which either there are no short curves, or on which D t .˛/ dominates K t .˛/ for all short curves ˛, the surfaces L t and G t remain a bounded distance apart. However the path L t may deviate arbitrarily far from G t along time intervals on which K t .˛/ is large and dominates D t .˛/. To prove Theorem 5.2, in addition to Theorems 5.3 and 5.4, we therefore need to control distance along intervals along which K t .˛/ is large. Let S˛ be the surface obtained by cutting S along a short curve ˛ and replacing the two resulting boundary components by punctures. The following somewhat surprising result shows that in this situation, the distance in T .S / is dominated by the distance in the thick part T .S˛ /. Proposition 5.10 ([7], Theorem D). If K t .˛/ is sufficiently large for all t 2 Œa; b, the distance in T .S˛ / between the restrictions of Ga and Gb to S˛ is equal to b a, up to an additive error that is bounded by a constant depending only on the topology of S . The proof of Theorem 5.2 also requires a detailed comparison of the rates of change of D t .˛/ and K t .˛/ with t . The situation is further complicated because the family of


147

curves which are short on L t varies with t , so that the intervals along which different curves ˛ are short may overlap. One needs a somewhat involved induction to complete the proof. Proof of Theorem 5.4. The main tool in the proof of Theorem 5.4 is the formula (2.1) for the variation of length with respect to Fenchel–Nielsen twist, together with the extension proved by Series [35] for variation with respect to the lengths of pants curves. To explain, if 2 , its hyperbolic length l is a real analytic function of the Fenchel–Nielsen coordinates .l˛ ; t˛ / for T .S / relative to a set A of pants curves of S . Series’ formula is an expression for @l =@l˛ , analogous to the formula (2.1) for @l =@t˛ . As in Section 3.4, homotope to run alternately along and perpendicular to the boundaries of the pairs of pants in the decomposition defined by A. The formula is a sum similar to one in (2.1), but involving additionally both the distance twisted by round the pants curves and the complex distance between a given segment of across a pair of pants P and the common perpendicular joining the corresponding components of @P . Imposing the condition that @.l t Cl t /=@l˛ D 0 for short curves ˛ gives additional constraints which need to be satisfied at the minimum of the function l t C l t . It requires considerable work to bring these results into the form in the statement of Theorem 5.4. The proof of Theorem 5.3 is explained in Section 6.4 below.

6 Short curves on Teichmüller geodesics In [25], Minsky analyses the curves which are hyperbolically short in a surface on which one has a metric defined by a quadratic differential q as in Section 5.1. In a series of papers [32], [33], [34], Rafi has developed this into a technique for estimating the lengths of curves which are short on a surface on a Teichmüller geodesic, in particular implying a proof of Theorem 5.3. Since this work contains some important ideas, we conclude this chapter with a brief summary of their results.

6.1 Flat and expanding annuli Let q be a finite area quadratic differential on a Riemann surface p 2 T .S / and let A S be an annulus with piecewise smooth boundary. We say A is regular if its boundary components @0 , @1 are q-equidistant and monotonically curved in the sense that their acceleration vectors always points into A, with suitable definition at singular points, namely the zeros of q. The total curvature of @i is Z X .@i / D .p/ C Œ .P /; @i

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where the sum is over all singular points P 2 @i and .P / is the interior angle at P . The annulus A is called flat if .@i / D 0 for each i and expanding if .@i / ¤ 0 for i D 0; 1. By the Gauss–Bonnet theorem, X ordP (6.1) .@0 / C .@1 / D where the sum is over the singularities of q in the interior of A and ordP is the order of the zero at P . A regular annulus is primitive if it contains no singularities of q in its interior. It follows from (6.1) that a primitive annulus is either flat or expanding. A flat annulus is necessarily primitive, and is foliated by Euclidean geodesics homotopic to the boundaries. Thus a flat annulus is isometric to a cylinder obtained as the quotient of a Euclidean rectangle in R2 . Expanding annuli are exemplified by an annulus bounded by a pair of concentric circles in R2 . In this case, with a suitable choice of sign convention, .@0 / D 2 , .@1 / D 2 . Any expanding annulus is coarsely isometric to this example [25]. Theorem 6.1 ([25], Theorem 4.5, [32]). Let A S be an annulus that is primitive with respect to q and with boundaries @0 and @1 . Let d be the q-distance between @0 and @1 . Then either (i) A is flat and mod A D d= l@0 .q/ or (ii) A is expanding and mod A logŒd= l@0 .q/. Theorem 6.2 ([25], Theorem 4.6). Let p 2 T .S / be a Riemann surface and let q be a quadratic differential on p. Let A be any homotopically non-trivial annulus whose modulus on p is sufficiently large. Then A contains an annulus B that is primitive with respect to q and such that mod A mod B. (The statement of Theorem 4.6 in [25] should read mod A m0 not mod A m0 .)

6.2 Moduli of annuli and hyperbolically short curves One can link the hyperbolic and quadratic differential metrics on a surface using annuli of large modulus. Let h be a hyperbolic metric on S . If ˛ is short in h, Maskit [22] showed that the extremal length Ext.˛/ and hyperbolic length l˛ .h/ are comparable, up to multiplicative constants. Moreover, there is an embedded collar C.˛/ around ˛ whose modulus is comparable to 1= l˛ .h/ (see [27] for an explicit calculation), and therefore also to 1= Ext.˛/. Combining with the results in Section 6.1, this gives Theorem 6.3 ([6], Theorem 5.2). If ˛ is a simple closed curve which is sufficiently short in a hyperbolic metric h on S, then for any compatible quadratic differential q, there is an annulus A that is primitive with respect to q with core homotopic to ˛ such that 1 mod.A/: l˛ .h/


149

6.3 Rafi’s thick-thin decomposition for the q-metric Rafi [33] used the above results to analyse the relationship between the q-metric on a Riemann surface p 2 T .S/ and the uniformising hyperbolic metric h in the thick components of the standard thick-thin decomposition of h. The main result is that on the hyperbolic thick parts of .S; h/, the two metrics are comparable, up to a factor which depends only on the moduli of the expanding annuli around the short curves in the boundary of the thick components. For a subsurface Y of S , let Yy be the unique subsurface of .S; q/ with q-geodesic boundary in the homotopy class of Y that is disjoint from all the maximal flat annuli containing boundary components of @Y . Theorem 6.4 (Rafi [33]). Let p 2 T .S/ be a Riemann surface, let h be the hyperbolic metric that uniformises p and let Y be a thick component of the hyperbolic thick-thin decomposition of .S; h/. Then there exists Y > 0 such that (i) diamq Yy Y ; (ii) for any non-peripheral simple closed curve Y , we have

l .q/ Y l .h/: In fact Y is the length of the q-shortest non-peripheral simple closed curve contained in Yy unless Y is a pair of pants, in which case Y D maxflq .i /g where i , i D 1; 2; 3, are the boundary curves of Yy .

6.4 Proof of Theorem 5.3 This is an application of Theorem 6.1. Let R be a Riemann surface and let q be a quadratic differential on R. It follows as above from Equation (6.1) that every simple closed curve on .R; q/ either has a unique q-geodesic representative, or is contained in a family of closed Euclidean geodesics which foliate a flat annulus. Denote by F . / the maximal flat annulus, which necessarily contains all q-geodesic representatives of . (If the geodesic representative of is unique, then F . / is taken to be the degenerate annulus containing this geodesic alone.) Denote the (possibly coincident) boundary curves of F ./ by @0 , @1 and consider the q-equidistant curves from @i outside F . /. Let @O i denote the first such curve which is not embedded. If @O i ¤ @i , then the pair @i , @O i bounds a region Ei ./ whose interior is an annulus with core homotopic to , and which by its construction is regular and expanding. Let h be the uniformising hyperbolic metric on R. Combining Theorems 6.1, 6.2 and 6.3 we have: Corollary 6.5. If ˛ is an extremely short curve on .R; h/, then 1 max f mod F .˛/; mod E0 .˛/; mod E1 .˛/g : l˛ .h/

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The proof of Theorem 5.3 then follows from the following two propositions. Proposition 6.6 ([6], Proposition 5.6). Let q D q t denote the quadratic differential on the surface G t whose horizontal and vertical foliations are t , t respectively. Let ˛ 2 be a curve that is neither vertical nor horizontal. Then mod F t .˛/ D t .˛/: Proposition 6.7 ([6], Proposition 5.7). Let p 2 T .S / be a Riemann surface with compatible quadratic differential q. Suppose that ˛ is extremely short in the uniformising hyperbolic metric h and let Y be a thick component of the hyperbolic thick-thin decomposition of .S; h/, one of whose boundary components is ˛. Let ˛O be the q-geodesic representative of ˛ on the boundary of Yy and let E.˛/ be a maximal expanding annulus on the same side of ˛O as Yy . Then mod E.˛/ log

Y : l˛ .q/

Proposition 6.6 is an exercise in Euclidean geometry and Proposition 6.7 follows from Theorems 6.1 and 6.4.

6.5 Rafi’s combinatorial formula for Teichmüller distance Rafi [34] has developed the above ideas further into a combinatorial expression for the Teichmüller distance between any two surfaces p; p 0 2 T .S / in which different families of curves may be short. Let h, h0 be the hyperbolic metrics uniformising p, p 0 respectively, and let Mp , Mp0 be short markings for h, h0 (see 5.1). Then: Theorem 6.8 ([34], Theorem 6.1). X X log d˛ .Mp ; Mp0 / k dY .Mp ; Mp0 / k C dT .S/ .p; p 0 / Y

˛…

1 1 C max log C max dH˛2 .p; p 0 /; C max log ˛2

lˇ .h/ ˇ 2 p0 lˇ .h0 / ˇ 2 p

(6.2)

where p , p0 , and are defined as follows: p D f˛ 2 W l˛ .h/ < ; l˛ .h0 / > g; p0 D f˛ 2 W l˛ .h0 / < ; l˛ .h/ > g; D f˛ 2 W l˛ .h/ < ; l˛ .h0 / < g: Here dY .Mp ; Mp0 / is the distance between the projections of the union of the curves in the markings Mp ; Mp0 in the curve complex of Y and d˛ .Mp ; Mp0 / is the relative twist of the curves in Mp ; Mp0 round ˛. For X 0, the function ŒX k takes the value 0 when X < k and X when X k. This formula is not needed in the proofs of Theorems 5.1 and 5.2.


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References [1]

L. Bers,An inequality for Riemann surfaces. In Differential geometry and complex analysis (I. Chavel and H. Farkas, eds.), Springer-Verlag, Berlin 1985, 87–93. 143

[2]

F. Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Ann. Fac. Sci. Toulouse Math. 5 (1996), 233–297. 138, 139

[3]

F. Bonahon, Kleinian groups which are almost Fuchsian. J. Reine Angew. Math. 10 (2005), 1–15. 124, 138, 139, 140

[4]

F. Bonahon and J-P. Otal, Laminations mesurées de plissage des variétés hyperboliques de dimension 3. Ann. of Math. 160 (2004), 1013–1055. 135

[5]

J. Brock, R. Canary, and Y. Minsky, The classification of Kleinian surface groups, II: The Ending Lamination Conjecture. Ann. of Math., to appear; arXiv:math/0412006v2 [math.GT]. 142

[6]

Y. Choi, K. Rafi, and C. Series, Lines of minima and Teichmüller geodesics. Geom. Funct. Anal. 18 (2008), 698–754. 124, 134, 140, 141, 142, 143, 145, 146, 148, 150

[7]

Y. Choi, K. Rafi, and C. Series, Lines of minima are uniformly quasi-geodesic. Pacific J. Math. 237 (2008), 21–44. 124, 140, 141, 146

[8]

R. Díaz and C. Series, Examples of pleating varieties for twice punctured tori. Trans. Amer. Math. Soc. 356 (2003), 621–658. 124, 132, 133, 137

[9]

R. Díaz and C. Series, Limit points of lines of minima in Thurston’s boundary of Teichmüller space. Algebr. Geom. Topol. 3 (2003), 207–234. 124, 134, 135

[10] D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. In Analytical and geometric aspects of hyperbolic space (D. B. A. Epstein, ed.), London Math. Soc. Lecture Note Ser. 111, Cambridge University Press, Cambridge 1987, 112–253. 126, 135, 137, 138 [11] A. Fathi, F. Laudenbach, and V. Poénaru (eds.), Travaux de Thurston sur les surfaces. Astérisque 66–67 (1979). 125, 127, 130, 135 [12] F. Gardiner and H. Masur, Extremal length geometry of Teichmüller space. Complex Variables Theory Appl. 16 (1991), 209–237. 124, 128, 130, 131 [13] J. Hu, Earthquakes on the hyperbolic plane. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2012, 71—122. 126 [14] J. Hubbard and H. Masur, Quadratic differentials and foliations. Acta Math. 142 (1979), 221–274. 130 [15] L. Keen and C. Series, How to bend pairs of punctured tori. In Lipa’s Legacy (J. Dodziuk and L. Keen, eds.), Contemp. Math. 211, Amer. Math. Soc, Providence, R.I., 1997, 359–388. 131, 137, 139 [16] L. Keen and C.Series, Pleating invariants for punctured torus groups. Topology 43 (2004), 447–491. 133, 136, 138, 139 [17] S. Kerckhoff, The asymptotic geometry of Teichmüller space. Topology 19 (1980), 2–41. 144 [18] S. Kerckhoff, The Nielsen realization problem. Ann. of Math. 117 (1983), 235–265. 125, 126 [19] S. Kerckhoff, Lines of minima in Teichmüller space. Duke Math J. 65 (1992), 187–213. 123, 125, 126, 127, 128, 129, 130, 132, 134

152

Caroline Series

[20] A. Lenzhen, Teichmüller geodesics that do not have a limit in P MF . Geom. Topology 12 (2008), 177–197. 134 [21] A. Marden, Outer circles: An introduction to hyperbolic 3-manifolds. Cambridge University Press, Cambridge 2007. 135 [22] B. Maskit, Comparison of extremal and hyperbolic lengths. Ann. Acad. Sci. Fenn. 10 (1985), 381–386. 135, 148 [23] H. Masur, Two boundaries of Teichmüller space. Duke Math. J. 49 (1982), 183–190. 134 [24] H. Masur and Y. Minsky, Geometry of the complex of curves II: Hierarchical structure. Geom. Funct. Anal. 10 (2000), 902–974. 142 [25] Y. Minsky, Harmonic maps, length, and energy in Teichmüller space. J. Differential Geom. 35 (1992), 151–217. 125, 143, 147, 148 [26] Y. Minsky, Teichmüller geodesics and ends of hyperbolic 3-manifolds. Topology 32 (1993), 625–647. 144 [27] Y. Minsky, Extremal length estimates and product regions in Teichmüller space. Duke Math. J. 83 (1996), 249–286. 135, 142, 143, 145, 148 [28] J-P. Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3. Astérisque 235 (1996). 127, 130 [29] A. Papadopoulos, Geometric intersection functions and Hamiltonian flows on the space of measured foliations on a surface. Pacific J. Math. 124 (1986), 375–402. 126 [30] J. Parker and C. Series, Bending formulae for convex hull boundaries. J. d’Analyse Math. 67 (1995), 165–198. 133 [31] R. C. Penner with J. Harer, Combinatorics of train tracks. Ann. of Math. Stud. 125, Princeton University Press, Princeton, N.J., 1992. 128 [32] K. Rafi, A characterization of short curves of a Teichmüller geodesic. Geom. Topol. 9 (2005), 179–202. 125, 141, 142, 147, 148 [33] K. Rafi, Thick-thin decomposition of quadratic differentials. Math. Res. Lett. 14 (2007), 333–342. 125, 142, 143, 146, 147, 149 [34] K. Rafi, A combinatorial model for the Teichmüller metric. Geom. Funct. Anal. 17 (2007), 936–659. 125, 147, 150 [35] C. Series, An extension of Wolpert’s derivative formula. Pacific J. Math. 197 (2001), 223–239. 147 [36] C. Series, On Kerckhoff minima and pleating varieties for quasifuchsian groups. Geom. Dedicata 88 (2001), 211–237. 124, 130, 131, 132, 135 [37] C. Series, Limits of quasifuchsian groups with small bending. Duke Math. J. 128 (2005), 285–329. 124, 135, 136, 138 [38] C. Series, Thurston’s bending measure conjecture for once punctured torus groups. In Spaces of Kleinian groups (Y. Minsky, M. Sakuma and C. Series, eds.), London Math. Soc. Lecture Note Ser. 329, Cambridge University Press, Cambridge 2006, 75–90. 136 [39] S. P. Tan, Complex Fenchel-Nielsen coordinates for quasifuchsian structures. Internat. J. Math. 5 (1994), 239–251. 136, 139 [40] K. Strebel, Quadratic differentials. Ergeb. Math. Grenzgeb. (3) 5, Springer-Verlag, Berlin 1984. 130, 143


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[41] W. Thurston, Earthquakes in two dimensional hyperbolic geometry. In Low-dimensional geometry and Kleinian groups (D. B. A. Epstein, ed.), London Math. Soc. Lecture Note Ser. 112, University Press, Cambridge 1986, 91–112. 126 [42] S. Wolpert, The Fenchel-Nielsen deformation. Ann. of Math. 115 (1982), 50–528. 126, 139 [43] S. Wolpert, On the symplectic geometry of deformations of a hyperbolic surface. Ann. Math. 117 (1983), 207–234. 126, 129, 139 [44] S. Wolpert, The Weil–Petersson metric geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 47–64. 139

Part B

The group theory, 3

Chapter 4

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2

3

4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 General questions on discrete groups and discrete transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summaries of properties of groups and spaces in this chapter . . . . . . . 2.1 Properties of arithmetic groups . . . . . . . . . . . . . . . . . . . 2.2 Properties of actions of arithmetic groups on symmetric spaces X 2.3 Properties of mapping class groups Modg;n . . . . . . . . . . . . . 2.4 Properties of actions of mapping class groups Modg;n on Teichmüller spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Properties of outer automorphism groups Out.Fn / . . . . . . . . . . 2.6 Properties of outer space Xn and the action of Out.Fn / on Xn . . . 2.7 Properties of Coxeter groups . . . . . . . . . . . . . . . . . . . . . 2.8 Properties of hyperbolic groups . . . . . . . . . . . . . . . . . . . . How discrete groups and proper transformation groups arise . . . . . . . 3.1 Finitely generated groups, Cayley graphs and Rips complexes . . . 3.2 Rational numbers and p-adic norms . . . . . . . . . . . . . . . . . 3.3 Discrete subgroups of topological groups . . . . . . . . . . . . . . 3.4 Fundamental groups and universal covering spaces . . . . . . . . . 3.5 Moduli spaces and mapping class groups . . . . . . . . . . . . . . 3.6 Outer automorphism groups . . . . . . . . . . . . . . . . . . . . . 3.7 Combinatorial group theory . . . . . . . . . . . . . . . . . . . . . 3.8 Symmetries of spaces and structures on these spaces . . . . . . . . Arithmetic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . 4.2 Generalizations of arithmetic groups: non-arithmetic lattices . . . . 4.3 Generalizations of arithmetic groups: S -arithmetic subgroups . . . 4.4 Generalizations of arithmetic groups: non-lattice discrete subgroups and Patterson–Sullivan theory . . . . . . . . . . . . . . . . . . . . 4.5 Symmetric spaces and actions of arithmetic groups . . . . . . . . . 4.6 Fundamental domains and generalizations . . . . . . . . . . . . . . 4.7 Fundamental domains for Fuchsian groups and applications to compactification . . . . . . . . . . . . . . . . . . . . . . . . . .

. 158 . 158 . . . . .

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182 187 189 190 192 194 194 195 196 196 197 197 198 199 199 200 204 205

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4.8 Minkowski reduction theory for SL.n; Z/ . . . . . . . . . . . . . 4.9 Reduction theory for general arithmetic groups . . . . . . . . . . 4.10 Precise reduction theory for arithmetic groups . . . . . . . . . . . 4.11 Combinatorial properties of arithmetic groups: finite presentation and bounded generation . . . . . . . . . . . . . . . . . . . . . . . 4.12 Subgroups and overgroups . . . . . . . . . . . . . . . . . . . . . 4.13 Borel density theorem . . . . . . . . . . . . . . . . . . . . . . . . 4.14 The Tits alternative and exponential growth . . . . . . . . . . . . 4.15 Ends of groups and locally symmetric spaces . . . . . . . . . . . 4.16 Compactifications and boundaries of symmetric spaces . . . . . . 4.17 Baily–Borel compactification of locally symmetric spaces . . . . 4.18 Borel–Serre compactification of locally symmetric spaces and cohomological properties of arithmetic groups . . . . . . . . 4.19 The universal spaces E and E via the Borel–Serre partial compactification . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Mapping class groups Modg;n . . . . . . . . . . . . . . . . . . . . . . 5.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . 5.2 Teichmüller spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Properties of Teichmüller spaces . . . . . . . . . . . . . . . . . . 5.4 Metrics on Teichmüller spaces . . . . . . . . . . . . . . . . . . . 5.5 Compactifications and boundaries of Teichmüller spaces . . . . . 5.6 Curve complexes and boundaries of partial compactifications . . . 5.7 Universal spaces for proper actions . . . . . . . . . . . . . . . . . 5.8 Cohomological properties of Modg;n . . . . . . . . . . . . . . . . 5.9 Pants decompositions and the Bers constant . . . . . . . . . . . . 5.10 Fundamental domains and rough fundamental domains . . . . . . 5.11 Generalized Minkowski reduction and fundamental domains . . . 5.12 Compactifications of moduli spaces and a conjecture of Bers . . . 5.13 Geometric analysis on moduli spaces . . . . . . . . . . . . . . . . 6 Interactions between locally symmetric spaces and moduli spaces of Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 The Jacobian map and the Schottky problem . . . . . . . . . . . . 6.2 The coarse Schottky problem . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 217 . . 221 . . 224 . . . . . . .

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1 Introduction 1.1 Summary In this chapter, we discuss similarities, differences and interaction between two natural and important classes of groups: arithmetic subgroups of Lie groups G and mapping class groups Modg;n of surfaces of genus g with n punctures. We also mention

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similar properties and problems for related groups such as outer automorphism groups Out.Fn /, Coxeter groups and hyperbolic groups. Since groups are often effectively studied by suitable spaces on which they act, we also discuss related properties of actions of arithmetic groups on symmetric spaces and actions of mapping class groups on Teichmüller spaces, hoping to get across the point that it is the existence of actions on good spaces that makes the groups interesting and special, and it is also the presence of large group actions that also makes the spaces interesting. Interaction between locally symmetric spaces and moduli spaces of Riemann surfaces through the example of the Jacobian map will also be discussed in the last part of this chapter. Since reduction theory, i.e., finding good fundamental domains for proper actions of discrete groups, is crucial to transformation group theory, i.e., to understand the algebraic structures of groups, properties of group actions and geometry, topology and compactifications of the quotient spaces, we discuss many different approaches to reduction theory of arithmetic groups acting on symmetric spaces. These results for arithmetic groups motivate some results on fundamental domains for the action of mapping class groups on Teichmüller spaces. For example, the Minkowski reduction theory of quadratic forms is generalized to the action of Modg D Modg;0 on the Teichmüller space Tg to construct an intrinsic fundamental domain consisting of finitely many cells, solving a weaker version of a folklore conjecture in the theory of Teichmüller spaces. On several aspects, more results are known for arithmetic groups, and we hope that discussion of results for arithmetic groups will suggest corresponding results for mapping class groups and hence increase interactions between the two classes of groups. In fact, in writing this survey and following the philosophy of this chapter, we noticed the natural procedure in §5.12 of constructing the Deligne–Mumford compactification of the moduli space of Riemann surfaces from the Bers compactification of the Teichmüller space by applying the general procedure of Satake compactifications of locally symmetric spaces, i.e., how to pass from a compactification of a symmetric space to a compactification of a locally symmetric space by making use of the reduction theory for arithmetic groups. The layout of this chapter is as follows. In the rest of this introduction, we discuss some general questions about discrete groups, group actions and transformation group theories. In §2, we summarize results on arithmetic subgroups of semisimple Lie groups G and mapping class groups Modg;n of surfaces of genus g with n punctures, and their actions on symmetric spaces of non-compact type and Teichmüller spaces respectively.1 For comparison and for the sake of completeness, we also discuss corresponding properties of three related classes of groups: outer automorphism groups of free groups, Coxeter groups and hyperbolic groups. In §3, we describe several sources where discrete groups and discrete transformation groups arise. In §4 and §5 we give definitions and details of some of the properties listed earlier in §2 for arithmetic groups and mapping class groups. In the last section, §6, we deal with the coarse Schottky problem, a large-scale geometric generalization of the classical Schottky problem of characterizing the Jacobian varieties among abelian varieties, 1 The

lists are certainly not complete and only results known to us are listed.

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i.e., the image of the Jacobian map from the moduli space Mg of compact Riemann surfaces of genus g to the Siegel modular variety Ag , an important Hermitian locally symmetric space, which is equal to the moduli space of principally polarized abelian varieties of dimension g.

1.2 General questions on discrete groups and discrete transformation groups Groups are fundamental objects and they describe symmetry in mathematics and sciences. Basically, there are two kinds of groups: (1) discrete groups, i.e., groups with the discrete topology, (2) non-discrete (or continuous) groups, in particular, Lie groups. These two classes of groups are closely related in many ways and embeddings of discrete groups into non-discrete groups give rise to interesting transformation groups, as seen in various results about the fundamental pair of groups Z R. Of course, any group can be given the discrete topology and hence considered as a discrete group. On the other hand, as far as discrete groups are concerned, it is probably most natural and fruitful to study groups that occur naturally as discrete subgroups of topological groups or as discrete transformation groups, i.e., discrete groups acting properly discontinuously on topological spaces that have some reasonable properties, for example, being locally compact. In this expository chapter, we discuss two important classes of infinite discrete groups: (1) arithmetic subgroups of linear algebraic groups such as SL.n; Z/ and GL.n; Z/ and discrete subgroups of Lie groups G; (2) mapping class groups Modg;n of compact orientable surfaces Sg;n of genus g with n points removed (i.e., with n punctures). We will discuss similarities and differences between these two classes of groups and their properties. There have been several excellent surveys on properties of mapping class groups from different perspectives and comparison with arithmetic groups, for example, [123], [169], [188], [190], [317], [54], [164], [321]. We hope that the current survey is complementary to the existing ones. We will also discuss similar properties of some related groups. For example, the family of outer automorphism groups Out.Fn / of free groups Fn is closely related to the two classes of groups mentioned above, and we will also discuss briefly their properties. There are also excellent surveys on these topics such as [73], [344], [412]. Other closely related families of groups discussed briefly in this chapter include Coxeter groups and hyperbolic groups. Since groups first arose as symmetries or transformation groups of number fields and differential equations, and since many properties of groups can be understood by studying their actions on suitable spaces, we will emphasize the point of view of geometric transformation groups. We will also study two classes of spaces naturally associated with the above two classes of groups: (1) symmetric spaces of semisimple Lie groups and more general homogeneous spaces; (2) Teichmüller spaces of marked


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Riemann surfaces. Furthermore, we will study actions of these groups on such spaces and their quotients, which are locally symmetric spaces, and moduli spaces of Riemann surfaces (or algebraic curves) respectively. Besides being important for understanding properties of the groups, these spaces are also interesting in themselves. In some sense, the groups are studied in order to understand the spaces on which the groups act and also to understand the quotients of the actions. The groups themselves have sometimes played a secondary role in some applications. For example, Teichmüller spaces and actions of mapping class groups on them were originally studied in order to understand the moduli spaces of Riemann surfaces. It is often the case that a group action contains more information than the quotient, as seen, for example, in the context of equivariant cohomology theory. Given a discrete group , the following problems are natural: (1) Finite generation of and some variants, for example, bounded generation. In general, it is much more difficult to find explicit generators than to prove existence. (2) Finite presentation of and derivation or understanding other properties from the presentation. (3) Internal structures such as finite subgroups, subgroups of finite index, normal subgroups of , and overgroups of finite index (i.e., groups that contain as subgroups of finite index). Existence of torsion-free subgroups of finite index is important and allows one to define virtual properties of . (4) Combinatorial properties of such as the word problem, the conjugacy problem, and the isomorphism problem for classes of groups that contain . (5) Other finiteness properties of such as FP1 , FP, FL in homological algebra and the existence of CW-models of classifying (or universal) spaces E for proper and fixed point free actions of and E for proper actions of , which satisfy various finiteness conditions, for example, existence of only finitely many cells in all dimensions modulo , of finitely many cells in each dimension modulo , or E and E being of finite dimension. (6) Cohomology groups and cohomological invariants of such as cohomological dimension and the Euler characteristic, and properties such as the Poincaré duality and generalized Poincaré duality properties. (7) Other algebraic invariants of and its associated group ring Z such as K-groups and L-groups of the group ring Z in the algebraic and geometric topology. (8) Large-scale properties of endowed with word metrics (or equivalently its Cayley graphs with each edge of length 1) such as growth rate, quasi-rigidity properties, asymptotic dimension, the rationality of the growth series, and large-scale properties of infinite subgroups of such as bounds on the distortion function. (9) External properties: existence of linear representations and their properties such as Property T for -actions on Hilbert spaces, and existence of actions on topological spaces and manifolds (i.e., non-linear representations) and their properties such as Property FA of Serre for actions on trees.

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(10) Realizations of as subgroups of linear groups, discrete subgroups of Lie groups and other topological groups. (11) Ends, compactifications and boundaries of and related -spaces when the group is infinite. As pointed out earlier, a notion closely related to the one of discrete group is that of topological transformation group. Many of the above properties of groups can be studied and understood by using actions of on suitable topological spaces. On the other hand, finding the right space is often not easy, and general groups probably do not act on spaces with desirable properties since such actions usually impose some conditions on the groups. The groups studied in this chapter do act on good spaces in the sense that the spaces have rich structures that can be described and understood, and hence they are special and interesting from this point of view. For discrete subgroups of Lie groups G, it is relatively easy to find spaces on which these groups act. For example, there are natural classes of homogeneous spaces associated with the Lie groups G on which the discrete groups act. But for other discrete groups such as the mapping class groups of surfaces Modg;n and the outer automorphism groups of free groups Out.Fn /, the situation is more complicated and the construction of analogous spaces is less direct. For the mapping class groups of general manifolds and for automorphism groups of general discrete groups, the construction of analogous spaces are not known and might not be possible. This makes Modg;n and Out.Fn / really special. We also note that the idea of transformation groups was motivated and occurred before the concept of group was introduced. For example, translations and rotations in the plane R2 , in the space R3 and their compositions were known in the ancient times, though not in the language of group theory. When the concept of group was formally introduced, it was also introduced for transformation groups on roots of algebraic equations. Let X be a topological space. Assume that a group acts on it by homeomorphisms. Then the pair .; X/ is called a topological transformation group. We always assume that the action is proper, i.e., for any compact subset K X, the subset f 2 j K \ K ¤ ;g is compact. When is given the discrete topology, this is equivalent to the fact that acts properly discontinuously on X , i.e., for any compact subset K X, the subset f 2 j K \ K ¤ ;g is finite. In the literature and also in the following, if a discrete group acts properly discontinuously on a topological space, we often say that the group acts properly on the space.2 2 Non-proper actions of infinite discrete groups occur naturally and also play an important role in understanding

structures of the groups. For example, the action of SL.2; Z/ on H2 extends continuously to the boundary H2 .1/ D S 1 , and the extended action on the boundary S 1 is not proper. This is a special case of the action of an arithmetic subgroup G on the Furstenberg boundaries of a semisimple Lie group G. This action on the Furstenberg boundaries has played a crucial role in understanding rigidity properties of arithmetic groups. Another type of problems related to understanding quotients of non-proper actions, or rather the non-proper actions themselves, occurs in non-commutative geometry.


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In the following, we assume that is a discrete group and that acts properly on X. Such a -space X is called a proper -space, and the pair .; X / is called a proper transformation group, or a discrete transformation group. For any proper -space, the following questions are natural: (1) The structure of each orbit of and its relation with the ambient space X . This is closely related to structures of finite subgroups of . Suppose that X is a metric space and acts isometrically. Then another natural question is whether endowed with the word metric is quasi-isometric to the -orbits endowed with the induced subspace metric from X. (2) The nature of fixed point sets X F in X of finite subgroups F of . For example, is the fixed point set X F nonempty? If X is contractible, is the fixed point set X F contractible? (3) The structure of the quotient nX. For example, when is nX compact? Suppose that nX is noncompact. What is its end structure? How to compactify it? Suppose that X has a measure which is invariant under the left action of . When does nX have finite measure? Suppose that X is a geodesic space. What is the structure of geodesics of nX that go to infinity?3 What are the properties of the geodesic flow of nX ? For example, is it ergodic? Another natural problem concerns existence and distribution of closed geodesics of nX , which correspond to periodic orbits of the geodesic flow. (4) Suppose that X is a Riemannian manifold. What are the spectral properties of the Laplace operator of nX? For example, one can ask about the existence of continuous spectrum and discrete spectrum (or L2 -eigenvalues) of nX , the existence of generalized eigenfunctions for the continuous spectrum, the existence of discrete spectrum inside the continuous spectrum, the asymptotic behavior of the counting function of the discrete spectrum, and the connection between the spectral theory and the geometry of nX , for example, the lengths of closed geodesics. (5) Let X be a Riemannian manifold and be an infinite discrete group. What are the L2 -invariants of X with respect to the action of , and what are the relations with the corresponding invariants of nX?4 (6) Fundamental domains for the -action on X and rough fundamental domains for satisfying various finiteness conditions. How to construct them and how to understand their shapes at infinity if nX is noncompact? How are these fundamental domains related to the structure at infinity of X and nX ? (7) Assume that is infinite and hence X is noncompact. What kind of compactifications does X have? How does act on the boundaries of these compactifications? 3A striking application of understanding structures going to infinity is the McShane identity in [301], Corollary 5, Theorems 4 & 2. 4 The basic point of view is that instead of taking -invariant functions or differential forms, one considers L2 .X / as a representation of and takes the von Neumann dimension. See [270] for details.

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What are the properties of limit sets of orbits of on the boundaries and what are their relations to other properties of or to the quotient nX? (8) Relations between structures of X and nX such as compactifications and function theory if X is a Riemannian manifold. For example, how can compactifications of X be used to compactify noncompact quotients nX ? How are eigenfunctions and spectra of X and nX related? In studying these problems, constructing good fundamental domains is a crucial step. The best example to explain this fact is the action of Fuchsian groups on the Poincaré upper half plane. Because of this consideration, in this chapter, we will discuss various aspects of fundamental domains and rough fundamental domains for actions of arithmetic groups on symmetric spaces in some detail. This is an important part of the foundational reduction theory of arithmetic groups. These results motivate results on fundamental domains for the action of Modg;n on the Teichmüller space Tg;n . If X is a metric space and if acts cocompactly and isometrically on X , then with any word metric associated to a finite generating set is quasi-isometric to X, and boundaries of X can often naturally be thought of as boundaries of . On the other hand, in many instances, the quotient of nX is not compact, and boundaries of X can also be considered as boundaries of , and they have played a fundamental role in the study of . This is the case for non-uniform arithmetic groups discussed below, especially in their rigidity properties. As mentioned before, for general groups , it is not easy to find suitable -spaces. On the other hand, there are some general constructions of such spaces. A particularly important class of -spaces is the class of universal spaces for proper actions of . These spaces are unique up to homotopy equivalence and are usually denoted by E. They are the terminal spaces in the category of proper -spaces. An E-space is characterized by the following conditions: (a) acts properly on E. (b) For any finite subgroup F , the set of fixed points .E/F is nonempty and contractible. In particular, E is contractible. If is torsion-free, then E is the universal space E for proper and fixed point free actions of , which is also the universal covering space of a classifying space B of , where B is characterized uniquely up to homotopy equivalence by the conditions: 1 .B/ D , and i .B/ D f1g for i 2. It is known that if is virtually torsion-free, i.e., if there exists a finite index torsion-free subgroup of , then for any model X of E, the virtual cohomological dimension vcd satisfies the upper bound vcd dim X:

(1)

It was proved by Serre [384] that if is virtually torsion-free and vcd is finite, then there exists a finite-dimensional model of E. Some natural questions concerning E are the following: (1) Assume that is virtually torsion-free. Does there exist a model X of E such that dim X D vcd ?


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(2) Assume that is virtually torsion-free and vcd is finite. Is there a model X of E such that nX is compact? Furthermore, can X satisfy the condition dim X D vcd ? (3) Assume that is virtually torsion-free and vcd is finite. Given a natural model X of E such that nX is noncompact, is it possible to find a -equivariant compactification Xx of X such that the inclusion X ,! Xx is a homotopy equivalence? Is it possible to find a -stable subspace with a compact quotient (or rather a finite CW complex) under such that it is an equivariant deformation retraction of X ? If dim X > vcd , is there a -equivariant deformation retract S of X such that dim S D vcd and nS is compact? (Such deformation retracts S are often called spines of X. More generally, any cocompact deformation retract is also called a spine of X .) (4) Assume that is torsion-free and satisfies the Poincaré duality. When does admit a model of B given by a compact manifold without boundary? If such a manifold model of B exists, is it unique up to homeomorphism? (5) For some important arithmetic groups such as SL.n; Z/, find explicit models of E and E given by CW-complexes so that we can compute cohomology and homology groups of . Questions (2) and (3) are closely related to the problem of existence of fundamental domains for that are unions of simplices, or -equivariant cell decompositions of X . In fact, once X has a -equivariant simplicial complex structure, then it is easier to construct deformation retracts and spines. The second part of question (4) is called the Borel conjecture for , which asserts that two closed aspherical manifolds with the same fundamental group are homeomorphic to each other. (Note that if M is an aspherical manifold, i.e., i .M / D f1g for i 2, then it is a model of B for D 1 .M /.) See [131], [135] for precise statements and references. Though arithmetic subgroups of semisimple Lie groups and mapping class groups are virtually torsion-free, some other natural groups, for example S-arithmetic subgroups of algebraic groups of positive rank over function fields such as SL.n; Fp Œt /, where Fp is a finite field and t is a variable, are not virtually torsion-free. Then their cohomological dimensions are equal to infinity. For such groups, finding good models of the universal spaces for proper actions is still important. In this chapter, we will discuss results addressing the above questions for arithmetic groups and mapping class groups. Many results are also known for related groups such as the outer automorphism group Out.Fn / of the free group Fn , Coxeter groups and hyperbolic groups. Besides these common questions and properties shared by these different classes of groups and spaces, we will also discuss the Jacobian (or period) map between the moduli space Mg of compact Riemann surfaces of genus g and the Siegel modular variety Ag , which is an important arithmetic locally symmetric space and is also the moduli space of principally polarized abelian varieties of dimension g, to show that there are interactions between these different spaces.

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We hope that the results presented in this chapter will justify the rather unusual title of this chapter: as in the famous book “A tale of two cities” by Charles Dickens, what happened in London, Paris and on ways between them made the whole story interesting and exciting. Arithmetic groups and mapping class groups are interesting in their own, but analogies and relations between them have motivated many new problems and results for each of these groups. The Jacobian map between Mg and Ag has also played an important role in understanding Mg . Therefore, various results and perspectives on arithmetic groups, mapping class groups and their associated spaces are all different parts of one tale! Acknowledgments. I would like to thank A. Papadopoulos for the invitation to contribute to this volume of Handbook of Teichmüller theory, which motivated me to write this chapter with an unusual title, for reading this chapter very carefully several times and making many helpful suggestions, and for pointing out the references [82], [415]. I would also like to thank Lixin Liu for pointing out the conjecture of Bers ([42], Conjecture IV), S. Prassidis for valuable information about Coexter groups, R. Spatzier for the references [240] and [329], and Weixu Su for pointing out the presence of some half Dehn twists in the stabilizer of ordered pants decompositions. A part of the writing of this chapter was carried out during my visit to MSC, Tshinghua University, 2010 and I would like thank them for providing a very good working environment. Finally I would like to thank two referees for reading carefully a preliminary version of this chapter and making many helpful suggestions. This work was partially supported by NSF grant DMS 0905283.

2 Summaries of properties of groups and spaces in this chapter To emphasize similarities and differences between arithmetic groups and mapping class groups on the one hand, and between symmetric spaces and Teichmüller spaces on the other hand, we make four lists of properties for them in parallel. The paper of Harer [169] is a valuable reference on comparing the properties of arithmetic groups and mapping class groups, mainly concentrating on cohomological properties. The surveys [188] and [123] are also extensive and cover many different topics. Besides studying arithmetic groups and mapping class groups, the surveys [75], [344] also compare the similarities between these groups and the outer automorphism groups of free groups. We try to include some other properties and hope to provide a different perspective. On the other hand, it is clear from the table of contents that the current survey is not comprehensive, and many results on arithmetic groups and mapping class groups are not mentioned. For more results about mapping class groups, see also the books [123], [334], [335]. For more references on arithmetic groups, see [198], [39], [411].


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After these four lists, for comparison, we also list some similar properties of Out.Fn / and two other important classes of groups: Coxeter groups and hyperbolic groups. We will be less exhaustive about properties of these latter three classes of groups. We state properties to be discussed in this chapter in the following order: (1) arithmetic groups (§2.1), (2) symmetric spaces and locally symmetric spaces (§2.2), (3) mapping class groups Modg;n (§2.3), (4) Teichmüller spaces and moduli spaces (§2.4), (5) outer automorphism groups Out.Fn / (§2.5), (6) outer spaces Xn (§2.6); (7) Coxeter groups (§2.7), (8) hyperbolic groups (§2.8).

2.1 Properties of arithmetic groups The following notation will be used in this chapter. Let G GL.n; C/ be a linear semisimple algebraic group defined over Q, and G D G .R/ the real locus of G , a real Lie group with finitely many connected components.5 Let K G be a maximal compact subgroup. Then the homogeneous space X D G=K with an invariant metric is a symmetric space of noncompact type, in particular it is a simply connected and nonpositively curved complete Riemannian manifold. Let G .Q/ be an arithmetic subgroup. An important example of an arithmetic subgroup is SL.n; Z/. (See §4 below for a precise definition of arithmetic groups.) Then acts properly on X and the quotient nX is called an arithmetic locally symmetric space. The rank of X is defined to be the maximal dimension of flat geodesic subspaces of X, which is equal to the R-rank of G , i.e., the maximal dimension of R-split tori of G , and it plays a very important role in the study of X and of lattice subgroups of G acting on X. The Q-rank of G is equal to the maximal dimension of Q-split tori in G and plays an important role in understanding the geometry of nX . For example, nX is compact if and only if the Q-rank of G is equal to 0. Because of this, we also call it the Q-rank of nX, or rather the Q-rank of for convenience, though the terminology may not be so standard. It follows from the definition that the Q-rank of nX is less than or equal to the rank of X . For the sake of the discussion below, we can keep the example G D SL.n; C/ GL.n; C/, n 2, in mind. Then the Q-rank and R-rank of G are both equal to n 1. 5 We assume that G is a semisimple linear algebraic group for simplicity. Most of the results stated here work for reductive algebraic groups as well, or with suitable modification. For various applications and induction involving parabolic subgroups, it is important to consider reductive but non-semisimple algebraic groups. For example, GL.n; C/ is a reductive, non-semisimple algebraic group.

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For the arithmetic subgroup D SL.n; Z/, the quotient nX is noncompact. This is consistent with the fact that the Q-rank of SL.n; Z/ is positive. In the rest of this subsection, we list some properties of arithmetic groups roughly according to the following categories: (a) combinatorial properties, (b) group theoretical properties, (c) cohomological properties, (d) large-scale (or asymptotic) properties, (e) ridigity properties, (f) classifying spaces and properties of actions. More details on some of these properties will be provided later. (a) Combinatorial properties (1) is finitely generated. (2) In many cases, for example if D SL.n; Z/, n 3, or if is equal to the integral points of Chevalley groups6 of rank at least 2, is also boundedly generated. The bounded generation of arithmetic subgroups is closely related to a positive solution of the congruence subgroup problem (see §4.11 and §6 in [354], Theorem D in [265], [349] for the definition and for more details.) But if the R-rank of G is equal to 1, is not boundedly generated. (3) is finitely presented. (See §4.11.) (4) The word problem is solvable for , and the conjugacy problem of is also solvable, but the solvability of the isomorphism problem for arithmetic groups is not known in general. See [159], [160], [161], [4] for related results and references. Some related results on Dehn functions, isoperimetric functions are also known. See [71] for definitions and §17.9 in [197] for other references. (b) Group theoretical properties (5) is residually finite. (See §4.12.) (6) admits torsion-free subgroups of finite index. (See §4.12.) (7) has only finitely many conjugacy classes of finite subgroups. (See §4.12.) (8) is contained in only finitely many other arithmetic subgroups of G. (See §4.12.) 6 Roughly speaking, a Chevalley group is a semisimiple linear algebraic group defined over Z in the sense that its Lie algebra has a basis, a Chevalley basis, whose structure constants are integers. Once such a Chevalley basis is constructed, the Chevalley group can be defined as the identity component of automorphism group of the Lie algebra and its Z-structure is determined by the Z-structure of the Lie algebra.


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(9) If G D SL.n/ and G .Z/ D SL.n; Z/ and n 3, then every arithmetic subgroup of SL.n; Z/ is a congruence subgroup, i.e., contains a principal congruence subgroup. This property fails for arithmetic subgroups of SL.2; Z/. In general, for a linear algebraic group G GL.n; C/ defined over Q, the original congruence subgroup problem asks if every arithmetic subgroup G .Z/ contains a congruence subgroup. This is equivalent to the condition that the congruence subgroup kernel associated with G .Z/ (or with G .Q/) is trivial. More generally, if the congruence subgroup kernel of G .Z/ is finite, we say that the congruence subgroup problem has a positive solution for G .Z/, though not every arithmetic subgroup of G .Z/ contains a congruence subgroup. It is known that if G is simply connected and absolutely almost simple, if the R-rank of G (or the rank of the symmetric space X D G=K) is at least 2, and if the Q-rank of G is positive, then the congruence subgroup kernel is finite. If the R-rank of G is equal to 1, then the congruence subgroup kernel of G .Z/ is infinite in general. The congruence subgroup problem, or the congruence subgroup kernel, is usually formulated and studied for the more general class of S-arithmetic subgroups of linear algebraic groups defined over number fields. (See [361], [362], p. 303–304, and §4.4 in [197] for the most general statements, complete results and more references.) (10) The Tits alternative holds for : every subgroup of is either virtually solvable or contains a subgroup isomorphic to the free group F2 on two generators. (See §4.14.) For related results on maximal subgroups of infinite index, see the paper [280]. (11) is irreducible, i.e., it is not a product of two infinite groups up to finite index, if and only if G is almost simple over Q. (See [195], Remark 2.5, and [20].) (12) Assume that is irreducible and the R-rank of G (or the rank of X ) is at least 2, then every normal subgroup of is either finite or of finite index (Margulis normal subgroup theorem). (See [278] and §4.12.) If the R-rank of G is equal to one, there are in general infinite normal subgroups of infinite index. (See [109].) (13) The rank of as an abstract group is equal to the real rank of the Lie group G, or equivalently the rank of the symmetric space X D G=K. (See [22] and references therein.) (c) Cohomological properties (14) The virtual cohomological dimension of is finite and is equal to dim X rQ , where rQ is the Q-rank of (or G ). (See §4.18.) (15) The cohomology groups H i .; Z/ and the homology groups Hi .; Z/ are finitely generated in every degree. Cohomology and homology of arithmetic groups have been extensively studied and there is a huge literature on them. (See the most recent survey [379] and the references there.) (See also §4.19.) (16) The cohomology ring H .; Z/ is finitely generated, which is an analogue of Evens–Venkov theorem for finite groups. (See [357], [213].)

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(17) The Euler characteristic of can be computed and often expressed in terms of special values of the Riemann zeta function and more general L-functions, and Bernoulli numbers in some cases. (See [168], [384], §3.7, [385], §3.1, pp. 253–256 of [79], [352].) Basically, this follows from the Gauss–Bonnet formula for the locally symmetric space nX and the formula for the volume of nX since the Euler–Poincaré measure in the Gauss–Bonnet formula is also invariant under G and hence is proportional to the Haar measure. (18) is a virtual duality group of dimension dim X rQ , where rQ is the Q-rank of . The dualizing module of is the only nonzero reduced homology group of the Q-spherical Tits building Q .G /, which is an infinite simplicial complex of dimension rQ 1, whose simplices are parametrized by proper Q-parabolic subgroups of G . is a virtual Poincaré duality group if and only if rQ D 0 is equal to 0, i.e., if and only if the quotient nX is compact. (See §4.19.) (19) The cohomology of families of classical arithmetic groups such as SL.n; Z/ stabilizes as n ! 1. This has important implications for K-groups of Z and rings of integers of number fields. (See [59] and also [197] for references.) (d) Large-scale properties (20) If G has no compact factor, then is Zariski dense in G (Borel density theorem). (See §4.13.) (21) grows exponentially, i.e., the number of elements in a ball with respect to any word metric of grows exponentially with the radius. (See §4.14.) (22) The asymptotic dimension of is finite. Recall that for any metric space .M; d /, its asymptotic dimension is the smallest integer n such that for every R > 0, there exists a covering of M by uniformly bounded sets fU˛ g such that every geodesic ball of radius R meets at most n C 1 sets from fU˛ g. This notion was introduced by Gromov. (See [200] for references.) (23) If is cocompact, then the tangent cone (or asymptotic cone) of with respect to any word metric is the same as the cone at infinity of the symmetric space X D G=K, which is an affine R-building. If is irreducible and the rank of G (or X D G=K) is at least 2, then its tangent cone is not determined yet but is homeomorphic by a Lipschitz map to a subset of an affine R-building, by a result of [267]. (See also [344].) See also [31], [112] for other quasi-isometry invariants of . (24) If is cocompact, then clearly has one end. Otherwise, the number of ends of depends on the real rank of G. When G has real rank at least 2, has one end. This follows from the Property T of and a characterization of groups with infinitely many ends by Stallings [394], which implies that a group having Property FA has one end. (See [197], p. 154, for more detail.)


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(e) Rigidity properties (25) Assume that is irreducible and G has trivial center and is not isomorphic to PSL.2; R/. Then for any arithmetic subgroup 1 of a semisimple Lie group G1 with trivial center, any isomorphism ' W ! 1 extends to an isomorphism ' W G ! G1 (Mostow strong rigidity). (See [322].) Furthermore, also satisfies the Margulis superrigidity. There is also a lot of work on rigidity for nonlinear actions in the Zimmer program. There are also other rigidity results for measure equivalence and orbit equivalence, and for lattices in more general locally compact groups. (See [278], [432], [315] and the references in [197].) For rigidity results on von Neumann algebras related to arithmetic groups, see [350] and the references therein. (26) When is irreducible and X is of rank at least 2, is quasi-isometry rigid in the sense that any group quasi-isometric to is isomorphic to up to finite index. (See [121] and the references there.) (27) The Borel conjecture and the integral Novikov conjectures in L-theory (i.e., surgery theory), K-theory, and C -algebra theory hold for . The original Borel conjecture states that if two closed aspherical manifolds are homotopy equivalent, then they are homeomorphic. The Borel conjecture is equivalent to the condition that certain assembly maps are isomorphisms. The isomorphism of an assembly map in each theory is sometimes called the Borel conjecture in that theory as well. (See [132], [200] and references therein.) (28) If G has trivial center and no nontrivial compact factor, then is C -simple, i.e., the reduced C -algebra Cr ./ of is simple. (See [34].) (29) is Hopfian, i.e., every surjective homomorphism ' W ! is an isomorphism, by the general result of Malcev for linear groups. (See §4.4 in [417]). (30) is co-Hopfian, i.e., every injective homomorphism ' W ! is an isomorphism (See [351].) (31) If the rank of G is equal to 1, then has Property RD and the Baum–Connes conjecture in the theory of C -algebras holds for (see [95]) (note that the Baum–Connes conjecture is an analogue of the Borel conjecture and also of the Farrell–Jones conjecture in geometric topology; it asserts that an assembly map for topological K-groups in C -algebras is an isomorphism), and is also weakly amenable (see [98]). On the other hand, if G is a simple Lie group of rank greater than or equal to 2 and is non-uniform, then does not have Property RD (see [219], [267]). (f) Classifying spaces and properties of actions (32) There exist -cofinite universal spaces E for proper actions of . (See §4.19.) If the associated symmetric space X D G=K is linear in the sense that it is a homothety section of a self-adjoint homogeneous cone, then admits -cofinite universal spaces E of dimension equal to the virtual cohomological dimension

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of , which is realized by a spine of X , i.e., a -equivariant deformation retract of X . (See [14] and §4.19.) (33) satisfies Property T when is irreducible and G has real rank at least 2. Recall that a group satisfies Property T if the trivial representation is isolated in the unitary dual of , or, equivalently, if whenever acts on a Hilbert space unitarily with an almost fixed point, then it has a fixed point. (See [35] for detailed discussions and applications. See also the papers [118], [115].) (34) satisfies Property FA of Serre and hence does not split when is irreducible and when G has real rank at least 2. Recall that a group has Property FA if every action on a tree has at least one fixed point. If acts on a tree but does not have Property FA, then it splits, i.e., admits an amalgam (free product with amalgamation). If has Property T, then it has Property FA. (See [6], [279], [416].) Remark 2.1. In the above discussion, we have assumed that is an arithmetic subgroup of a semisimple Lie group G. A natural generalization of the class of arithmetic subgroups is the class of lattice subgroups of Lie groups, where a discrete subgroup of a Lie group G is called a lattice subgroup (or a lattice) if the volume of nG is finite with respect to an invariant measure. Then most of the above properties for arithmetic subgroups also hold for lattice subgroups of Lie groups with finitely many connected components. Arithmetic subgroups of semisimple Lie groups are lattice subgroups. Conversely, by the famous theorem of Margulis on arithmeticity of lattices [278], if is an irreducible lattice of a semisimple Lie group G and if the rank of G (or X D G=K) is at least 2, then is an arithmetic subgroup of G. For more details, see §4.2. Remark 2.2. Another important and natural generalization of arithmetic subgroups consists of the class of S -arithmetic subgroups. Many of the above properties hold for them as well. See §4.3 for definitions and details. Though the above list is long, it is sketchy and it is only a list of properties of arithmetic groups known to the author. It is almost surely incomplete. Besides the references for the above properties that were already given, we mention below some books on arithmetic subgroups, discrete subgroups of Lie groups, and related locally symmetric spaces in the order they were first published. We hope that such a list will also give a historical perspective on subjects related to arithmetic subgroups and discrete subgroups. (1) R. Fricke, F. Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen, 1897 [140]. (2) J. Lehner, Discontinuous groups and automorphic functions, 1964 [255]. (3) J. Wolf, Spaces of constant curvature, 1967 [422].


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(4) A. Borel, Introduction aux groupes arithmétiques, 1969 [57]. (5) I. M. Gel’fand, M. Graev, I. Pyatetskii-Shapiro, Representation theory and automorphic functions, 1969 [146]. (6) G. Shimura, Introduction to the arithmetic theory of automorphic functions, 1971 [387]. (7) M. Raghunathan, Discrete subgroups of Lie groups, 1972 [360]. (8) G. Mostow, Strong rigidity of locally symmetric spaces, 1973 [322]. (9) W. Magnus, Noneuclidean tesselations and their groups, 1974 [282]. (10) A. Ash, D. Mumford, M. Rapoport, Y. Tai, Smooth compactification of locally symmetric varieties, 1975 [15]. (11) H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, Crystallographic groups of four-dimensional space, 1978 [78]. (12) J. Humphreys, Arithmetic groups, 1980 [184]. (13) M. F. Vignéras, Arithmétique des algèbres de quaternions, 1980 [409]. (14) A. Beardon, The geometry of discrete groups, 1983 [29]. (15) R. Zimmer, Ergodic theory and semisimple groups, 1984 [432]. (16) S. Krushkal, B. Apanasov, N. Gusevskii, Kleinian groups and uniformization in examples and problems, 1986 [249]. (17) V. Nikulin, I. Shafarevich, Geometries and groups, 1987 [331]. (18) B. Maskit, Kleinian groups, 1988 [283]. (19) G. van der Geer, Hilbert modular surfaces, 1988 [408]. (20) G. Margulis, Discrete subgroups of semisimple Lie groups, 1991 [278]. (21) B. Apanasov, Discrete groups in space and uniformization problems, 1991 [10]. (22) R. Benedetti, C. Petronio, Lectures on hyperbolic geometry, 1992 [37]. (23) S. Katok, Fuchsian groups, 1992 [225]. (24) B. Iversen, Hyperbolic geometry, 1992 [194]. (25) E. Vinberg, O. Shvartsman, Discrete groups of motions of spaces of constant curvature, in Geometry, II, pp. 139–248, 1993 [410]. (26) A. Lubotzky, Discrete groups, expanding graphs and invariant measures, 1994 [266]. (27) V. Platonov, A. Rapinchuk, Algebraic groups and number theory, 1994 [348]. (28) J. Ratcliffe, Foundations of hyperbolic manifolds, 1994 [363]. (29) W. Thurston, Three-dimensional geometry and topology. Vol. 1, 1997 [406]. (30) J. Elstrodt, F. Grunewald, J. Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory, 1998 [117].

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(31) K. Matsuzaki, M. Taniguchi, Hyperbolic manifolds and Kleinian groups, 1998 [294]. (32) W. Goldman, Complex hyperbolic geometry, 1999 [148]. (33) E. Vinberg, V. Gorbatsevich, O. Shvartsman, Discrete subgroups of Lie groups, in Lie groups and Lie algebras, II, pp. 1–123, 2000 [411]. (34) B. Apanasov, Conformal geometry of discrete groups and manifolds, 2000 [11]. (35) M. Kapovich, Hyperbolic manifolds and discrete groups, 2001 [223]. (36) K. Ohshika, Discrete groups, 2002 [332]. (37) W. Fenchel, J. Nielsen, Discontinuous groups of isometries in the hyperbolic plane, 2003 [134]. (38) C. Maclachlan, A. Reid, The arithmetic of hyperbolic 3-manifolds. 2003 [274]. (39) B. Sury, The congruence subgroup problem: an elementary approach aimed at applications, 2003 [398]. (40) L. Keen, N. Lakic, Hyperbolic geometry from a local viewpoint, 2007 [231]. (41) A. Marden, Outer circles. An introduction to hyperbolic 3-manifolds, 2007 [277]. (42) F. Bonahon, Low-dimensional geometry. From Euclidean surfaces to hyperbolic knots, 2009 [56]. The book [197] contains a lot of references for other topics related to arithmetic groups. We will define and discuss a portion of these properties in §4 below, showing how actions on symmetric spaces, reduction theories for arithmetic groups, and compactifications of symmetric and locally symmetric spaces are used to prove some of the above properties, hence emphasizing the importance of interaction between groups and spaces.

2.2 Properties of actions of arithmetic groups on symmetric spaces X Let G be an arithmetic subgroup as above, and X D G=K be a symmetric space of noncompact type with an invariant metric. (a) Orbits and fixed point sets (1) acts properly and isometrically on X , and nX is an orbifold with finitely many singular loci. The orbifold nX admits finite smooth covers. This follows from the existence of torsion-free finite index subgroups of . (2) For any finite subgroup F , the fixed point set X F is nonempty and contractible. Hence X is a model of an E-space.


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(3) When is cocompact, with a word metric is clearly quasi-isometric to any -orbit in X with the induced subspace metric. When is irreducible and the rank of X is at least 2, with a word metric is quasi-isometric to any -orbit in X with the induced subspace metric. (See [267].) (4) Distributions and growths of -orbits in X can be described. (See [150] and references therein.) (b) Boundaries of compactifications of symmetric spaces and -actions (5) As a simply connected nonpositively curved Riemannian manifold, X admits a natural geodesic compactification X [ X.1/ by adding the set of equivalence classes of directed geodesics X.1/ at infinity, where two directed unit speed geodesics .t /; 0 .t/ are equivalent if the distance between them near C1 is bounded, i.e., lim sup t!C1 d..t /; 0 .t // < C1. The -action on X extends continuously to X [ X.1/, and the stabilizers of the boundary points in G are exactly the proper parabolic subgroups of G. X also admits several other compactifications such as the Satake compactifications on which the -action extends continuously, and the stabilizers of the boundary points in G are smaller than the parabolic subgroups of G in general. (See §4.16.) (6) Compactifications of X such as the geodesic and Satake comapactifications contain distinguished boundary subsets, called the Furstenberg boundaries, that are stable under . Note that the -action on the Furstenberg boundaries is not proper. The -action on the maximal Furstenberg boundary is amenable with respect to the Haar measure and also with respect to any -quasi-invariant measure. (See [432], Theorem 3.1 in [329].) The Furstenberg boundaries and -actions on them have played a foundational role in the rigidity of arithmetic subgroups and more general lattices of G. (See [432], [278].) (c) Volumes of the quotient space and fundamental sets (7) With respect to the invariant Riemannian metric on X , nX has finite volume. (See §4.9.) (8) nX is compact if and only if the Q-rank of nX , i.e., of G , is equal to 0, or equivariantly, if does not contain any nontrivial unipotent element. (See §4.9). (9) If nX is compact, then its simplicial volume is positive. (See [251] and [83] for the definition and the precise statement of the result). On the other hand, if the Q-rank of nX is at least 3, then the simplicial volume of nX is zero. (See [263], Theorem 1.1.) When the Q-rank of G is equal to 1 or 2, it is not known whether the simplicial volume of nX is positive or not. For some Q-rank 1 locally symmetric spaces including the Hilbert modular varieties, it is known that the simplicial volume is positive [262], and for the Hilbert modular surfaces, the simplicial volume can be computed explicitly using the result in [84].

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(10) In the special case where D SL.n; Z/ and X D SL.n; R/=SO.n/, there is a Mahler compactness criterion for subsets of nX. This is a foundational result in the geometry of numbers and in the reduction theory of arithmetic groups. (See [92].) (11) Fundamental sets (or rough fundamental domains) of the -action on X can be described in terms of Siegel sets associated with representatives of -conjugacy classes of Q-parabolic subgroups of G . In some special cases, fundamental domains can be described explicitly, for example, in the theory of Minkowski reduction for GL.n; Z/. (See §4.8 and §4.9.) (12) The volume of nX can be computed and can be expressed in terms of special values of the Riemann zeta function or more generally L-functions. (See [352].) (13) The Gauss–Bonnet formula holds for nX and can be used to compute the Euler characteristic of arithmetic groups. (See [168] and §3 in [385].) (d) Rigidity properties (14) Suppose that is irreducible and X is not isometric to the Poincaré upper half plane. For any locally symmetric space 1 nX1 of finite volume, if nX and 1 nX1 are homotopy equivalent, then they are isometric up to a suitable scaling of the metrics (Mostow strong rigidity). (See [322].) (15) There are many generalizations of Mostow strong rigidity. In [23], one locally symmetric space is replaced by a nonpositively curved Riemannian manifold without changing the conclusion that the two spaces must be isometric up to scaling. In [254], one locally symmetric space is further replaced by a geodesically complete CAT.0/-space.7 In [43], for a compact locally symmetric space of negative sectional curvature, its invariant metric is characterized by the property that it has minimal entropy among all negatively curved metrics of the same volume on it. For other generalizations of Mostow strong rigidity, see the references in [198] and [205]. (16) In [89] and [21], rank rigidity says that any finite volume irreducible nonpositively curved Riemannian manifold of rank at least 2 is a locally symmetric space. (17) There is also a characterization of irreducible higher rank locally symmetric spaces of finite volume among all irreducible nonpositively curved manifolds of finite volume in terms of bounded cohomology, i.e., vanishing of the vector space of quasi-homomorphisms of the fundamental group. (See [52], [88].) (e) Compactifications of locally symmetric spaces and ends (18) Suppose that nX is noncompact. Then it admits a Borel–Serre compactification nX BS such that the inclusion nX ,! nX BS is a homotopy equivalence. 7A CAT.0/-space is a geodesic space such that every triangle in it is thinner than a corresponding triangle of the same side lengths in R2 .


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nX BS is the quotient of a partial Borel–Serre compactification Xx BS which is a real analytical manifold with corners, and acts real-analytically and properly on it. If is torsion-free, then nX BS is a real analytic manifold with corners. (See [62] and [61].) If contains torsion elements, then nX BS is a real analytic compact orbifold. (19) The locally symmetric space nX has one end when the Q-rank of nX , i.e., of G , is at least 2. When the Q-rank of nX is equal to 1, there are only finitely many ends, and the ends are in one-to-one correspondence with the -conjugacy classes of Q-parabolic subgroups of G . (See §4.9.) By passing to subgroups of large finite index, we can get locally symmetric spaces of Q-rank 1 with as many ends as desired. x BS is a cofinite model of E. (See (20) The partial Borel–Serre compactification X [206].) (21) nX also admits other compactifications such as the reductive Borel–Serre comx RBS , and the Baily–Borel compactification nX BB when nX is pactification X Hermitian. (See §4.17 and [61] for references.) (22) nX admits deformation retracts that are compact submanifolds with corners, i.e., the thick part where the injectivity radius is bounded uniformly from below. This gives a realization of the partial Borel–Serre compactification of X by a subspace of X, i.e., the inverse image in X of the thick part of nX. (See [371].) (23) If the Q-rank rQ of nX is equal to 1 or if X is a linear symmetric space, then nX admits deformation retracts that are of dimension dim X rQ , i.e., the virtual cohomological dimension of . (See §4.19 for more details and references.) On the other hand, X is a contractible complete Riemannian manifold of smallest dimension on which can act by isometries, or is a contractible manifold on which acts properly. (See [50].) (24) When nX is a Hermitian arithmetic symmetric space, the Baily–Borel compactification nX BB is a normal projective variety defined over specific number fields. Let D be the unit disc of C. The Borel extension theorem says that every holomorphic map D f0g ! nX extends to a holomorphic map D ! nX BB . (See [60].) (f) Large-scale geometry (25) The asymptotic cone (or the tangent space at infinity) of nX is a cone over a finite simplicial complex, which is the quotient nQ .G / of the Tits building Q .G /. (See [177], [215], [256].) (26) The rays or EDM (eventually distance minimizing) geodesics of nX can be classified and suitable equivalence classes of such geodesics can be identified with boundary points of various compactifications. (See [215].)

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(27) The logarithm law for geodesics holds for nX . This says roughly that from any fixed basepoint, for almost all unit speed geodesics c.t / starting at this point, lim sup t!C1 d.c.t /; x0 /= log t exists and is a positive constant depending only on X (See [239], [396].) (g) Spectral theory of nX (28) When nX is noncompact, the Laplace operator associated with the invariant metric has a nonempty continuous spectrum that consists of a union of half lines Œa; C1/, and their generalized eigenfunctions are given by Eisenstein series. This is the famous theory of Eisenstein series that has played a fundamental role in the celebrated Langlands program. The square integrable eigenfunctions are Maass forms and their existence is a subtle problem. (See [252].) (29) There is a Selberg (or Arthur–Selberg) trace formula relating the spectral data of nX to the geometric data of nX . When nX is a hyperbolic surface, the trace formula relates the spectra (both the continuous and the discrete) of the Laplace operator to the lengths of closed geodesics. For general locally symmetric spaces, the geometric side is in terms of orbital integrals. The original motivation of the Selberg trace formula is to prove existence of Maass forms, and the Arthur–Selberg trace formula has been used to prove the Langlands functoriality principle. (See [383], [12].) (30) A generalized Poisson relation connects sojourn times of scattering geodesics and the singularities of the Fourier transform of the spectral measure. (See [218].) This relation was motivated by the Poisson relation for compact Riemann manifolds in [114]. (h) L2 -cohomology and intersection cohomology (31) When nX is a Hermitian locally symmetric space, the Baily–Borel compactification nX BB is usually a singular projective variety. The intersection cohomology of nX BB with the middle perversity can be canonically identified with the L2 -cohomology of nX , which is known as the Zucker conjecture and was proved independently in [264], [373]. See the survey [217] in this volume. (32) For p 1, the Lp -cohomology group of nX can be canonically identified with the cohomology group of the reductive Borel–Serre compactification nX RBS . (See [433].) We will not discuss in detail most of the above properties, except several properties related to arithmetic groups in §4.

2.3 Properties of mapping class groups Modg;n Let Sg be a compact oriented surface of genus g, and Sg;n be a noncompact surface obtained from Sg by removing n points. For simplicity, Sg;0 is defined to be Sg . Let Modg;n D DiffC .Sg;n /=Diff0 .Sg;n / be the mapping class group of Sg;n ,


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where DiffC .Sg;n / is the group of orientation preserving diffeomorphisms of Sg;n , and Diff0 .Sg;n / is the identity component of DiffC .Sg;n /, which is also a normal subgroup. Modg;n only depends on the pair g; n. Let HomeoC .Sg;n / be the group of orientation preserving homeomorphisms of Sg;n , and Homeo0 .Sg;n / the identity component. Then the quotient group HomeoC .Sg;n /=Homeo0 .Sg;n / is isomorphic to Modg;n . (See [313].) Assume that 2g 2 C n > 0. Then Sg;n admits complete hyperbolic metrics of finite area. Let Tg;n be the Teichmüller space of marked complete hyperbolic metrics of finite area (or equivalently complex structures) on Sg;n . Then Modg;n acts on Tg;n by changing the markings of the hyperbolic metrics, and the quotient Modg;n nTg;n is the moduli space Mg;n of complete hyperbolic metrics of finite area on Sg;n , or equivalently the moduli space of compact Riemann surfaces of genus g with n punctures, i.e., the moduli space in algebraic geometry of projective curves over C with n marked points. In this subsection, we list some properties of mapping class groups Modg;n . In the next subsection, we list properties of the action of Modg;n on Tg;n and its quotient Modg;n nTg;n Š Mg;n . (a) Nonisomorphism with arithmetic groups (1) Modg;n is not isomorphic to any arithmetic subgroup of a semisimple Lie group G (see [188]) or more generally to any lattice subgroup of a Lie group with finitely many connected components (see [195] for references.) Stronger rigidity results on characterization of locally compact second countable topological groups that contain Modg;n as a lattice were obtained in [238], Corollary 1.5. (2) If is an irreducible lattice of a semisimple Lie group of rank at least 2, then any homomorphism W ! Modg;n has finite image. (See [167] for references.) On the other hand, the symmetric statement that any homomorphism W Modg;n ! has finite image is not true.8 (b) Combinatorial properties (3) Modg;n is finitely generated. In fact, it is generated by finitely many suitable Dehn twists. (See [188], [126], [54], [67].) But Modg;n is not boundedly generated. (See [125].) (4) Modg;n is finitely presented. (See [188], [126].) (5) Modg;n is automatic and hence the word problem is solvable. The conjugacy problem for Modg;n is also solvable (See [320], [179], [122].) (c) Group theoretical properties (6) Modg;n admits torsion-free subgroups of finite index. (See [188].) 8 For example, there is a natural surjective homomorphism Mod ! Sp.2g; Z/. For n > 0, there is also a g surjective homomorphism Modg;n ! Modg;n1 . (See [169], p. 144.) Given the super-rigidity of higher rank arithmetic irreducible subgroups, it may be natural to conjecture that the first example mentioned is essentially the only case of a homomorphism of Modg;n into a semisimple algebraic group with a Zariski dense image.

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(7) Modg;n is residually finite. (See [188].) (8) Modg;n has only finitely many conjugacy classes of finite subgroups. (See [216].) (9) Every finite subgroup of Modg;n can be realized as a subgroup of the automorphism group of a Riemann surface (the Nielsen realization problem). (See [233], [424].) (10) Modg;n satisfies a strong version of the Tits alternative: every subgroup of Modg;n is either virtually abelian or contains a subgroup isomorphic to the free group F2 on two generators. (See [295], [55], [190] and references.) For more results on subgroups of Modg;n and similarities with results for finitely generated linear groups, see [190] and also [232], [340], [321]. (11) Modg;n is irreducible, i.e., it is not isomorphic to a product of two infinite groups up to finite index. (See [188].) (12) The rank of Modg;n as an abstract group is equal to 1 [188], but its geometric rank is equal to 3g 3 C n [33]. (13) Modg;n contains infinite normal subgroups that are of infinite index, i.e., the analogue of the Margulis normal subgroup theorem does not hold. In the case of Modg , there is a surjective homomorphism Modg ! Sp.2g; Z/ with an infinite kernel, the Torelli group, which is an infinite normal subgroup. (See [122] for more discussion.) (14) The congruence subgroup problem for Modg;n is solved for some cases when g 2 and is still open for higher values of g. (See [191] for a precise definition of the congruence subgroup problem and applications, and [364] and references for known results on that problem.) (d) Cohomological properties (15) The virtual cohomological dimension of Modg;n is finite and can be written down explicitly. For example, the virtual cohomological dimension of Modg is equal to 4g 5. (See [169].) (16) The cohomology and homology groups of Modg;n are finitely generated in every degree. (See [169], [216].) A lot of work has been done about these groups. See [169] for a good summary and [123] for some motivation and applications. (17) The cohomology ring H .Modg;n ; Z/ is finitely generated, which is an analogue of the Evens–Venkov theorem for finite groups. (See [357], [213].) (18) The Euler characteristic of Modg;n can be computed in terms of Bernoulli numbers, or special values of the Riemann zeta function. (See §8 in [169], [171].) (19) Modg;n is a virtual duality group, but not a virtual Poincaré duality group. Its dualizing module is the only nonzero reduced homology group of the curve complex C .Sg;n / of Sg;n , which is an infinite simplicial complex with vertices corresponding to homotopy classes of essential simple closed curves of Sg;n and is an analogue of the Tits building of a linear semisimple algebraic group. (See [169], [188], [192].)


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(20) The cohomology groups of families of Modg;n stabilize as g; n ! C1. (See [169], Theorem 6.1.) The stable rational cohomology ring of Modg;n is a polynomial ring as conjectured by Mumford. (See [276], [275].) (21) There exist cofinite universal spaces for proper actions of Modg;n . (See [216].) (e) Large-scale properties (22) Modg;n grows exponentially. The fact that Modg;n is not virtually abelian and the Tits alternative imply that it grows at least exponentially. By general results ([105], p. 181, Remark 53 (iii)) it grows exponentially. (23) The asymptotic dimension of Modg;n is finite. (See [47].) (24) The maximal topological dimension of locally compact subsets of any asymptotic cone of Modg;n and the geometric rank of Modg;n are determined in [33]. Every point of any asymptotic cone of Modg;n is a global cut-point and the cone is tree-graded. (f) Rigidity properties (25) Modg;n is quasi-isometry rigid in the sense that any group quasi-isometric to Modg;n is isomorphic to Modg;n up to finite index. (See [166], [32].) It also satisfies the measure equivalence rigidity. (See [236], [237] for detailed statements.) (26) The analogue of Mostow strong rigidity holds: For any two subgroups of finite index, i Modgi ;ni , i D 1; 2, any isomorphism ' W 1 ! 2 extends to an isomorphism ' W Modg1 ;n1 ! Modg2 ;n2 . (See [195] for references.) This gives a solution to a conjecture in [191], §14. There are also analogues of superrigidity results. (See [123], [188] and references there.) (27) Modg;n satisfies the Hopfian and co-Hopfian properties. (See [193].) (28) Modg;n does not have Property T. (See [9] for the general case and [399] for the genus 2 case.) (29) Modg;n satisfies Serre’s Property FA and hence does not split. (Recall that a group has Serre’s Property FA if every action of this group on a tree has a fixed point.) (see [102].) (30) Modg;n and its subgroups of finite index have one end, using the fact that a group having Property FA does not split and hence has one end, which follows from a characterization of groups with infinitely many ends by Stallings [394]. (31) The integral Novikov conjecture in both K- and L-theories, i.e., on the injectivity of the assembly map, holds for Modg;n . This follows from a general result stating that finite asymptotic dimension implies the validity of integral Novikov conjectures in K-, L-theories, which also implies the stable Borel conjecture. (See [200], [201] for explanations and the definition of the stable Borel conjecture.)

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(32) Modg;n is C -simple, i.e., the reduced C -algebra Cr .Modg;n / is simple. (See [72].) Since Modg;n and its finite index subgroups are not virtual Poincaré duality groups and hence cannot be realized as fundamental groups of closed aspherical manifolds, the usual Borel conjecture stating that two closed aspherical manifolds with the same fundamental group are homeomorphic automatically holds for them. (See [135] for an explanation of the Borel conjecture.) In the above list, we have tried to follow the order of corresponding properties of arithmetic groups. Since the emphasis here is more on global properties of the group Modg;n , there are many results on properties of individual elements of Modg;n that we have not listed here. See [188], [403], [3] and the references there. Some of the properties related to structures of Teichmüller spaces will be explained in §5, but they will be even less detailed than for arithmetic groups.

2.4 Properties of actions of mapping class groups Modg;n on Teichmüller spaces Let Tg;n be the Teichmüller space of surfaces of genus g with n punctures, and Modg;n the associated mapping class group defined as above. (a) Orbits and quotients by the mapping class groups (1) The Teichmüller space Tg;n is a contractible complex manifold of dimension 3g 3 C n, and Modg;n acts holomorphically and properly on Tg;n . (See [327], [187].) (2) The quotient Modg;n nTg;n is the moduli space Mg;n of hyperbolic metrics on Sg;n , or equivalently the moduli space of projective curves over C with n marked points. Mg;n is an orbifold and admits finite smooth covers. The first statement follows from the fact that Modg;n acts transitively on the markings of complex structures on Sg;n , and the second from the existence of torsion-free finite index subgroups of Modg;n . F (3) For any finite subgroup F Modg;n , the fixed point set Tg;n is nonempty and contractible. Hence, Tg;n is a model of EModg;n .

(4) Modg;n is not quasi-isometric to any of its orbits in Tg;n with respect to the Teichmüller metric ([125], Theorem 2.1). (5) Distributions of Modg;n -orbits and their asymptotic behavior can be described. (See [119] and references therein.) (b) Volumes of quotients (6) Tg;n admits several different metrics that are invariant under Modg;n : the incomplete Weil–Petersson metric, and the complete Teichmüller metric, the Bergman


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metric, the Kobayashi metric, the Carathéodory metric, the Kähler–Einstein metric, the McMullen metric, the Ricci metric, the perturbed Ricci metric (or Liu– Sun–Yau metric). All these complete metrics are quasi-isometric to each other. (See [260], [425]–[427], [299] and references.) There is also the Thurston (or Lipschitz) asymmetric metric. (See [338] for an exposition and references.) The introduction [336] gives brief definitions of most of these metrics. (7) There are many different properties of metrics on Tg;n . The Weil–Petersson is an incomplete Kähler metric which is also geodesically convex in the sense that every two distinct points are connected by a unique geodesic. For a comprehensive study of the Weil–Petersson metric of Teichmüller space, see [427]. The Teichmüller metric is a Finsler metric and is not a CAT.0/-space or Gromov hyperbolic space but it has some properties that resemble a CAT.0/-space, for example, every two points can be connected by a unique Teichmüller geodesic. See [306], [113], [359], [16], [292] and the surveys [290], [338] for references. For the Thurston Lipschitz metric, see the survey [338] and [404]. For the McMullen metric, the bottom of the spectrum of the Teichmüller space is positive [299]. For the Ricci and other related metrics, see [260], [261]. See also [336] for an overview of these metrics. (8) With respect to any of the above metrics, the volume of Mg;n is finite. This follows from the asymptotics (or rather quasi-isometry class) of the metrics near the boundary of Mg;n . In fact, the complete metrics mentioned above are quasiisometric to a Poincaré type metric near the boundary divisors of the Deligne– DM Mumford Mg;n compactification. (See [260].) The asymptotic behavior of the Weil–Petersson metric is also known. (See [425], [426].) (9) For any finite index torsion-free subgroup of Modg;n , the simplicial volume of nTg;n is zero when g 2, or g D 1 and n 3, and g D 0 and n 6. (See [209], [210].) (c) Symmetries and compactifications of Teichmüller spaces (10) The isometry group of Tg;n with respect to any of the metrics in (6) is discrete and contains Modg;n as a subgroup of finite index when 3g 3 C n 2. More generally, for any complete, finite covolume Modg;n -invariant Finsler (or Riemannian) metric on Tg;n , when 3g 3 C n 2, its isometry group contains Modg;n as a subgroup of finite index. (See [130], Theorem 1.2.) Except for a few exceptions, the isometry group of the Teichmüller metric and the Weil–Petersson metric of Tg;n is equal to Modg;n , which is the degree-two extension of Modg;n obtained by including the diffeomorphisms that do not necessarily preserve the orientation of the surface Sg;n . This is related to the fact that the automorphism group of the curve complex C .Sg;n / is equal to Modg;n except for a low genus few cases. (See [188], [271], [425] and the references there.) (11) Tg;n admits several compactifications: (a) the Thurston compactification, (b) the Bers compactification, (c) the Teichmüller ray compactification [234], (d) the

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harmonic map compactification, (e) the Weil–Petersson visual compactification with respect to the Weil–Petersson metric, (f) the compactification via extremal lengths of essential simple closed curves [144], (g) the horofunction (or Gromov) compactification with respect to the asymmetric Thurston metric [415], (h) the real spectrum compactification of Tg as a semi-algebraic set [82], and there are other compactifications, for example, the compactification in [316] associated with ƒ-trees. (a) The Thurston compactification is defined intrinsically by lengths of simple closed geodesics of the marked hyperbolic Riemann surfaces and the Modg;n action on Tg;n extends continuously to the Thurston compactification (see [403], [405]). (b) The Bers compactification is the closure of Tg;n under the Bers embedding which realizes Tg;n as a bounded domain in C 3g3Cn (see [235]). It depends on the choice of a basepoint and the action of Modg;n on Tg;n does not extend continuously to this compactification. (c) The Teichmüller ray compactification is obtained by adding a point to each ray in the Teichmüller metric from a fixed basepoint in Tg;n . It depends on the basepoint and the action of Modg;n on Tg;n does not extend continuously to this compactification either (see [234]). It is important to point out that the boundary of the Teichmüller ray compactification is not equal to the visual sphere (i.e., the space of equivalence classes of Teichmüller rays, where two rays are said to be equivalent if they stay at a bounded distance), since the latter is non-Hausdorff [296]. (d) There is also a compactification of Tg;n by harmonic maps, which is equivariantly homeomorphic to the Thurston compactifcation (see [423] for the definition and the proof of the homeomorphism). (e) The Weil–Petersson visual compactification of Tg;n by adding the visual sphere, i.e., the set of geodesics from a fixed base-point [76] depends on the base-point, and the Modg;n -action on Tg;n does not extend continuously to the boundary when 3g 3 C n 2. This visual compactification strictly contains the completion of the Weil–Petersson metric of Tg;n , which turns out to be the augmented Teichmüller space [1], a partial compactification of Tg;n . (See [291] and [425].) (f) The boundary of the compactification by extremal lengths in [144] (see also [312]) is contained in the boundary of the Thurston compactification. (g) The horofunction (or Gromov) compactification with respect to the Thurston asymmetric metric is equivariantly homeomorphic to the Thurston compactification [415]. (h) The real spectrum compactification is only defined for Tg and dominates the Thurston compactification and has the property that the action of Modg on Tg


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extends continuously to the compactification, and every element in Modg has at least one fixed point in the compactification [82]. (12) The action of Modg;n on the compact metrizable Hausdorff space of complete geodesic laminations for Sg;n is topologically amenable and hence the Novikov conjecture in surgery theory holds for Modg;n . (See [165] and also [236]). (d) Fundamental sets (13) The quotient Modg;n nTg;n is noncompact if its dimension is positive. The reason is that simple closed curves on hyperbolic surfaces can be pinched and the resulting surfaces have smaller genus and more punctures. (14) There is a Mumford compactness criterion for subsets of Mg;n , which is an analogue of the Mahler compactness criterion for subsets of the locally symmetric space SL.n; Z/nGLC .n; R/=SO.n/. (See [324].) (15) Fundamental sets (or rough fundamental domains) of the Modg;n -action on Tg;n can be described by Bers sets associated with representatives of Modg;n -orbits of pants decompositions of Sg;n . The Minkowski reduction theory for SL.n; Z/ acting on the space of positive definite quadratic forms can be generalized to give an intrinsic fundamental domain for Modg;n that is a union of finitely many cells (§5.11.) (16) Tg;n has one end. For any finite index subgroup of Modg;n , nTg;n has also one end if 3g 3 C n > 0. The former is clear since Tg;n is diffeomorphic to R6g6C2n , and the latter follows from the fact that the curve complex C .Sg;n / is connected. (See §5.6.) (17) When n > 0, the existence of a Modg;n -equivariant intrinsic simplicial decomposition of Tg;n and of a Modg;n -equivariant spine of the right dimension, i.e., equal to the virtual cohomological dimension of Modg;n , is known [169]. On the other hand, when n D 0, the existence of a Modg -equivariant intrinsic cell decomposition of Tg and a spine of Tg of the right dimension is not known in general. When g D 2, it is known [367]. A Modg -equivariant, cocompact deformation retract of Tg is known [216], and a Modg -equivariant, cocompact deformation retract of Tg of positive codimension is also known [203]. (18) Tg admits a Modg -equivariant deformation retract, which is a cocompact submanifold with corners of Tg and gives a cofinite model of the universal space EModg for proper actions of Modg . (See [216].) (e) Large-scale geometry (19) The asymptotic cone of Modg;n nTg;n with respect to the Teichmüller metric exists and is equal to the metric cone over a finite simplicial complex, which is the quotient of the curve complex C.Sg;n / by Modg;n . (See [257], [128].)

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(20) The eventually distance minimizing (EDM) geodesics of Modg;n nTg;n in the Teichmüller metric can be classified and the boundary of the Deligne–Mumford compactification of Modg;n nTg;n can be described in terms of equivalence classes of these geodesics. (See [127].) (21) The logarithmic law for geodesics holds for Modg;n nTg;n in the Teichmüller metric. It corresponds to the logarithmic law for noncompact finite volume hyperbolic manifolds. (See [289].) (f) Compactifications of the quotient (22) Modg;n nTg;n admits a Borel–Serre type compactification Modg;n nTg;n

BS

such

BS

that the inclusion Modg;n nTg;n ,! Modg;n nTg;n is a homotopy equivalence. It BS can be taken as the quotient of a Borel–Serre type partial compactification Tg;n BS on which acts properly. The boundary of Tg;n is homotopy equivalent to the curve complex C.Sg;n /. (See [188], [210] and §5.6). (23) The curve complex C.Sg;n / is an infinite simplicial complex. It has infinite diameter and is a hyperbolic space in the sense of Gromov. (See [129], [188], [307].) (24) The Borel extension theorem for holomorphic maps from the punctured disk to Mg;n , f W D f0g ! Mg;n , holds for the Deligne–Mumford compactification DM DM Mg;n , i.e., f extends to a holomorphic map fO W D ! Mg;n . (See [186].) If the map f is algebraic, then the extension of f after passing to a finite covering of D f0g is the stable reduction of curves in algebraic geometry. (See [108].) (g) Cohomological properties (25) The L2 -cohomology group of Mg;n D Modg;n nTg;n with respect to any of the canonical complete metrics in (6) is isomorphic to the cohomology group of the Deligne–Mumford compactification of Mg;n . In fact, for all p < C1, the Lp -cohomology group of Mg;n with respect to any of the above canonical complete metrics is also isomorphic to the cohomology group of the Deligne– Mumford compactification of Mg;n . (They all define the same Lp -cohomology groups since they are quasi-isometric; see [372], [217].) With respect to the incomplete Weil–Petersson metric, when p 4=3, the Lp -cohomology group of Mg;n is also isomorphic to the cohomology group of the Deligne–Mumford compactification of Mg;n , but for p < 43 , it is isomorphic to the cohomology group of Mg;n . (26) The Gauss–Bonnet formula holds for Mg;n in all the canonical metrics defined in (6) above. (See [214].) (27) The stable rational cohomology ring of Mg;n is a polynomial ring generated by the Miller–Morita–Mumford i -classes of dimension 2i as conjectured by Mumford. (See [276], [275].)


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There are many results on dynamics and properties of elements of Modg;n acting on Teichmüller spaces and their boundaries which are not mentioned here. See the papers [188], [32], [166] and the books [334], [335], [123], [403], [3] and the extensive references there for problems and results related to Teichmüller spaces and mapping class groups.

2.5 Properties of outer automorphism groups Out.Fn / Besides the above two classes of groups, there is a closely related class of groups, Out.Fn / D Aut.Fn /=Inn.Fn /, where Fn is the free group on n generators, n 2. A lot of recent work on Out.Fn / has been motivated by results on arithmetic groups and Modg;n . When n D 2, it is known that Out.Fn / Š GL.2; Z/, and Modg D SL.2; Z/. This explains the common roots of these three important classes of groups. We will also mention some properties of two further classes of groups: Coxeter groups and hyperbolic groups. Since these three classes of groups, Out.Fn /, Coxeter groups and hyperbolic groups, are not our main objects of study, the lists of their properties will not be as exhaustive as the previous two. The following is a partial list of properties of Out.Fn /. (a) Nonisomorphism with arithmetic groups and mapping class groups (1) When n 3, Out.Fn / is not isomorphic to any arithmetic subgroup of a linear algebraic group (more generally of a lattice subgroup of a Lie group with finitely many connected components) or to a mapping class group Modg;n . (See [195] for references.) (2) When n 4, Out.Fn / is not linear. (See [136].) (b) Combinatorial properties (3) Out.Fn / is finitely generated, and explicit generators are known. (See [412].) (4) Out.Fn / is not boundedly generated. (See [125], Theorem 3.5.) (5) Out.Fn / is finitely presented, and explicit relations are known. (See [413].) (6) Out.Fn / is not an automatic group but the word problem for it is solvable. (See [73].) (c) Group theoretical properties (7) Out.Fn / has only finitely many conjugacy classes of finite subgroups. (See [412].) (8) Out.Fn / admits torsion-free subgroups of finite index. (See [272], p. 25–27.) (9) Out.Fn / is residually finite. (See [27], [26].) (10) A strong version of the Tits alternative holds for Out.Fn /: every subgroup of Out.Fn / is either virtually abelian or contains a free subgroup isomorphic to F2 . (See [51].)

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(11) Out.Fn / is irreducible, i.e., it is not a product of two infinite groups up to finite index. (See [195] for references). (d) Cohomological properties (12) The virtual cohomological dimension of Out.Fn / is equal to 2n 3. (See [101].) (13) The cohomology and homology groups of Out.Fn / are finitely generated in every degree. (See [414].) (14) The cohomology ring H .Out.Fn /; Z/ is finitely generated, which is an analogue of the Evens–Venkov theorem for finite groups. (See [357], [213].) (15) The Euler characteristic of Out.Fn / is known for small n. On the other hand, there is no simple formula for general n. (See §2.2.6 in [412], §7.2 in [245]) (16) The cohomology group of the family Out.Fn / stabilizes as n ! C1: when n 2i C 4, Hi .Out.Fn // is independent of n. (See [175], and [143] for stability of Hi .Aut.Fn //.) (17) Out.Fn / is a virtual duality group, but not a virtual Poincaré duality group. (See [49], [196].) The dualizing module of Out.Fn / is not known yet as in the cases of arithmetic groups and mapping class groups. There are several candidates for the analogue of the curve complex and spherical Tits buildings for Out.Fn /. These simplicial complexes have the homotopy type of a bouquet of spheres, i.e., the analogue of the Solomon–Tits theorem for Tits buildings holds [176], but the problem of whether their homology group can realize the dualizing module of Out.Fn / is not clear. (See also [221], [204]). (18) There exist cofinite universal spaces of proper actions of Out.Fn / of dimension equal to the virtual cohomological dimension of Out.Fn /, which is equal to 2n3. (See [101] for the contractibility of outer space and its equivariant deformation retraction to its spine, and [420], [248] for the contractibility of fixed point sets of finite subgroups of Out.Fn /.) (e) Rigidity properties (19) For any two finite index subgroups i Out.Fni /, i D 1; 2, every isomorphism ' W 1 ! 2 extends to an isomorphism ' W Out.Fn1 / ! Out.Fn2 /. (See [124] and also [195] for references). (20) Out.Fn / has Property FA of Serre when n 3. (See [102].) (21) Out.Fn / and its finite index subgroups are co-Hopfian. (See [74], [124].) (22) Out.Fn / is C -simple, i.e., the reduced C -algebra Cr .Out.Fn // is simple. (See [72].) (f) Large-scale properties (23) Out.Fn / has exponential growth. Since Out.Fn / is not virtually abelian, the Tits alternative implies that it contains non-abelian free groups and hence it grows at least exponentially. By the general results ([105] (p. 181, Remark 53 (iii)) it grows exponentially.


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(24) Out.Fn / and its subgroups of finite index have one end when n 3. See Theorem 3.9 in [413]. This also follows from the fact that Out.Fn / has Property FA as for arithmetic groups and Modg;n . There are many results on the dynamics of elements of Out.Fn / and their actions on the outer spaces and their boundaries which are not listed here. See the survey articles [45], [412], [414] and the paper [220]. Comparing with the lists of properties for arithmetic groups and mapping class groups Modg;n , it is clear that the following conjectures are reasonable: (1) The rank of Out.Fn / as an abstract group is equal to 1. (2) The asymptotic dimension of Out.Fn / is finite and hence the integral Novikov conjectures in various theories hold for Out.Fn /. Since Out.Fn / and its finite index subgroups are not virtual Poincaré duality groups and hence cannot be realized as fundamental groups of closed aspherical manifolds, the Borel conjecture stating that two closed aspherical manifolds with isomorphic fundamental groups are homeomorphic is automatically satisfied by them.

2.6 Properties of outer space Xn and the action of Out.Fn / on Xn The analogue of symmetric spaces and Teichmüller spaces for Out.Fn / is the outer space Xn of marked metric graphs with fundamental group isomorphic to Fn , which was introduced in [101]. This is an infinite simplicial complex with some vertices and simplicial faces missing, and Out.Fn / acts on it simplicially by changing markings of the metric graphs. Though Out.Fn / had been extensively studied earlier in combinatorial group theory, the introduction of outer space Xn and the action of Out.Fn / on it has changed the perspective on Out.Fn /. This is an instance which shows the importance of transformation group theory in understanding properties of groups. The following is a partial list of properties of outer space Xn . (a) Orbits of action (1) Xn is a contractible infinite simplicial complex of dimension 3n 4. When n D 2, the underlying space of Xn can be identified with the upper-half plane H2 . (See [101].) (2) Out.Fn / acts simplicially and properly on Xn . (3) There are only finitely many Out.Fn /-orbits of simplices in Xn . (See [101].) (b) Classifying spaces (4) For every finite subgroup of Out.Fn /, its fixed point set in Xn is nonempty and contractible. Hence Xn is a model of the universal space EOut.Fn / for proper actions of Out.Fn /. (See [420], [248].)

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(5) Xn admits an Out.Fn /-equivariant deformation retraction onto its spine, which is of dimension 2n3, equal to the virtual cohomological dimension of Out.Fn /, and gives a cofinite model of the universal space EOut.Fn / for proper actions of Out.Fn / of the smallest possible dimension. (c) Compactifications (6) Xn admits a compactification on which the Out.Fn /-action on Xn extends continuously. This is an analogue of the Thurston compactification of the Teichmüller space Tg;n . (See [100], [101], p. 93, [412], [220].) (7) Xn admits an analogue of the Borel–Serre partial compactification on which Out.Fn / acts properly and which is .2n 5/-connected at infinity. This was used to prove that Out.Fn / is a virtual duality group. (See [49].) In [204], there is also a realization of this partial compactification of Xn by a truncated subspace Xn ."/ as in the case of the realization of the partial Borel–Serre compactification of a symmetric space by its thick part. Since the outer space Xn is not a manifold, many differential geometric and functional theoretical results for symmetric spaces and Teichmüller spaces have no analogues for Xn . So far, there is no natural complete metric on Xn yet. There are many other results about Out.Fn / and outer spaces that we have not mentioned here. See the survey articles [412], [414], [45] and the paper [220].

2.7 Properties of Coxeter groups Coxeter groups form a large class of groups that often provide interesting examples or counter-examples for various facts. They are also test groups for important properties. Though other groups discussed in this chapter are also special and serve a similar purpose, many properties of Coxeter groups can be determined explicitly from their generators and relations, i.e., their presentations. In this subsection, we list some of the properties of Coxeter groups related to those discussed earlier for arithmetic groups and Modg;n . Briefly, a Coxeter matrix is a symmetric matrix .mij /, i; j D 1; : : : ; n, with entries in N [ f1g satisfying the conditions: mi i D 1, and mij 2 if i ¤ j . The associated Coxeter group is the group defined by the presentation hr1 ; : : : ; rn j .ri rj /mij D 1; i; j D 1; : : : ; ni: In this presentation, when mij D 1, no relation of the form .ri rj /mij is imposed. See [104] for precise definitions and detailed discussions. See also [339] for discussions on related braid groups and Artin groups. Let W be a Coxeter group. Then it satisfies the following properties. (a) Combinatorial properties (1) W is finitely generated, by definition.


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(2) W is finitely presented, by definition. (3) The word problem is solvable for W . (See [104], Theorem 3.4.2, [71], Theorem 1.4, p. 441, [104], p. 5, and Theorem 12.3.3.) (4) The conjugacy problem is solvable for W . (See [246], [18].) (b) Group theoretical properties (5) W is virtually torsion-free. (See [104], Corollary D.1.4.) (6) W is residually finite. (See [104], Proposition 14.1.11.) (7) A strong version of the Tits alternative holds for W : any subgroup of W either contains a subgroup isomorphic to the free group F2 on two generators or is virtually abelian. (See [104], Proposition 17.2.1.) (8) W is a CAT.0/-group, i.e., it acts properly, isometrically and co-compactly on a CAT.0/-space. This procedure provides many CAT.0/-groups. (See Chapter 12 in [104].) (c) Cohomological properties (9) The cohomology and homology groups of W are finitely generated in every degree. (10) The cohomology ring H .W; Z/ is finitely generated, which is an analogue of Evens–Venkov theorem for finite groups. (See [357], [213].) (11) The Euler characteristic of W can be computed explicitly in terms of the presentation. (See Chapter 16 in [104].) (12) Whether W is a virtual Poincaré duality group or not can be determined explicitly. (See [104], Theorem 10.9.2.) (13) There exist cofinite spaces EW for proper actions of W . (See [104], p. 5.) (d) Large-scale properties (14) W has either polynomial growth or exponential growth. (See [104], Proposition 17.2.1.) (15) The growth series of W is a rational function. (See §17.1 in [104].) (e) Rigidity properties (16) For torsion-free subgroups of W , the Borel conjecture stating that assembly maps are isomorphisms holds, since they can be realized as discrete subgroups of GL.n; R/ for some n. (See [132].) A more refined version of the Borel conjecture is the relative Borel conjecture for groups containing torsion elements, and the relative Borel conjecture holds for the whole group W . (See [355], [323].) (17) Infinite Coxeter groups do not have Property T. (See [66].) (18) Every Coxeter group acts amenably on a compact space. (See [110].) The natural spaces associated with W are CW complexes and buildings [104]. They give rise to models of the universal spaces EW of proper actions of W .

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2.8 Properties of hyperbolic groups Another important class of groups consists of hyperbolic groups, which were introduced by Gromov [155] to characterize combinatorially phenomena (or properties) of negative curvature, i.e., fundamental groups of compact negatively curved Riemannian manifolds. Hyperbolic groups are generic groups among all finitely generated groups in some sense and enjoy many good properties. Arithmetic subgroups of real Lie groups of rank at least 2 belong to the opposite ends of the spectrum of finitely generated groups. Another important related class of groups is the class of CAT.0/-groups. See [71]. See [68], [69] for a discussion on various important classes of groups and relations between them. Assume that is a hyperbolic group. (a) Combinatorial properties (1) is finitely generated by assumption. (2) Any non-elementary hyperbolic group, i.e., not containing a cyclic subgroup of finite index, is not boundedly generated. (See [125], Proposition 3.6.) (3) is finitely presented. (4) The word problem is solvable for . (5) The conjugacy problem is solvable for . (See [71] for details and references.) (6) The isomorphism problem is solvable for torsion-free hyperbolic groups. (See [382].) (b) Group theoretical properties (7) A strong version of the Tits alternative holds for : every subgroup of a hyperbolic group is either virtually cyclic or contains a subgroup isomorphic to F2 . (See [71].) (8) admits only finitely many conjugacy classes of finite subgroups. (9) The cohomology and homology groups H i .; Z/, Hi .; Z/ are finitely generated in every degree. Furthermore, is of type FP1 . (10) If contains a torsion-free subgroup of finite index, then the cohomology ring H .; Z/ is finitely generated, which is an analogue of Evens–Venkov theorem for finite groups. (See [357], [213].) (11) The geometric rank of is equal to 1. (See [77] for the definition of the geometric rank.) The algebraic rank of as an abstract group is also equal to 1. (See [22] for the definition of the algebraic rank). (c) Large-scale properties (12) The group with a word metric, or, equivalently, its associated Cayley graph, admits a compactification by adding a boundary consisting of equivalence classes


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of geodesics, called the Gromov boundary. The -action on by multiplication extends continuously to the Gromov compactification. See [222] for an extensive summary on structures of the boundary and actions of on this boundary. (13) The Martin boundary of , which describes the cone of positive harmonic functions, is known to be equal to the Gromov boundary. The asymptotic behavior of random walks on is known. (See [8], [421].) (14) Except for the trivial case of finite and virtually cyclic groups, has exponential growth. (See [13].) (15) The growth series of is a rational function. (See [72], Theorem 3.1, p. 459.) (16) The asymptotic dimension of is finite. (See [368].) (d) Cohomological properties (17) If is virtual torsion-free, then the virtual cohomological dimension of is finite, and it is equal to the dimension of the Gromov boundary of . (18) There exist cofinite models of universal spaces E for proper actions of . (See [302].) (19) If is torsion-free, then is a Poincaré duality group of dimension n if and only if its Gromov boundary has the integral Cech cohomology of S n1 , and is a duality group of dimension n if its Gromov boundary has the integral Cech cohomology of a bouquet of spheres of dimension n 1. (See [53].) (e) Rigidity properties (20) The Borel conjecture and the Farrell–Jones conjecture hold for . (See [25], [24].) (21) If is torsion-free, then is Hopfian, i.e., every epimorphism ' W ! is an isomorphism. (See [381], [85].) (22) satisfies the Kadison–Kaplansky conjecture. (See [356].) (23) satisfies the Baum–Connes conjecture. (See [304].) (24) is weakly amenable. (See [333] and also [98].) There are detailed lists of properties of hyperbolic groups and of the closely related CAT.0/ groups in [269]. The natural spaces associated with hyperbolic groups , which are analogues of symmetric spaces for arithmetic groups and Teichmüller spaces for mapping class groups, are the Rips complexes (or Vietoris–Rips complexes). The Rips complexes have played an important role in studying hyperbolic groups. In the above discussion, we have emphasized similarities between the five classes of groups. On the other hand, there is an important difference between them: the first two classes of groups (i.e., arithmetic groups and mapping class groups Modg;n ) act properly on manifolds naturally associated with them while the latter three classes of groups do not have natural proper actions on manifolds.

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The orbit spaces in the first two cases are some of the most important manifolds (or rather orbifolds) in mathematics: arithmetic locally symmetric spaces and moduli spaces of Riemann surfaces (or algebraic curves), and we can study analysis, topology and geometry of these spaces. The interaction between geometry, topology and analysis on these spaces makes them very special and interesting. We hope that the above lists of properties have conveyed some similarities, differences and interaction between these spaces as well.

3 How discrete groups and proper transformation groups arise In this section, we discuss briefly several sources from which discrete groups arise, either as discrete subgroups of topological groups or as discrete transformation groups (i.e., groups acting properly discontinuously on topological spaces). In some sense, the notion of discrete transformation group is more important than that of discrete group. It is the existence of interesting actions which makes the groups interesting. Group actions also make the spaces interesting. Of course, group actions can also be studied for their own sake.

3.1 Finitely generated groups, Cayley graphs and Rips complexes Probably the most direct way to get discrete groups is to start with a group and endow it with the discrete topology. In general, this does not lead to an interesting discrete group since there are no natural topological spaces with reasonable properties9 on which such a group acts properly. As emphasized at the beginning of this chapter, group actions are needed to understand the groups and also to make the groups interesting. But there are exceptions, and these exceptions often give rise to interesting examples. The first important general case is when is a finitely generated group. Let S be a finite set of generators that is symmetric in the sense that 2 S if and only if 1 2 S . We assume that S does not contain the identity element. Associated with S , there is a word metric dS on defined by dS .x; y/ D jx 1 yjS , where x; y 2 , and jx 1 yjS is the word length of the shortest expression of x 1 y in terms of the generating set S. Clearly left multiplication of on leaves this metric dS invariant, and acts isometrically and properly on .; dS /. On the other hand, .; dS / is a totally disconnected topological space. A closely related connected space is the Cayley graph G .; S /. The vertices of G .; S / are the elements of , and two elements x; y 2 are connected by an edge if and only if x 1 y 2 S . Then G .; S/ is a 1-dimensional -CW complex. Assume that each edge is given length 1. Then G .; S/ becomes a locally compact geodesic length space, 9 Some natural properties we expect from these spaces include the fact that they are CW-complexes, locally compact topological spaces, or manifolds.


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and the natural inclusion ,! G .; S/ gives an isometric embedding of .; dS / into G .; S /. The left multiplication of on extends to an isometric and proper action on G .; S /. If is a free nonabelian group Fn and S is a minimal symmetric set of generators, then G .; S / is a tree and hence contractible. Otherwise, G .; S / is in general noncontractible. There is a fattened version of the Cayley graph, called the Rips complex, a finitedimensional -CW complex,10 which gives rise to cofinite models of E. (See [71], pp. 468–470.) It is constructed as follows. For any positive integer d , define a simplicial complex Rd .; S/ whose k-simplexes consist of .k C 1/-tuples .0 ; 1 ; : : : ; k / of pairwise distinct elements of such that for all 0 i j k, dS .i ; j / d . The 1-skeleton of R1 .; S/ is equal to the Cayley graph G .; S /. It is clear that the action of on G .; S/ extends to an action on Rd .; S / with a compact quotient. It is easy to see that when is finitely presented and d 1, Rd .; S / is simply connected. In some cases, for example, when is a hyperbolic group in the sense of Gromov, it was proved in [155] (see [332], Proposition 2.68) that Rd .; S / is contractible for d 1. It was proved in [302] that for any finite subgroup F , the fixed point set .Rd .; S//F is nonempty and contractible. Therefore, Rd .; S / is a cofinite model of the universal space E for proper actions of . From the point of view of large-scale geometry, there is no difference between finitely generated discrete groups and their Cayley graphs. For a systematic study of large-scale geometry (or asymptotic geometry) of infinite groups, see [156].

3.2 Rational numbers and p-adic norms Let Q be the field of rational numbers. Under the standard embedding Q ,! R, the subspace topology on Q induced from the usual topology of R is not discrete. With respect to addition, Q is not a discrete topological group. On the other hand, there is a natural topological group A which is locally compact and contains Q as a discrete subgroup. Briefly, for every prime number p, there is a p-adic metric dp on Q. The completion of Q with respect to dp is called the field of p-adic numbers and denoted by Qp . Let Y Qp : ADR p

As a ring, A is called the ring of adeles. Then under the diagonal embedding, Q ,! A, Q becomes a discrete subgroup. Similarly, the multiplicative group of nonzero rational numbers Q can be embedded into a locally compact, totally disconnected group , called the ring of ideles, as a discrete subgroup. Both embeddings are important in number theory. See [146]. 10 The Rips complex is also called the Vietoris–Rips complex, or the Vietoris complex. It was first introduced by Vietoris in 1927. After Rips applied the same complex to the study of hyperbolic groups in the sense of Gromov, it was called the Rips complex and popularized by Gromov in 1987.

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The rings A of adeles and of ideles are also related to groups that we are discussing here. For example, given a linear semisimple algebraic group G defined over Q, we can define G .A/, a locally compact group. Then G .Q/ G .A/ is a discrete subgroup. For a compact open subgroup C of G .A/, the quotient G .Q/nG .A/=C is a finite union of locally symmetric spaces nX D nG .R/=K discussed in this chapter.

3.3 Discrete subgroups of topological groups A natural way to produce discrete groups is to take subgroups of topological groups G such that the induced subspace topology on is discrete. As already said, such subgroups are called discrete subgroups. An important example is Zn Rn . Another important and related example is SL.n; Z/ SL.n; R/. These are examples of arithmetic subgroups of linear algebraic groups. The action of on G is proper. More generally, for any compact subgroup K G, the natural left action of on the homogeneous space G=K is also proper. If G=K admits a left G-invariant metric or distance function, then the action is also isometric. A particularly important example is when G is a real Lie group and G=K admits a left G-invariant Riemannian metric. Then nG and nG=K provide many important examples of manifolds and orbifolds.

3.4 Fundamental groups and universal covering spaces Another important source of proper transformation groups comes from fundamental groups of topological spaces. Let M be a connected topological manifold or more generally a connected and locally path connected topological space. Assume that its z be the universal covering space of fundamental group 1 .M / is nontrivial. Let M z , and the quotient nM z is equal M . Then the group D 1 .M / acts properly on M to M . If M is a finite connected graph, then 1 .M / is a free group. If M is a surface of negative Euler characteristic, then 1 .M / is an infinite surface group. If M is a complex algebraic variety, then 1 .M / provides a large natural class of groups. Though algebraic varieties can be constructed easily, properties of their fundamental groups are not easy to describe. It is not easy either to decide whether a group can be realized as such a fundamental group. See the book [7] and the paper [224] for details and references. Instead of smooth manifolds, we can also consider orbifolds and their fundamental groups in the category of orbifolds. For algebraic varieties (or rather schemes), we can also take their algebraic (or étale) fundamental groups. See [303]. The monodromy group of some differential equations with regular singularities also gives rise to interesting discrete subgroups [181], [182], [429], [430], [107].


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Some standard operations on topological spaces such as direct products, smashed products, connected sums also produce direct products of groups, free products of groups, and amalgamated products of groups.

3.5 Moduli spaces and mapping class groups Let S be a topological space. A natural topological space associated with S is the space of self-homeomorphisms of S , Homeo.S /.11 It is a topological group and is often not discrete. Its identity component Homeo0 .S / is a normal subgroup, and the quotient Homeo.S/=Homeo0 .S/ is called the mapping class group of S and denoted by Mod.S /. Since each element of Mod.S / represents a connected component, it seems natural to give Mod.S/ the discrete topology. In general, it is not easy to find a good space on which Mod.S / acts properly. But some special cases provide important examples. (a) Let S D S 1 S 1 , a torus of dimension n. Consider the space of isotopy classes of all marked flat Riemannian metrics on S with total volume 1, where a marking is a choice of a basis of 1 .S/. Any such marked flat manifold corresponds to a marked lattice ƒ of Rn of covolume 1, where a marking on a lattice ƒ is a choice of a basis. It can easily be seen that this moduli space can be identified with the symmetric space SL.n; R/=SO.n/. Then Mod.S / corresponds to SL.n; Z/ and acts properly on this moduli space, and the quotient is the moduli space of flat Riemannian metrics on S with total volume 1. (b) Let Sg be a compact orientable surface of genus g. Assume that g 2. Let Tg be the space of isotopy classes of marked hyperbolic metrics on †, where a marking is also a choice of a basis of 1 .Sg /. Then Modg D Mod.Sg / acts properly on Tg and the quotient is the moduli space Mg of hyperbolic metrics on Sg .

3.6 Outer automorphism groups In group theory, a natural question is this: starting from a group , how to produce new groups from it? There are several natural groups associated with besides taking products. The first group is the group of automorphisms of , Aut./. Similarly, we can consider the group of inner automorphisms Inn./ and the group of outer automorphisms Out./ D Aut./=Inn./. For a countable group , Aut./ and Out./ are countable groups and hence can be reasonably considered as discrete groups. Let Sg be a compact orientable surface of genus g 2, and D 1 .Sg /. Then by the Dehn–Nielsen theorem, Out./ D Modg , the mapping class group. (See [188], [126].) 11 Other spaces that can be derived from S are products and quotients of S and various combinations, for example, the symmetric product.

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Let Fn be the free group on n generators. Then Out.Fn / was mentioned before, and the group Aut.Fn / is also important. The automorphism groups of right-angled Artin groups are closely related to the groups discussed in this chapter. See [94] and references therein. If D Zn , then Out.Zn / D Aut.Zn / D GL.n; Z/. This point of view provides one link between the classes of groups discussed in this chapter.12 On the other hand, it is not obvious how to find a space on which Out./ acts properly besides its Cayley graphs. There are some special cases. One particularly interesting case is when D Fn . Then Out.Fn / acts properly on the outer space of marked metric graphs whose fundamental group is isomorphic to Fn . Another important case is that of Out.1 .Sg // acting properly on the Teichmüller space Tg , the space of marked hyperbolic metrics on the surface Sg .

3.7 Combinatorial group theory Another natural way to construct a discrete group is to specify generators and relations between them. Any group defined by finitely many generators and finitely many relations is countable and hence giving it the discrete topology is natural. On the other hand, given a finitely presented group, it cannot be decided whether the group is finite or not, nor can it be decided whether it is trivial or not. Besides Cayley graphs and Rips complexes, it is not easy to construct spaces on which finitely generated groups act. For most arithmetic groups, it is difficult to find explicit generators and relations. This brings up a natural question: how to effectively describe a group. Probably the most important class of groups constructed by generators and relations is the class of Coxeter groups. It is probably a miracle that many properties can be deduced from the generators and relations of these groups. Furthermore, there are natural spaces with desirable properties on which Coxeter groups act. See [104] for details. See also [99]. There are other important groups whose properties are understood due to their actions on suitable spaces, for example the Thompson group in [81] and the important right angled Artin groups constructed in [46]. See also the papers [80] and [139], and the book [147] for a more systematic study on how topological methods, in particular, actions on CW-complexes, are used to study groups.

12 Unlike the Dehn–Nielsen theorem for compact surfaces, it we take a connected graph whose fundamental group is equal to Fn , for example, the rose Rn with n petals, and apply the construction of mapping class groups in the previous subsection, we will not get Mod.Rn / Š Out.Fn /. It seems natural to consider the following generalization. Let Homtp.Rn / be the group of all homotopy self-equivalences of Rn , and the Homtp0 .Rn / its identify component. Then Homtp.Rn /=Homtp0 .Rn / is a group and should be isomorphic to Out.Fn /.


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3.8 Symmetries of spaces and structures on these spaces Since symmetries in various contexts are described by groups, one reasonable way to construct groups is to consider symmetry group of spaces. The above discussion about Aut./ and Out./ fits well this idea. Different groups arise when different conditions are imposed, i.e., when different kind of symmetries are considered. Considering the vector space Rn and all linear transformations on Rn , we get the general linear group GL.n; R/. If we consider only those linear transformations that preserve the lattice Zn , then we get the discrete subgroup GL.n; Z/. Clearly GL.n; R/ acts on Rn . On the other hand, this action is not proper. To obtain a natural space on which GL.n; R/ acts properly, we note that linear transformations in GL.n; R/ map Zn to other lattices of Rn , and for any lattice ƒ in Rn , its stabilizer in GL.n; R/ is infinite (in fact, it is an arithmetic subgroup with respect to a suitable Q-structure on the algebraic group GL.n; C/) and permutes bases of ƒ. Therefore, the space of lattices with distinguished bases is a natural space on which GL.n; R/ acts properly and transitively, and GL.n; Z/ also acts properly on this space. If we consider the group of all isometries of Rn , which is generated by translations and rotations, then it acts properly on Rn . The subgroup stabilizing the lattice Zn is a discrete subgroup, and it acts properly discontinuously on Rn . More generally, given a metric space .X; d /, its group of isometries I.X/ is a topological group, and I.X/ acts properly on X . There are two special cases. The first case is when I.X/ acts transitively on X , and X is a homogeneous space. Symmetric spaces considered above are special examples of homogeneous Riemannian manifolds. The second case is that of a generic metric space X, where I.X/ is a trivial or at most a discrete group. For example, suppose that M is a manifold and is a discrete group acting properly discontinuously on it by diffeomorphisms. Assume that it acts without fixed points. Take a generic metric on the quotient manifold nM and lift it up to M . Clearly, this metric on M is invariant under and its isometry group is in general equal to . It is not easy to find explicit and natural examples of metric spaces for which I.X/ is an infinite discrete group. In this sense, it is an interesting fact that the isometry group of the Teichmüller space Tg with respect to the Teichmüller metric or Weil–Petersson metric is equal to the mapping class group Modg when g 3.

4 Arithmetic groups In this section, we give a formal definition of arithmetic groups or rather arithmetic subgroups, explain concepts related to the properties of arithmetic groups introduced in §2.1, and indicate briefly how their actions on symmetric spaces can be used to prove some of these properties.

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4.1 Definitions and examples The most basic example of an arithmetic group is the subgroup Z of integers in R. However, R is not a semisimple Lie group. Groups we study in this chapter are natural generalizations of the arithmetic subgroup SL.2; Z/ SL.2; R/. Recall that a subgroup G of GL.n; C/ is called a linear algebraic group if it is an algebraic variety and if its group operations, i.e., the multiplication G G ! G , .x; y/ 7! xy, and the inverse G ! G , x 7! x 1 , are morphisms between algebraic varieties. If the variety G and the group operations are defined over Q, then G is said to be defined over Q, and G is also called a Q-linear algebraic group. The notion of being defined over R can be defined similarly. We note that linear algebraic groups are always defined over C. Important examples of linear algebraic groups include SL.n; C/, the orthogonal group O.Q/ associated with a quadratic form Q in n variables, O.Q/ D fg 2 GL.n; C/ j Q.gx; gx/ D Q.x; x/; x 2 C n g; and the symplectic group Sp.2n; C/ associated with a skew-symmetric bilinear form. The usual special linear group SL.n; C/ is defined over Q. If the quadratic form Q is defined over Q, then O.Q/ is defined over Q. The same thing is true for the symplectic group. These examples indicate that algebraic groups often arise from linear transformations that preserve a certain algebraic structure, i.e., the symmetry group of the algebraic structure. This also supports the basic point of transformation group theory in this chapter. If the algebraic structure is defined over Q, then the algebraic group that preserves it is defined over Q. Given a Q-linear algebraic group G , its Q-locus G .Q/ is well defined. Since it is also defined over R, its real locus G .R/ is a real Lie group with finitely many connected components and is denoted by G, i.e., G D G .R/. For every embedding G GL.n; C/ defined over Q, we can define G .Z/ D G .Q/ \ GL.n; Z/. We emphasize that G .Z/ depends on the embedding of G . Let K G be a maximal compact subgroup. Then the homogeneous space X D G=K is diffeomorphic to Rn , where n D dim X . If G is a reductive algebraic group, for example, G D GL.n; C/, then G D GL.n; R/, and X D GL.n; R/=O.n/ with any invariant Riemannian metric is a symmetric space of nonpositive curvature. If G is a semisimple algebraic group, then X is a symmetric space of noncompact type. Clearly any discrete subgroup G acts properly on G. However, for some applications, it is more convenient to consider the proper action of on X. For example, it is known that X is a model of the universal E-space for proper actions of whether G is semisimple or not, but G is not simply connected and hence not contractible and cannot serve as a universal space for . See [269] for details.


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Definition 4.1. A subgroup G D G .R/ is called an arithmetic subgroup if it is contained in G .Q/ and commensurable with G .Z/, i.e., the intersection \ G .Z/ is of finite index in both and G .Z/. Natural examples of arithmetic subgroups include G .Z/ and its subgroups of finite index. We note that given any linear algebraic group G GL.n; C/ defined over Q, for any g 2 GL.n; Q/, gG g 1 gives another Q-linear algebraic group isomorphic to G , and their Q- and R-loci are isomorphic. On the other hand, gG g 1 .Z/ and G .Z/ are not isomorphic in general. Remark 4.2. A more general definition of arithmetic subgroups of the real Lie group G D G .R/ is as follows. A subgroup G, which is not necessarily contained in G .Q/, is called an arithmetic subgroup if it is commensurable with G .Z/. The following fact is clear from the definition. Proposition 4.3. Let G D G .R/ be the real locus of a linear algebraic group G as above. If is an arithmetic subgroup of G according to Definition 4.1 (or according to the more general one in the above remark), then it is a discrete subgroup of G. If G is a reductive Lie group, then nX is usually called an arithmetic locally symmetric space. It should be emphasized that for a Lie group G, its arithmetic subgroups depend on the Q-structure of G, i.e., on the existence of a Q-linear algebraic group G whose real locus is equal to G. Different Q-structures usually give rise to non-commensurable arithmetic subgroups. For example, SL.2; R/ admits arithmetic subgroups , for example, SL.2; Z/, such that nSL.2; R/ is noncompact, and other arithmetic subgroups 0 such that 0 nSL.2; R/ is compact. Remark 4.4. One good example to illustrate the notion of Q-structure is to consider lattices ƒ of Rn and Q-structures on Rn . Each lattice ƒ defines a Q-structure on Rn , i.e., a Q-linear subspace of dimension n. Let v1 ; : : : ; vn be a basis of ƒ. Then Qv1 C C Qvn defines a Q-linear subspace V of Rn such that V ˝Q R D Rn . Two lattices ƒ1 and ƒ2 define the same Q-structure if and only if ƒ1 \ ƒ2 is also a lattice. Remark 4.5. Another definition of arithmetic groups is as follows. It looks more general at first sight, but turns out to be the same by using the functor of restriction of scalars (see [348] for example). Let k be a number field, i.e., a field that is a finite extension of Q. Let Ok be its ring of integers. Suppose that G GL.n; C/ is a linear algebraic group defined over k. Then any subgroup of G .k/ commensurable with G .Ok / is called an arithmetic subgroup of G . To realize as a discrete subgroup of a real Lie group, we need to use the product of G .k /, where runs over all real and

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complex embeddings, or Archimedean places of k. Embedding into any one of the factors will not give a discrete subgroup in general. The arithmetic subgroups defined in this more general case are also commensurable with “integral” elements. Remark 4.6. Given any Lie group H with finitely many connected components, it is in general not true that H is the real locus of a Q-linear algebraic group G . For example, any Lie group that is not linear, i.e., that cannot be embedded into GL.n; R/ will provide such an example. Alternatively, suppose that G is a Q-simple linear algebraic group such that its real locus G can be written as a product G D G1 G2 such that G1 is noncompact and G2 is compact and has positive dimension. Take H D G1 . Then H is often not the real locus of a Q-linear algebraic group. We can also take H to be the product of G with a compact Lie group that is not linear. For such a Lie group H that differs from the real locus G D G .R/ of a linear algebraic group G by compact Lie groups, arithmetic subgroups are defined as follows. A discrete subgroup H of H is called an arithmetic subgroup if there exists a Q-linear algebraic group G and a Lie group homomorphism ' W G ! H with compact kernel and compact cokernel and an arithmetic subgroup G G such that '.G / is commensurable with H . For convenience, we call such a Q-linear algebraic group G and a Lie group homomorphism ' W G ! H a Q-structure on H . In general, different Q-structures on H give rise to non-commensurable classes of arithmetic groups. For example, the discussions in Remark 4.4 about Q-structures and lattices in Rn illustrate this point. A natural question concerns the size of arithmetic subgroups relative to the ambient Lie groups G. For this purpose, we introduce some definitions. Definition 4.7. A discrete subgroup of a Lie group G with finitely many connected components is called a lattice (or a lattice subgroup) if with respect to any left invariant Haar measure on G, the volume of nG is finite. If is a lattice of G, then the locally homogeneous space nX with respect to any invariant metric has finite area, where X D G=K as above. Definition 4.8. A discrete subgroup or lattice of a Lie group G is called a cocompact (or uniform) lattice if the quotient nG is compact. The arithmetic subgroup Z is a cocompact lattice of R. We note that in order to view C as a linear algebraic group, we identify it with the unipotent linear algebraic group of upper triangular 2 2 matrices with 1s along the diagonal. Then its real locus is R and Z is an arithmetic subgroup. More generally, every lattice of Rn is cocompact. The arithmetic subgroup f˙1g of GL.1; R/ D R f0g is not a lattice. The arithmetic subgroup GL.2; Z/ is not a lattice of GL.2; R/ either. The following results hold (see [360]).


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Proposition 4.9. If G is a nilpotent Lie group, then every arithmetic subgroup of G is a uniform lattice. For the semisimple case, the situation is more complicated and hence more interesting. Proposition 4.10. If G is a semisimple Lie group, then every arithmetic subgroup of G is a lattice. This is an important consequence of the reduction theory for arithmetic groups discussed below. We note that GL.n; Z/ is not a lattice of GL.n; R/, but SL.n; Z/ is a lattice in SL.n; R/. The basic question of when an arithmetic subgroup of a semisimple Lie group is a uniform lattice is answered by the following result (see [57], [197] for references). Proposition 4.11. Assume that G is a semisimiple linear algebraic group. Then an arithmetic subgroup of the real locus G D G .R/ is a uniform lattice of G if and only if does not contain any nontrivial unipotent element, which is equivalent to the condition that the Q-rank of G is equal to 0. For example, SL.2; Z/ contains many unipotent elements such as 10 b1 , where b 2 Z, and hence SL.2; Z/ is a non-uniform arithmetic subgroup of SL.2; R/. Similarly, SL.n; Z/ is a non-uniform arithmetic subgroup of SL.n; R/. Though it is not easy to see it explicitly, SL.n; R/ admits uniform arithmetic subgroups with respect to different Q-structures on SL.n; R/ (or SL.n; C/). In fact, we have the following result of Borel [58]. Proposition 4.12. Every connected semisimple Lie group G contains uniform arithmetic subgroups with respect to suitable Q-structures on G. The basic idea of Proposition 4.12 is to make use of Q-bases of the Lie algebra g of G, i.e., bases such that the structure constants are rational numbers, to construct a form of g over a totally real number field E of degree strictly greater than 1 such that under any non-identity embedding of E into R, g is a compact form of the complex Lie algebra g ˝ C. Then the compactness criterion in Corollary 4.36 below shows that the arithmetic subgroups defined with respect to the number field E are uniform. See [58], p. 116 and Proposition 3.8, for more details. It is perhaps worthwhile to point out that G also admits different Q-structures which admit non-uniform arithmetic subgroups. They are easier to see for classical groups such as SL.n; C/ and Sp.2n; C/ etc. In general, they can be constructed by the Chevalley basis of the Lie algebra g, or rather from the arithmetic subgroups of the Chevalley group associated with the Lie algebra g. For example, consider the two quadratic forms Q1 .x1 ; : : : ; xn / D x12 C C 2 2 axn2 , where a is a positive integer xn1 xn2 and Q2 .x1 ; : : : ; xn / D x12 C C xn1

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such that Q2 .x1 ; : : : ; xn / D 0 has no nontrivial integral solution. They define two Q-linear algebraic groups G1 D O.Q1 / and G2 D O.Q2 /. The quadratic forms are not isomorphic over Q, but G1 .R/ Š G2 .R/. Let G D G1 .R/ Š G2 .R/. Arithmetic subgroups of G with respect to the Q-structure from G1 are not uniform discrete subgroups, but arithmetic subgroups of G with respect to the Q-structure from G2 are uniform discrete subgroups.

4.2 Generalizations of arithmetic groups: non-arithmetic lattices Arithmetic subgroups of Lie groups are natural and provide a large class of lattice subgroups. On the other hand, the class of lattices is strictly larger than the class of arithmetic subgroups of semisimple Lie groups. Recall that a Fuchsian group is said to be of the first kind if its limit set is equal to the whole boundary H2 .1/. Otherwise it is said to be of the second kind. A lattice of SL.2; R/ or PSL.2; R/ is a Fuchsian group of the first kind. For a finitely generated Fuchsian group, the converse is also true. On the other hand, most Fuchsian groups of the first kind are not arithmetic subgroups for the reason that there are uncountably many Fuchsian groups of the first kind, but only countably many arithmetic subgroups of SL.2; R/. In some sense, Teichmüller theory was created to study these nonarithmetic Fuchsian groups. This adds another link between the two classes of groups in the title of this chapter. Though there is abundant supply of non-arithmetic Fuchsian groups, it is not obvious how to construct them explicitly. One important class consists of Hecke triangle groups. In fact, most of the Hecke triangle groups are not arithmetic groups. Recall that for every integer q 3, there is a Hecke triangle subgroup q of SL.2; R/ generated by Sq D 10 2 cos1=q and T D 01 1 0 . Except for q D 3; 4; 6, is not an arithmetic subgroup, i.e., not commensurable with SL.2; Z/. (See, for example, [400], [178], [241]). For relations between Hecke triangle subgroups and Teichmüller theory, see [174]. The isometry group SO.n; 1/ of the real hyperbolic space Hn of dimension n also contains many non-arithmetic lattices [157]. Non-arithmetic lattices only occur in rank 1 semisimple Lie groups. More precisely, the famous arithmeticity theorem of Margulis (see [278], and [198] and the references there) says that if G is a semisimple Lie group of rank at least two and is an irreducible lattice of G, then is an arithmetic subgroup with respect to a suitable Q-structure on G. Among rank 1 semisimple Lie groups, the question of arithmeticity of lattices is open only for SU.n; 1/ when n is at least 4. For a survey of some geometric constructions of lattices in SU.3; 1/, see [341], and for some constructions of lattices, in SO.n; 1/, for example by reflections, see [410]. Lattices of semisimple Lie groups share many properties of arithmetic groups listed in §2.1. In fact, all the properties listed there hold for them. The basic reason is that an analogue of the reduction theory for arithmetic groups holds for lattices of rank 1


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semisimple Lie groups [145] and hence the structure at infinity of associated locally symmetric spaces can be understood. Remark 4.13. Let G be a semisimple Lie group, X D G=K be the associated symmetric space with an invariant Riemannian metric, and G be a lattice. Then nX is a locally symmetric space of finite volume. The spectral theory of the Laplace operator of nX , in particular the question of existence of square integrable eigenfunctions, depends on whether is arithmetic or not. This is an instance where whether a lattice is arithmetic or not makes a big difference. See [178], [241] and the references there for the Phillips–Sarnak conjecture on existence of square integrable eigenfunctions. Unlike the case of arithmetic locally symmetric spaces nX whose volumes can be computed in terms of special values of the Riemann zeta function or L-functions, there is no such formula for non-arithmetic locally symmetric spaces.

4.3 Generalizations of arithmetic groups: S -arithmetic subgroups Another generalization of arithmetic subgroups consists of S-arithmetic subgroups. The reason why it is a natural generalization is the following consideration. Take any set of finitely many elements 1 ; : : : ; m 2 GL.n; Q/ and let be the subgroup h1 ; : : : ; m i. If some of the matrix entries of 1 ; : : : ; m are not integral, then is not a discrete subgroup of GL.n; R/ in general. (Note that might be a discrete subgroup of GL.n; R/. For example, any hyperbolic element of SL.2; R/ generates a discrete cyclic subgroup of SL.2; R/, and this fact is independent of whether is integral or not.) As emphasized in the introduction and Section 3, it is important and fruitful to realize such natural groups as discrete subgroups of some locally compact topological groups which are similar to Lie groups in some sense. Let p1 ; : : : ; pk be the set of primes that occur in the denominators of the matrix entries of 1 ; : : : ; m . Each prime pi gives a completion Qpi of Q. Note that R is the completion of Q corresponding to 1. Let S D f1; p1 ; : : : ; pk g be a finite set of places of Q. (Note that a place of a field is an equivalence class of valuations of the field.) Define the ring ZS of S-integers to consist of rational numbers whose denominators contain only primes from p1 ; : : : ; pk . It is also denoted by ZŒ p11 ; : : : ; p1k . It is clear that D h1 ; : : : ; m i is contained in GL.n; ZŒ p11 ; : : : ; p1k /. It is also clear that under the diagonal embedding, GL.n; ZŒ p11 ; : : : ; p1k / is a discrete subgroup of GL.n; R/ GL.n; Qp1 / GL.n; Qpk /. Therefore, we have realized as a discrete subgroup of the locally compact topological group GL.n; R/ GL.n; Qp1 / GL.n; Qpk /. Given any linear algebraic group G GL.n; C/ defined over Q, there is a subgroup G .ZŒ p11 ; : : : ; p1k / D G .Q/ \ GL.n; ZŒ p11 ; : : : ; p1k / of G .Q/ and G .R/. Definition 4.14. A subgroup of G .Q/ is called an S-arithmetic subgroup if it is commensurable with G .ZŒ p11 ; : : : ; p1k /.

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Proposition 4.15. Under the diagonal embedding, every S-arithmetic subgroup of G is a discrete subgroup of the locally compact topological group G .R/ G .Qp1 / G .Qpk /. Remark 4.16. The set S of places of Q is exactly of the right size so that the product G .R/ G .Qp1 / G .Qpk / contains S-arithmetic subgroups as discrete subgroups. Clearly adding more places will still preserve the discreteness of the image of the diagonal embedding of . Note that if G is a semisimple linear algebraic group, then any S -arithmetic subgroup is a lattice of G .R/ G .Qp1 / G .Qpk / with respect to the Haar measure on the product, and hence adding more places will produce an ambient group which is too big in some sense. Since ZŒ p11 ; : : : ; p1k is not finitely generated, it is not true that for any Q-linear algebraic group G , its S-arithmetic subgroups, in particular, G .ZŒ p11 ; : : : ; p1k /, are finitely generated. Many other properties of arithmetic subgroups listed in §2.1 do not hold for them. If G is semisimple, then S-arithmetic subgroups are finitely generated and finitely presented, and all other finiteness properties, duality and many other properties listed in §2.1 also hold for them. As emphasized before, the action of arithmetic subgroups on symmetric spaces has played an important role in understanding arithmetic subgroups. For S -arithmetic subgroups, symmetric spaces are replaced by products of symmetric spaces and Bruhat– Tits buildings. Since the natural models of E of S-arithmetic subgroups are products of symmetric spaces and Bruhat–Tits buildings and hence are not manifolds, there are no natural Riemannian manifolds associated with S -arithmetic subgroups as locally symmetric spaces associated with arithmetic subgroups. There is no analogue of spectral theory of locally symmetric spaces either, though the notion of automorphic representations still makes sense or one can try to combine the usual Laplacian operator for symmetric spaces and the discrete Laplacian for Bruhat–Tits buildings. Remark 4.17. Qp is an important example of a local compact field arising from the completion of a global field Q. Another important example of global field is the function field of an algebraic curve over a finite field, for example Fp .t /, where Fp is a finite field with p elements, and t is a variable. We can also define linear algebraic groups G over Fp .t/ and S-arithmetic subgroups for any finite set of places S of the global field Fp .t/. Unlike the case of S -arithmetic subgroups of linear algebraic groups over Q, S-arithmetic subgroups of G .Fp .t // usually do not have many finiteness properties. For example, if the rank of G over Fp .t / is positive, then S -arithmetic subgroups of G .Fp .t// are not virtually torsion-free and do not admit a cofinite E-spaces. In fact, S-arithmetic subgroups of G .Fp .t // are not even FP1 . On the other hand, if the rank of G over Fp .t / is zero, S -arithmetic subgroups of G .Fp .t // are virtually torsion-free and admit cofinite model of E-spaces. Many other properties listed in §2.1 hold for them too. See [199] for references.


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4.4 Generalizations of arithmetic groups: non-lattice discrete subgroups and Patterson–Sullivan theory

As discussed before, arithmetic subgroups of linear algebraic groups provide natural examples of discrete subgroups of Lie groups that are lattices. On the other hand, there are many examples of discrete subgroups of Liegroups that are not lattices. For 1 1 example, the subgroup of SL.2; R/ generated by 0 1 is a discrete subgroup but the volume of nSL.2; R/ is not finite (or rather the area of the hyperbolic surface nH2 is not finite). A finitely generated Fuchsian group is of the first kind if and only if it is a lattice subgroup of SL.2; R/. The Fuchsian group constructed above is an elementary subgroup since its limit set ƒ./ contains only one point. There are also many non-elementary Fuchsian groups of the second kind, i.e., nH2 has infinite area. Recall that a Kleinian group is a discrete subgroup that acts isometrically on the real hyperbolic space H3 of dimension 3, i.e., a discrete subgroup of PSL.2; C/ (or SL.2; C/ for convenience). A Kleinian group is called elementary if its limit set in H3 .1/ contains at most 2 points. One interesting way to obtain a non-elementary Kleinian group is to take a cocompact Fuchsian group SL.2; R/. Then the inclusion SL.2; C/ gives a discrete subgroup of SL.2; C/ that it not a lattice, and hence the hyperbolic space nH3 has infinite volume. Its limit set in @H3 is a circle and hence it is not an elementary group. On the other hand, nH3 has finite topology. In general, for the hyperbolic spaces H3 , there is a large class of Kleinian groups that are geometrically finite, for example through combination theorems (see [283]). For a general simple Lie group of rank 1, we can also define geometrically finite discrete subgroups (see [86]). For these geometrically finite Kleinian groups, all the finiteness properties and cohomological properties for arithmetic groups listed in §2.1 hold. There are also important features that are more interesting for discrete subgroups that are not lattices. One particularly interesting example is the Patterson–Sullivan theory concerning measures supported on the limit sets of Kleinian groups . Briefly, the theory says that for any discrete group acting on Hn , there is a class of measures on the limit set of in the boundary at infinity of Hn which is determined by the distribution of points in each -orbit, and these measures reflect the spectral properties and the ergodic theory of the geodesic flow of the quotient manifold nHn . The Hausdorff dimension of the limit set is related to the bottom of the spectrum of the Laplacian on nHn . The bottom of the spectrum also has a positive eigenfunction. See the original papers [343], [397], and the book [330]. There are some generalizations to higher rank Lie groups and their discrete subgroups. See [5], [358], [259]. At one point, it seemed that one difficulty with the higher rank case was the lack of abundant examples of discrete subgroups that are not lattices and hence not too large, but not too small either. By the recent results in [250] and [87], the Hitchin representations of surface groups and maximal representations of surface groups for semisimple Lie

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groups of Hermitian type give classes of Zariski dense discrete subgroups of reductive Lie groups of higher rank that are not lattices.

4.5 Symmetric spaces and actions of arithmetic groups Let G be any Lie group with finitely many connected components, and K G be a maximal compact subgroup. Then the homogeneous space X D G=K is diffeomorphic to Rn , where n D dim X. Any arithmetic subgroup of G acts properly discontinuously on X. If G is a reductive Lie group, for example, G D GL.n; R/, then X with any invariant Riemannian metric is a symmetric space of nonpositive sectional curvature. If G is semisimple, then X is a symmetric space of noncompact type. Recall that a Riemannian manifold M is called a locally symmetric space if for every point x 2 M , the locally defined geodesic symmetry that reverses every geodesic passing though x is a local isometry. A Riemannian manifold X is called a symmetric space if it is locally symmetric, and for every point x 2 X , the local geodesic symmetry extends to a global isometry. We note that a symmetric space is automatically complete. On the other hand, locally symmetric spaces are not necessarily complete. For example, if X is a symmetric space, then for any point p 2 X, the complement X fpg is a locally symmetric space. It is known that if X is a symmetric space, then the identity component of its isometry group Isom0 .X/, denoted by G, acts transitively on X, hence X can be identified with G=K, where K is the stabilizer of any point in X. It is also known that we can always replace G by a reductive Lie group. (Note the isometry group of Rn is not reductive and this is the only exception among symmetric spaces.) It is known that the universal covering of a complete locally symmetric space is a symmetric space. This implies that any complete locally symmetric space can be written in the form nG=K, where G is a reductive Lie group, K is a proper maximal compact subgroup of G, and G is a discrete subgroup. According to the definition, a locally symmetric space should be a smooth manifold. Since many natural arithmetic groups contain torsion elements, the quotient spaces nX are often not smooth manifolds, but rather orbifolds. In view of this, whenever G is reductive, for any discrete subgroup G, nX is usually called a locally symmetric space as well. Of course, the most interesting class of locally symmetric spaces consists of locally symmetric spaces of finite volume. It is also known that locally symmetric spaces are characterized by the condition that the curvature tensor is parallel, i.e., the covariant derivative of the curvature tensor is zero. This immediately implies that if M is a Riemannian manifold of constant sectional curvature, then it is a locally symmetric space. In particular, hyperbolic manifolds are locally symmetric spaces. It is also known that a simply connected symmetric space X can be uniquely written as a product Rn X1 Xm , where each Xi is irreducible in the sense


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that whenever Xi is not a product of two Riemannian manifolds, or equivalently if Xi is written as Gi =Ki , where Gi is the identity component of the isometry group of Xi , then the associated involutive Lie algebra or the pair .gi ; ki / is irreducible. The factor Rn is called the flat factor of X . We note that in the above decomposition, the assumption that X is simply connected is important. For example, for any lattice ƒ Rn , the quotient ƒnRn is a symmetric space. If ƒ is irreducible, i.e., there is no isometric splitting Rn D Rn1 Rn2 such that ƒ D .ƒ \ Rn1 f0g/ .ƒ \ f0g Rn2 /, then ƒnRn is not isometric to a product, though Rn is reducible. The Euclidean space Rn is a flat symmetric space. A nonflat irreducible symmetric space X has either nonpositive sectional curvature or nonnegative sectional curvature. If the sectional curvature of X is nonpositive, X is called of noncompact type, and otherwise it is called of compact type. The two important examples are the real hyperbolic space Hn and the unit sphere S n in RnC1 . A symmetric space is called of compact type if it is simply connected, does not contain a nontrivial flat factor Rn , and if its irreducible factors are of compact type. Symmetric spaces of noncompact type can be defined similarly. A very important notion concerning the geometry of symmetric spaces is the notion of rank. A flat subspace of dimension r of a symmetric space X is an isometric immersion Rr ! X. When X is of compact type, the image is compact, isometric to a flat torus. If X is of noncompact type, then Rr ! X is an isometric embedding. The maximal dimension of flat subspaces of X is called the rank of X . The real hyperbolic space Hn is of rank 1, and the symmetric space SL.n; R/=SO.n/ is of rank n 1. Rank is additive in the sense that the rank of the product X1 X2 is the sum of the ranks of X1 and X2 . If G D G .R/ is the real locus of a linear algebraic group G defined over R, then the rank of X is equal to the R-rank of G , i.e., the maximal dimension of R-split tori contained in G . In fact, maximal flats in X are orbits of the real locus of such maximal split tori in G . The volume of a ball of radius R in a symmetric space X of noncompact type grows exponentially in R. In fact, let g D k C p be the Cartan decomposition of the Lie algebra g of G, and a p be a maximal abelian subalgebra. Then we have a root space decomposition of g: X g˛ : g D g0 C ˛2†.g;a/

Choose a positive chamber of a and hence a set of positive roots †C .g; a/. Let be the half sum of positive roots with multiplicity given by dim g˛ . The Killing form of g induces an invariant Riemannian metric on the symmetric space X D G=K. Let x0 be the basepoint of X corresponding to the identity coset K G. Let B.x0 ; R/ be the ball of radius R with center at x0 . Then it is well known that lim

R!C1

log vol.B.x0 ; R// D 2kk: R

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More precise information is known [240], Theorem A, and §6: as R ! C1, vol.B.x0 ; R// R

r1 2

e 2kkR ;

where r is the rank of X . For studying topological properties of arithmetic groups, the following result is important. Proposition 4.18 (Cartan fixed point theorem). Assume that G is semisimple, and X is a symmetric space of noncompact type. Then for any compact subgroup C of G, the set of fixed points of C in X is a nonempty totally geodesic submanifold. Proof. Since X is a simply connected and nonpositively curved Riemannian manifold, every compact subgroup C of G has at least one fixed point in X. In fact, for any point x 2 X, the center of gravity of the orbit C x exists and is fixed by C . Since C acts by isometries on X, its set of fixed points is a totally geodesic submanifold.

4.6 Fundamental domains and generalizations Suppose that a discrete group acts properly discontinuously on a topological space X. An effective way to understand the quotient nX and properties of is to find good fundamental domains for the -action on X. For example, suppose that X is a measure space and the -action preserves the measure. It is naturally expected that the measure descends to a measure on the quotient. It turns out that it can be defined using a measurable fundamental set (see the discussion after Proposition 4.20). Since there have been many different notions of fundamental sets, fundamental domains and fundamental regions, we will recall several definitions in order to clarify their meanings. Probably the most obvious definition of a fundamental set for a -action on X is a subset of X that meets every -orbit once. Its existence follows from the axiom of choice. In general, we impose some additional structures on fundamental sets so that they can be used to understand the quotient nX as a topological space or with another more refined structure. Due to the conventional meaning of fundamental sets in the reduction theory of arithmetic groups, we reserve the name fundamental set for something else in dealing with actions of arithmetic groups. Since X is a topological space and acts by homeomorphisms, a natural notion is that of fundamental domain. Recall that an open subset of X is called a fundamental domain of the -action on X if the following conditions are satisfied: S x x cover X, X D (1) The -translates of the closure 2 , (2) The -translates of are disjoint, and hence the map ! nX is injective. x is (3) The boundary @ is small in a certain sense, for example, the interior of equal to . If X is a measure space, it is natural to impose that the boundary x and nX have the same total @ has measure 0, so that we expect that , measure.


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x we will also call Since it is sometimes more convenient to describe the closure , x or even some subsets between and , x a fundamental domain for the -action. , Remark 4.19. If X is a smooth manifold, and is the interior of a submanifold with corners, then the conditions in (3) are certainly satisfied. But for general spaces X and -actions, fundamental domains have more complicated structures. Usually we require the boundary of to be not too complicated and small in some sense. But in general, the existence of such a fundamental domain in X is not clear. As shown below, if X is a Riemannian manifold and acts isometrically on X , then there always exist such fundamental domains (Proposition 4.22). If X is a measure space and the -action is measure preserving, then it is more natural to require that fundamental sets be measurable subsets, for example, Borel sets. Proposition 4.20. Let X be a second countable topological space, and let be a measure on X which is preserved by the -action. Then there exists a Borel subset F of X that meets every -orbit exactly once. When X is taken to be a topological group G, in [390], if a subset F of G satisfies the conditions: (1) F D G, (2) F meets every -orbit exactly once, i.e., for two different elements 1 ; 2 2 , 1 F \ 2 F D ;, (3) F is a Borel set, then F is called a fundamental set of the subgroup . If G is second countable, then such fundamental sets were constructed in [390] (Lemma 2). The same proof works in the above more general situation. Once we have constructed such a measurable fundamental set F , we can define a measure on nX as follows. Let W X ! nX be the projection. Then a subset S nX is defined to be measurable if 1 .S / \ F is measurable, and we define .S/ D . 1 .S / \ F /: It can be shown that this definition of the measure on nX is independent of the choice of F . It is often convenient and important to impose some finiteness conditions on fundamental domains. One such condition is local finiteness: for any compact subset x \ C ¤ ;g is finite, i.e., any compact subset C is covered C X, the set f 2 j x In [390], fundamental sets satisfying this local by only finitely many translates of . finiteness are called normal fundamental sets. The -action on X induces an equivalence relation on X, and it induces an equivx Denote the quotient by = x . Denote the projection alence relation on the closure . x defines a map = x ! nX, also denoted map X ! nX by . Its restriction to by .

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Proposition 4.21. Assume that is a locally finite fundamental domain for the x ! nX is a homeomorphism. action on X . Then the map W = See [29] for a proof, Theorem 9.2.4. This proposition says that up to homeomorx by identifying suitable points on phism, nX can be obtained from the closure the boundary. This is one instance where a fundamental domain can be used to understand the quotient nX as a topological space, i.e., by identifying some points on the boundary of . The best example to illustrate this result is to consider the action D Z2 on R2 by translation. Then the open unit square is a fundamental domain, and the quotient Z2 nR2 is obtained by identifying points on the boundary intervals of the unit square to get a torus. Another good example is to take the standard fundamental domain for the SL.2; Z/-action on the upper half plane H2 . Identifying the sides, we can show that SL.2; Z/nH2 is homeomorphic to C. Another important finiteness condition is the global finiteness condition: f 2 j x x ¤ ;g is finite, i.e, each translate of x meets only finitely many other translates \ by elements of , and the overlap on the boundary of these -translates is uniformly bounded. The importance of a fundamental domain satisfying the global finiteness condition is that its existence implies is finitely generated (see Proposition 4.39 below, and [390], [29] (Theorem 9.2.7) or [348]). This probably explains why it is not obvious that fundamental domains satisfying global or local finiteness conditions should exist for a general proper action of a discrete group. Rough (or coarse) fundamental domains. It is often difficult to find or construct fundamental domains. A subset R of X is called a rough (or coarse) fundamental domain for the -action on X if the following conditions are satisfied: x cover X , i.e., R x meets every -orbit. (1) The -translates of R (2) R meets every -orbit at most finitely many times. In this case, we usually do not impose conditions on the boundary of R, though many examples in applications do have small boundaries in some sense, for example, x is equal to R. we often take R to be an open subset, and the interior of the closure R It is often easier to construct and describe rough fundamental domains than fundamental domains, and their structures at infinity are simpler in general. Picking out a fundamental domain inside a rough fundamental domain might be complicated. When a symmetric space X D G=K is not a hyperbolic space, the action of arithmetic subgroups of G on X provides such examples. But for some applications, rough fundamental domains satisfying suitable conditions are sufficient. From the above definitions, it is clear that a fundamental domain is a rough fundamental domain. Usually there are some finiteness conditions imposed on rough fundamental domains as well. The local finiteness is satisfied by many known fundamental domains. But we often impose the stronger global finiteness condition requiring that the subset x\R x ¤ ;g is finite, i.e., each translate of R x meets only finitely many f 2 j R x ! nX is uniformly finite-to-one. other translates, which implies that the map R


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This condition is important in combinatorial properties of and its existence implies that is finitely generated (See Proposition 4.39). Rough fundamental domains constructed in the reduction theory of arithmetic groups acting on symmetric spaces satisfy such a global finiteness condition. The global finiteness condition is usually called the Siegel finiteness condition, and the rough fundamental domains are usually called fundamental sets in [57] and in literature on arithmetic groups and automorphic forms. We should emphasize that this is not the fundamental set defined at the beginning of this subsection and in other places such as [29] and [390]. Dirichlet fundamental domains. If X is a proper and complete metric space and acts isometrically and properly discontinuously, then a convenient way to obtain a fundamental domain is to take the Dirichlet fundamental domain. Suppose that there exist points in X that are not fixed by any nontrivial element of . For any basepoint x0 2 X not fixed by any nontrivial element of , define x 0 ; / D fx 2 X j d.x; x0 / d.x; x0 / for all 2 g: D.x x 0 ; / at least Assume that X is locally compact. Then every -orbit meets D.x once. One way to see this is as follows: in each -orbit, pick the set of points of minimal distance from x0 . Since acts properly discontinuously on X and X is a proper metric space, such points exist. The union of such points of minimal distance x 0 ; /. to x0 is equal to D.x Replacing the non-strict inequalities by strict inequalities, we obtain a domain D.x0 ; / D fx 2 X j d.x; x0 / < d.x; x0 / for all 2 g: This is usually called the Dirichlet domain of with center at x0 . x 0 ; / (or the It is natural to guess that the closure of D.x0 ; / is equal to D.x x interior of D.x0 ; / is equal to D.x0 ; /) and is a fundamental domain for the action. But this is not true for general metric spaces. The counterexample in [328] explores the following non-intuitive fact: there exists a metric space .X; d / such that for two different points p1 ; p2 , the bisector fx 2 X j d.x; p1 / D d.x; p2 /g contains open subsets of X. For example, take X D R2 with the L1 -metric, d..x; y/; .x 0 ; y 0 // D jx x 0 j C jy y 0 j; and the points p1 D .1; 1/; p2 D .1; 1/. Then the bisector contains both the second and fourth quadrants of the plane R2 shifted by the points .1; 1/ and .1; 1/ respectively. Proposition 4.22. Assume that X is a complete Riemannian manifold or a Euclidean simplicial complex (i.e., its metric restricts to the standard Euclidean metric on each simplex) and is complete. Then the following results hold: (1) The bisector of every pair of different points is a subset of X of codimension 1. x 0 ; /. (2) The closure of D.x0 ; / is equal to D.x

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(3) D.x0 ; / is a locally finite fundamental domain for the -action on X. In particular, the -action admits a fundamental domain in the sense defined above. (1) can be proved by contradiction and the fact that any minimizing geodesic segment connecting two points is smooth, and (2) follows from (1). (3) follows from the proof of a similar result in [29]. Under the above condition, D.x0 ; / is called the Dirichlet fundamental domain of x 0 ; / a Dirichlet the -action with center x0 . Sometimes, we also call its closure D.x fundamental domain for the -action as well. Recall that the property that a simply connected complete Riemannian manifold X has no conjugate point means that every pair of distinct points of X are joined by a unique geodesic segment up to parametrization. This condition is satisfied if X is a Hadamard manifold, i.e., a simply connected complete Riemannian manifold of nonpositive sectional curvature. If X is a simply connected complete Riemannian manifold without conjugate points, more structure of the boundary of the Dirichlet fundamental domain is known. See [116].

4.7 Fundamental domains for Fuchsian groups and applications to compactification Though the Dirichlet fundamental domain for any -action on X is canonically defined once the center x0 is fixed, it is usually useful only for special spaces X . For example, when X is the Euclidean space Rn , Dirichlet introduced this notion for lattices ƒ Rn in 1850. It is closely related to the more general Voronoi cells. Later Poincaré generalized the notion of Dirichlet fundamental domains to discrete isometric actions on hyperbolic spaces. When X is the hyperbolic plane H2 and is finitely generated, every Dirichlet fundamental domain D.x0 ; / is bounded by finitely many geodesics. In particular, D.x0 ; / satisfies both the local and global finiteness properties mentioned in the previous subsection. Dirichlet fundamental domains have played an important role in the study of Fuchsian groups . For example, assume that is a lattice. Then it is known that any Dirichlet fundamental domain D.x0 ; / is bounded by finitely many geodesic sides and hence is finitely generated [391]. The Poincaré upper half plane H2 admits a natural compactification by adding the boundary circle H2 .1/ D R [ f1g. The limit points of D.x0 ; / in the boundary circle are called cusp points of D.x0 ; /. If is a non-uniform lattice, or a Fuchsian group of the first kind, then D.x0 ; / has finitely many cusp points at infinity and they correspond to parabolic subgroups of defined below (or -rational parabolic subgroups of SL.2; R/). (Note that the cusps of the quotient nH2 correspond to -conjugacy classes of parabolic subgroups of , but some cusp points of D.x0 ; / may be projected to the same cusp of nH2 .)


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We note that the -action on H2 extends to the compactification H2 [ H2 .1/. We call a point in H2 .1/ a -rational boundary (or cusp) point if it is -equivalent to a cusp of D.x0 ; /. Define a subgroup of to be a parabolic subgroup if it is the stabilizer of a -cusp point. Then it can be shown that each parabolic subgroup consists of only parabolic elements, and every parabolic element is contained in some parabolic subgroup of . Since D.x0 ; / has only finitely many cusp points, it follows that contains finitely many conjugacy classes of parabolic subgroups, and parabolic elements of are conjugate to elements that fix some cusps of D.x0 ; /. The above notion of parabolic subgroups of is from the theory of Fuchsian groups. According to the general definition from the theory of Lie groups and algebraic groups, a closed subgroup P of SL.2; R/ is called a parabolic subgroup if and only if the quotient P nSL.2; R/ is compact. It can be proved that a subgroup of SL.2; R/ is a parabolic subgroup if and only if it fixes a boundary point in H2 .1/. We call a parabolic subgroup of SL.2; R/ -rational if it fixes a -cusp point. Then the following result holds and clarifies the relation between two definitions of parabolic subgroups. Proposition 4.23. For any -rational parabolic subgroup P of SL.2; R/, the intersection P \ is a parabolic subgroup of , and every parabolic subgroup of is of this form. Another characterization of -parabolic subgroups is the following one. Proposition 4.24. A parabolic subgroup P of SL.2; R/ is -rational if and only if the intersection P \ is a lattice of the unipotent radical NP of P , which is equivalent in this case to the condition that P \ is an infinite subgroup. When P is the parabolic subgroup consisting of upper triangular matrices, then NP is the subgroup consisting of upper triangular matrices with 1s on the diagonal. One consequence of this result is the following. Proposition 4.25. There is a one-to-one correspondence between the set of conjugacy classes of -rational parabolic subgroups of SL.2; R/ (or equivalently the set of conjugacy classes of parabolic subgroups of ) and the set of ends of nH2 . These results on relations between -parabolic subgroups and -cusp points can be used to construct compactifications of nH2 . For example, by adding all -cusps to H2 , we get an enlarged space that lies between H2 and H2 [ H2 .1/. Naturally, it has the subspace topology induced from the compactification of H2 . By strengthening this induced subset topology so that for every cusp point, it has a neighborhood basis, each of which is stabilized by the corresponding parabolic subgroup of , we obtain a S

partial compactification H2 . The strengthened topology is called the Satake topology.

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It is perhaps helpful to point out that the induced subspace topology does not contain any neighborhood basis of a cusp point that is stable under the stabilizer of the cusp. In fact, if we start with any neighborhood U of the cusp in the compactification H2 [ H2 .1/, then for in the stabilizer of the cusp, the translates U cover the compactification H2 [ H2 .1/, and the translates U \ H2 cover the whole space H2 . Using the Satake topology, it can be proved that acts continuously on the partial S

S

compactification H2 with a compact, Hausdorff quotient nH2 . S

The compactification nH2 is obtained from nH2 by adding one point to every end (or cusp neighborhood). This is the simplest example of Satake compactifications of locally symmetric spaces and also of the Baily–Borel compactification of Hermitian locally symmetric spaces. See §4.17 for the general case. The same procedure can be applied to construct the Borel–Serre compactification S

of nH2 . In the partial compactification H2 , blow up every cusp point to R, which is equal to NP , where P is the corresponding -rational parabolic subgroup and NP is the unipotent radical of P . The resulting space is the Borel–Serre partial compactification BS H2 . It is a real analytic manifold with boundary and acts on it real analytically and BS properly. The quotient nH2 is a compact manifold with boundary. It is mapped S

surjectively to the Satake compactification nH2 , and the inverse image of every S

boundary point of nH2 is equal to a circle. See §4.18 for the general case. S

The difference between these two compactifications is that nH2 admits a comBS plex structure as a compact Riemann surface, while nH2 is a manifold with boundary. Furthermore, the inclusion nH2 ! nH2

S

is not a homotopy equivalence S

since the loops around the cusps are homotopically trivial in nH2 , but the inclusion BS nH2 ! nH2 is a homotopy equivalence. BS

When is torsion-free, nH2 is a finite model of B-space. If contains BS torsion elements, then the Borel–Serre partial compactification H2 is a cofinite model of E. For other spaces, for example, symmetric spaces X D G=K that are not real hyperbolic spaces and of higher rank, there is no such nice structure of the Dirichlet fundamental domains as above. For example, D.x0 ; / is not bounded by totally geodesic hypersurfaces. If nX is noncompact, the notion of cusps is not defined to satisfy the above simple and clean relation with Q-parabolic subgroups of G , and the structure near infinity of D.x0 ; / is often complicated and not adapted to parabolic subgroups of G. Therefore, Dirichlet fundamental domains are not suitable for understanding analysis, geometry and compactifications of nX . In some sense, the reduction theory of arithmetic subgroups is about finding suitable fundamental domains or rough fundamental domains for actions of arithmetic subgroups on symmetric spaces that reflect structures of and G as in the case of Dirichlet fundamental domains for actions of Fuchsian groups on H2 . It turns out


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that fundamental sets (or rough fundamental domains) defined in terms of Siegel sets of parabolic subgroups serve such purposes well and hence are used in the reduction theory of arithmetic subgroups [57]. One major application of the reduction theory of arithmetic subgroups is to conS

BS

struct compactifications of nX similar to nH2 and nH2 . For example, they allow us to pick out “rational boundary points”. In the above discussion, we started with a Fuchsian group and obtained Dirichlet fundamental domains and used them to study the quotient space nH2 and parabolic subgroups of . Dirichlet fundamental domains are also useful in describing combinatorial properties of Fuchsian groups . In fact, there are elements of that pair geodesic sides of D.x0 ; /. These elements generate and relations between them can also be read off from their actions on the sides of D.x0 ; /. An important feature of Fuchsian groups is that we can reverse this process and construct Fuchsian groups from suitable hyperbolic polygons by giving generators and relations. This is called the Poincaré polygon theorem. There is also a higher dimensional generalization which replaces polygons by polyhedra. Probably the best examples are given by the Hecke triangle groups [400]. There is also the Klein combination theorem for Kleinian groups. The Klein combination theorem also works for groups acting on hyperbolic spaces in higher dimensions. See [29], Theorem 9.8.4, and [283]. It is perhaps worthwhile to point out that there is no analogue of the Poincaré polyhedron theorem or the Klein combination theorem for other symmetric spaces.

4.8 Minkowski reduction theory for SL.n; Z/ As discussed in the previous subsection, good fundamental domains for Fuchsian groups have played an important role in understanding the structure of Riemann surfaces and their compactifications, and also algebraic structures of Fuchsian groups. For arithmetic groups, the original motivation for reduction theory was slightly different. It was started by Lagrange and Gauss. Note that the Poincaré upper half plane H2 D SL.2; R/=SO.2/ can be identified with the space of binary positive definite quadratic forms of determinant 1, since SL.2; R/ also acts transitively on the latter, with the stabilizer of the quadratic form x 2 C y 2 equal to SO.2/. The quotient SL.2; Z/nH2 can be identified with the equivalence classes of such quadratic forms, where two quadratic forms Q1 .x; y/, Q2 .x; y/ are defined to be equivalent if they become equal under a change of variables by an element of SL.2; Z/. Consequently, two equivalent quadratic forms represent the same set of values over the integers. An important problem is to find “good” representatives in each equivalence class, and the notion of reduced quadratic form was introduced by Lagrange and Gauss. These representatives correspond to points of the usual fundamental domain fz 2 H2 j jzj 1; jRe.z/j 12 g for the SL.2; Z/-action on H2 .

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After that, the problem of finding fundamental domains (or rough fundamental domains) for arithmetic subgroups was called reduction theory. For n 3, a reduction theory for D SL.n; Z/ acting on SL.n; R/=SO.n/ was developed by Minkowski. Since this motivates directly a generalization for the action of the mapping class group Modg on the Teichmüller space Tg which we discuss below in §5.11, we briefly recall its definition. For the original papers of Minkowski, see [305]. See also the books [392], [402] for more detail. For various purposes, it will be easier to consider reduction theory for the action of SL.n; Z/ on GLC .n; R/=SO.n/, which can be identified with the space of positive quadratic forms in n-variables, denoted by Pn . The subspace SL.n; P R/=SO.n/ is denoted by SPn . For each positive quadratic form Q.x1 ; : : : ; xn / D ni;j D1 qij xi xj , denote its associated symmetric matrix .yij / by Q as well. Let e1 ; : : : ; en be the standard basis of Zn as above. Define the Minkowski reduction domain by DnM D fQ 2 Pn j qi i Q.v/; for all v 2 Zn f0g such that e1 ; : : : ; ei1 ; v can be extended to a basis of Zn g:

(1)

For each v 2 Zn , the condition qi i Q.v/ gives a linear equality on the coefficient matrix .qij /. Therefore, DnM is a convex subset of Pn or rather of the linear space n of symmetric n n-matrices. In particular, it is topologically a cell. This is one place where the linear and convexity structures of Pn are crucial, and hence Pn instead of the subspace SL.n; R/=SO.n/ is used. Proposition 4.26. The Minkowski reduction domain DnM is a fundamental domain for the action of SL.n; Z/ on Pn D GLC .n; R/=SO.n/: for 2 SL.n; Z/, the translates DnM cover the whole space Pn , and without overlap in the interior. To prove this, we need to show that for any Q 2 Pn , there exists A 2 SL.n; Z/ such that the symmetric matrix QŒA D At QA is contained in DnM . The idea for finding this matrix A is to find its column vectors v1 ; : : : ; vn one by one. For this purpose, we introduce the notion of reduced basis of Zn . An ordered basis v1 ; : : : ; vn of Zn is called a reduced basis with respect to the positive quadratic form Q if the following conditions are satisfied: (1) The first vector v1 is a nonzero vector v in Zn which minimizes the values Q.v/: Q.v1 / D

min

v2Zn f0g

Q.v/:

Clearly such a vector v1 has co-prime coordinates and can be extended to a basis of Zn .


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(2) For each i 2, vi is a vector among all vectors v such that v1 ; : : : ; vi1 ; v can be extended to a basis of Zn , and Q.vi / takes the minimum value: Q.vi / D

min

v2Zn ; v1 ;:::;vi 1 ;v forms part of a basis of Zn

Q.v/:

It is clear that for any positive definite quadratic form Q, there exists an associated reduced basis of Zn . On the other hand, there may exist more than one reduced basis. Given the above definition, a quadratic form Q 2 Pn is Minkowski reduced, i.e., Q 2 DnM , if and only if the standard basis e1 ; : : : ; en of Zn is a reduced basis. For any Q 2 Pn , to construct a matrix A 2 SL.n; Z/ such that QŒA 2 DnM , we take a reduced basis v1 ; : : : ; vn of Zn with respect to Q. Let A be the matrix whose column vectors are v1 ; : : : ; vn . By reversing the sign of one vector if necessary, we can see that A 2 SL.n; Z/. Then the standard basis e1 ; : : : ; en forms a reduced basis of QŒA. Therefore, QŒA is Minkowski reduced and contained in DnM In order to generalize this Minkowski reduction to the action of Modg;n on Tg , we formulate it in terms of lattices ƒ Rn , or equivalently tori Rn =ƒ. Given a marked lattice ƒ Rn with an ordered basis v1 ; : : : ; vn , let A D .v1 ; : : : ; vn / be the matrix formed from the basis. Then At A is a positive definite quadratic form. Conversely, any positive definite quadratic form Q can be written as At A, where A is uniquely determined up to multiplication by elements of O.n/. If we require det A D 1, then A is uniquely determined up to multiplication by elements in SO.n/. Let ƒ D AZn , v1 D Ae1 ; : : : ; vn D Aen . Then ƒ together with v1 ; : : : ; vn is a marked lattice of Rn . In terms of the torus (or flat manifold) Rn =ƒ, a marked lattice corresponds to a flat manifold M D Rn =ƒ together with the choice of an ordered minimal set of generators of the fundamental group 1 .M /. n Given any lattice ƒ R with the standard norm defined by the quadratic Pnendowed form Q0 .x1 ; : : : ; xn / D iD1 xi2 , an ordered basis v1 ; : : : ; vn is called a reduced basis if the following conditions are satisfied: (1) v1 is of shortest length among all nonzero vectors of ƒ. Clearly, such a vector v1 is not a nontrivial integral multiple of any vector of ƒ and hence can be extended to a basis of ƒ. (2) For every i 2, vi is of shortest norm among all vectors v 2 ƒ such that v1 ; : : : ; vi1 ; v can be extended to a basis of ƒ. We note that in defining a reduced basis of Zn with respect to a positive definite quadratic form Q, we use the standard lattice Zn and a general positive quadratic form Q. On the other hand, for a reduced basis of a lattice ƒ, we use the standard quadratic form Q0 . Of course, the two reduced bases correspond to each other under the identification between positive definite quadratic forms and marked lattices defined above. Naturally, a marked lattice .ƒI v1 ; : : : ; vn / is called Minkowski reduced if the ordered basis v1 ; : : : ; vn is a reduced basis of ƒ. Summarizing the above discussion, we have the following results:

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Proposition 4.27. The Minkowski fundamental domain DnM for the action of SL.n; Z/ on Pn D GLC .n; R/=SO.n/ is characterized by the following equivalent conditions: (1) A positive quadratic form Q 2 Pn is contained in DnM if and only if the standard basis e1 ; : : : ; en is a Minkowski reduced basis of Zn with respect to Q. (2) A marked lattice .ƒI v1 ; : : : ; vn / is contained in DnM if and only if the basis v1 ; : : : ; vn is Minkowski reduced. Proof. (1) follows from the definition of DnM , and (2) follows from the fact that if .ƒI v1 ; : : : ; vn / is a marked basis, then the standard basis e1 ; : : : ; en of Zn is reduced with respect to the positive definite quadratic form Q D At A, where A D .v1 ; : : : ; vn /, if and only if v1 ; : : : ; vn forms a reduced basis. Proposition 4.26 only gives the most basic properties of the Minkowski fundamental domain. By definition, the Minkowski reduction domain is defined by infinitely many inequalities: for every i, qi i Q.v/, where v is any vector v 2 Zn f0g such that e1 ; : : : ; ei1 ; v can be extended to a basis of Zn . Of course, there are in general infinitely many such vectors v. A natural question is whether finitely many inequalities are sufficient. The positive answer is an important result of Minkowski. We list this and other important results in the following proposition. Proposition 4.28. The Minkowski reduction domain DnM is defined by finitely many inequalities and is hence a convex polyhedral cone with finitely many faces. The tiling of the space Pn of positive definite quadratic forms by the translates DnM , 2 SL.n; Z/, is locally finite and each translate meets only finitely many others; hence DnM is a fundamental domain satisfying both the locally finite and globally finite conditions. The intersection of DnM with the subspace S Pn of Pn consisting of quadratic forms of determinant 1 is a cell, and the translates .DnM \ S Pn /, 2 SL.n; Z/, give an equivariant CW-complex structure of SL.n; R/=SO.n/ D SPn with respect to SL.n; Z/. In order to prove this, the fundamental theorems of Minkowski on the geometry of numbers are needed. See [305], [392], [402] for details. The (first) fundamental theorem of geometry of numbers is the following (see [392], p. 12, [158], [305]): Proposition 4.29. If a bounded convex domain K of Rn that contains the origin and is symmetric with respect to the origin has volume vol.K/ > 2n , then K contains at least one nonzero point of Zn . An immediate corollary is the following:


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Proposition 4.30. Let vol.B1 / be the volume of the unit ball in Rn with respect to the standard metric. Then for any lattice ƒ of Rn of covolume 1, there exists a nonzero vector v 2 ƒ such that 2n kvkn : vol.B1 / In particular, for any reduced basis v1 ; : : : ; vn of ƒ, kv1 kn

2n : vol.B1 /

The question of whether the norms of vectors in a reduced basis can be uniformly bounded is natural and it has also a positive answer [392] (First Finiteness Theorem, p. 99): Proposition 4.31. For any lattice ƒ Rn of covolume 1 and any reduced basis v1 ; : : : ; vn , the norms of the basis vectors satisfy the bounds: 1 3 2n 2n kv1 k : : : kvn k . /n.n1/=2 : vol.B1 / nŠ vol.B1 / 2 In some sense, this result says that a reduced basis tends to be as orthogonal as possible. In Propositions 4.30 and 4.31, we could have stated similar results for general lattices of Rn instead of covolume 1. The point of these results is that the bounds on norms of the vectors in a reduced basis are independent of lattices but only depend on the co-volumes of the lattices. These results seemed to motivate the existence of pants decompositions of hyperbolic surfaces such that lengths of geodesics in the pants decompositions are bounded by the Bers constants in Proposition 5.25 below. Remark 4.32. The second finiteness theorem of Minkowski in [392] (p. 127) refers to the fact that the Minkowski reduction domain is defined by finitely many inequalities, which was mentioned in Proposition 4.28. Determining these inequalities explicitly is very difficult and has been only carried out for small values of n. See [369], [370] for summaries and references.

4.9 Reduction theory for general arithmetic groups In the Minkowski reduction theory, an important role was played by the identification of the space SL.n; R/=SO.n/ with the space of positive definite quadratic forms of determinant 1, and also with the space of marked lattices of Rn of covolume 1. Such a moduli interpretation of points of the symmetric space SL.n; R/=SO.n/ and the locally symmetric space SL.n; Z/nSL.n; R/=SO.n/ is important in describing points in desired fundamental domains. For a general symmetric space X D G=K and an arithmetic group acting on it, there is no such moduli interpretation and hence there is no such notion of reduced

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points. As pointed out before, the Dirichlet fundamental domains are not suitable for various questions about nX. We recall some general statements on the reduction theory for arithmetic groups as developed by Siegel, Borel & Harish-Chandra, and Borel [57]. The key notion is that of Langlands decomposition of Q-parabolic subgroups and the induced horospherical decomposition of symmetric spaces. Fix a basepoint x0 D K 2 X D G=K. For every Q-parabolic subgroup P of G , its real locus P D P.R/ admits a Q-Langlands decomposition with respect to x0 , P D NP AP MP Š NP AP MP ; where NP is the unipotent radical of P , AP is the Q-split component of P , MP is a reductive group, and AP MP is the Levi factor of P invariant under the Cartan involution associated with K. Though NP is canonically defined, AP and MP depend on the choice of the base point x0 . Define the boundary symmetric space XP associated with P by XP D MP =.MP \ K/: Then the Langlands decomposition of P induces the horospherical decomposition of X with respect to P : (2) X Š N P AP XP : When X D SL.2; R/=SO.2/ Š H2 and P is the subgroup of upper triangular matrices, the horospherical decomposition corresponds to the x, y coordinates of the upper half plane H2 . Let nP be the Lie algebra of NP , and aP the Lie algebra of AP . The set of roots of the action of aP on nP is denoted by ˆ.AP ; P /. Then the subset aC P D fH 2 aP j ˛.H / > 0; ˛ 2 ˆ.AP ; P /g is called the positive chamber of aP determined by P . Similarly, C AC P D exp aP

is called the positive chamber of AP . For any t > 0, define AP;t D fa 2 AP j e ˛.log a/ > t g:

(3)

This is a shift of the positive chamber AC P. Definition 4.33. For any bounded subsets U NP and V XP , the subset of X corresponding to U AP;t V under the horopsherical decomposition in Equation (2) is called a Siegel set associated with P and denoted by P;t . The basic result in the reduction theory of arithmetic groups is the following. See [57] for a proof and more details.


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Proposition 4.34. Let G be a linear semisimple algebraic group defined over Q, and an arithmetic subgroup of G .Q/. Then there are only finitely many -conjugacy classes of Q-parabolic subgroups of G . Let P1 ; : : : ; Pk be representatives of these conjugacy classes. Then there are Siegel sets P1 ;t1 ; : : : ; Pk ;tk such that their union D P1 ;t1 [ [ Pk ;tk is a fundamental set for in the following sense: S (1) 2 D X. (2) For any g 2 G .Q/, the set f 2 j \g ¤ ;g is finite. The finiteness condition in (2) is called the Siegel finiteness condition and is a key result in the reduction theory for arithmetic groups. This basic result has many consequences and applications, which will be explained in later sections. We point out some immediate ones in this subsection. The first is the following. Corollary 4.35. Under the above assumption, the locally symmetric space nX has finite volume. The basic reason is that a Siegel set P;t is a product U AP;t V , and the invariant metric of X has a simple expression in horospherical coordinates. (See [59] or [60] for example.) Hence it can easily be shown that P;t has finite volume. For example, when D SL.2; Z/ and P is the subgroup of upper triangular matrices in SL.2; R/, then P is a Q-parabolic subgroup of SL.2; C/, and a Siegel set associated with P is a vertical strip fx C iy j a1 x a2 ; y > bg for some a1 ; a2 2 R, b > 0. Clearly such a region has finite hyperbolic area. Since any Siegel set corresponding to the improper Q-parabolic subgroup G is bounded and the existence of proper Q-parabolic subgroups of G is equivalent to the positivity of the Q-rank of G , we obtain the following consequence, which was a conjecture of Godement and proved independently by Borel & Harish-Chandra, and Mostow & Tamagama. Corollary 4.36. The locally symmetric space nX is compact if and only if the Qrank of is equal to 0, which is also equivalent to the fact that does not contain any nontrivial unipotent element. Recall that the Q-Tits building Q .G / of G is an infinite simplicial complex whose simplices are parametrized by proper Q-parabolic subgroups of G satisfying the following conditions: (1) For every Q-parabolic subgroup P, denote its corresponding simplex by P . Then P is of dimension 0 if and only if P is a maximal Q-parabolic subgroup of G . (2) For every pair of Q-parabolic subgroups P1 , P2 , P1 is a face of P2 if and only if P1 contains P2 . In particular, the vertices of any simplex P correspond to maximal Q-parabolic subgroups that contain P, and the intersection of these maximal Q-parabolic subgroups is equal to P.

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Since G .Q/ and hence act on the set of Q-parabolic subgroups by conjugation, they also act on the Tits building Q .G / by simplicial homeomorphisms. Another corollary of the reduction theory is the following. Corollary 4.37. The quotient nQ .G / is a finite simplicial complex. The theory of linear algebraic groups implies that Q .G / satisfies the axioms for Tits buildings. In particular, any two simplices are contained in an apartment, which is a finite simplicial complex and whose underlying space is the unit sphere in aP for a minimal Q-parabolic subgroup P of G . It follows that the Tits building Q .G / is connected if and only if the Q-rank is at least 2. Combining this with the reduction theory in Proposition 4.34 (or using the Borel–Serre compactification of nX defined later), we can prove the following result [204]. Proposition 4.38. The locally symmetric space nX is connected at infinity, i.e., has one end, if and only if the Q-rank of nX is at least 2. When the Q-rank is equal to 1, the ends of nX are in one-to-one correspondence with the -conjugacy classes of Q-parabolic subgroups.

4.10 Precise reduction theory for arithmetic groups Though the reduction theory in Proposition 4.34 suffices for many applications, it is an interesting and important problem to get fundamental domains, instead of rough fundamental domains (or fundamental sets in the sense of [57]), that can be conveniently described, for example, in terms of horopsherical decompositions with respect to Q-parabolic subgroups or as special subsets of Siegel sets. Another natural question is whether there is a generalization of Minkowski reduction for general arithmetic subgroups. In this subsection, we summarize cases of arithmetic groups for which more precise descriptions of and results on fundamental sets or fundamental domains are available and discuss approaches to obtain them. Linear symmetric spaces. For the symmetric space X D GLC .n; R/=SO.n/ and the arithmetic group SL.n; Z/, there is another reduction theory developed by Voronoi using perfect quadratic forms. (See [298] for a summary and references. See also [14] and Chapter 1 in [15]). The symmetric space GLC .n; R/=SO.n/ is special in that it is the self-adjoint homogeneous cone of positive definite quadratic forms in the vector space of all symmetric bilinear forms, and the symmetric space SL.n; R/=SO.n/ is a homothety section of the cone. The collection of perfect quadratic forms induces an SL.n; Z/-equivariant polyhedral cone decomposition of the cone GLC .n; R/=SO.n/, which is different from the equivariant decomposition arising from the translates of the Minkowski reduction domains.


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In general, a symmetric space X D G=K is called a linear symmetric space if it is a self-adjoint homogeneous cone or a homothety section of such a cone. For such a linear symmetric space X and an arithmetic subgroup G, X admits a -equivariant decomposition into simplicial cones such that there are only finitely many orbits of simplices. This implies that admits a fundamental domain that is a finite union of simplicial cones. Such simplicial decompositions are essential for toroidal compactifications of Hermitian locally symmetric spaces. See Chapter 1 in [15] and [14]. There are also generalizations of Minkowski reduction theory to other groups, for example, SL.n; Ok /, where k is a number field and Ok is the ring of integers of k. See [418], [419], [183], [247], Chapter I, §4. For GLC .n; R/=SO.n/, there is also theVenkov reduction. The idea is similar to the Dirichlet fundamental domain. Pick a positive definite matrix Q 2 GLC .n; R/=SO.n/. Then the inner product with Q defines a positive function on GLC .n; R/=SO.n/: dQ .A/ D Tr.AQ/;

A 2 GLC .n; R/=SO.n/:

For every SL.n; Z/-orbit in GLC .n; R/=SO.n/, consider the points where dQ takes minimum values. Then the union of such points gives the Venkov reduction domain. It is a convex polyhedron bounded by finitely many faces. See [369], [370]. A generalization of this reduction theory to general linear symmetric space is given in [243]. Symplectic groups. The Siegel upper half space hg D fX C iY j X; Y real g g symmetric matrices; Y > 0g Š Sp.2g; R/=U.g/ is not a linear symmetric space. But for D Sp.2g; Z/, a fundamental domain was explicitly determined by Siegel by making use of the reduction theory for SL.n; Z/. The proof is similar in some sense to that used in identifying the classical fundamental domain for SL.2; Z/. See [273] for details. For any finite index subgroup of Sp.2g; Z/, a finite union of suitable translates of the fundamental domain for Sp.2g; Z/ gives a fundamental domain for . For any other arithmetic subgroup of Sp.2g; Q/, a conjugate of by an element of Sp.2g; Q/ is contained in Sp.2g; Z/, and we can also obtain an explicit fundamental domain for by taking a finite union of suitable translates of the fundamental domain for Sp.2g; Z/. As in the case of arithmetic subgroups of SL.2; Q/, it is desirable to obtain a connected fundamental domain instead of a finite union of domains. Fundamental domains in complex hyperbolic spaces. As mentioned before, Dirichlet fundamental domains for Fuchsian groups acting on the Poincaré half plane have good properties and are useful in studying algebraic structures of Fuchsian groups. For other rank one symmetric spaces, Dirichlet fundamental domains are more complicated. For example, see [148] for Dirichlet fundamental domains in complex hy-

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perbolic spaces. Many results on explicit fundamental domains have been obtained, see the papers [120], [342], [137], [138], [341]. Equivariant tilings of symmetric spaces. The Minkowski and the Voronoi reduction theories induce equivariant cell decompositions of the symmetric spaces GLC .n; R/=SO.n/ and SL.n; R/=SO.n/. For general symmetric spaces X D G=K and arithmetic groups acting on them, coarser equivariant tilings (or decompositions) are known (see [371] for precise statements of the results and references). Such an equivariant decomposition is important for the Arthur–Selberg trace formula but does not give rise to a well-defined fundamental domain. The reason is that each tile admits an infinite stabilizer in , and finding a fundamental domain of such a stabilizer in each tile is not obvious but it might be less difficult than finding a fundamental domain for since it might be reduced to lower dimensional cases. On the other hand, by picking a fundamental domain for each tile, we can get a fundamental domain for which is a union of pieces parametrized by representatives of -conjugacy classes of Q-parabolic subgroups. The piece corresponding to the improper Q-parabolic subgroup G is bounded, and each piece for a proper Q-parabolic subgroup P is contained in a Siegel set of P. Though not canonically defined, such a fundamental domain for is useful for many applications. This reduction theory is usually called precise reduction theory. (See [61] for example.) In the classical reduction theory described in Proposition 4.34, Siegel sets of nonminimal Q-parabolic subgroups are not really needed. In fact, the union of suitable Siegel sets associated with representatives of -conjugacy classes of minimal Q-parabolic subgroups of G gives a fundamental set of . On the other hand, in constructing a fundamental domain from the above equivariant tiling, we do need all Q-parabolic subgroups of G , including the improper parabolic subgroup G . When the Q-rank of G is equal to 1, the fundamental domain constructed in this way is related to the one constructed in the next paragraph, though the point of view is slightly different and the latter is more intrinsic in some sense. Intrinsic fundamental domains for Q-rank 1 arithmetic subgroups. If the Q-rank of G is equal to 1, then we can get a fundamental domain of by using the height functions on X associated with Q-parabolic subgroups. To explain the idea, we interpret the Dirichlet fundamental domain D.x0 ; / in a slightly different way. By assumption, the base-point x0 is not fixed by any nontrivial element of . For each point x0 in the orbit x0 , define a function on X: dx0 .x/ D d.x; x0 /: Using this function, for each point x0 , we define a region .x0 / D fx 2 X j dx0 .x/ dy .x/; for all y 2 x0 g:


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Clearly, .x0 / D .x0 /. Then these subsets give a -equivariant decomposition x 0 ; /. Since no nontrivial element of of X , and .x0 / is the Dirichlet domain D.x fixes x0 or any point of the orbit x0 , acts simply transitively on these subsets. Suppose that there is a family of functions fi , i 2 I , on X that is stable under , i.e., for any 2 and fi , fi . x/ is equal to fj .x/ for some j 2 I . There are two cases. Case 1. Assume that for every x 2 X , inf i2I fi .x/ is realized. For each function fi , define a region z i D fx 2 X j fi .x/ fj .x/; j 2 I g: Case 2. Assume that for every x 2 X, supi2I fi .x/ is realized, then for each function fi , define a region z i D fx 2 X j fi .x/ fj .x/; j 2 I g: z i , i 2 I , form a family which is stable under . On the other These subsets hand, unlike the previous case, each of these subsets is not yet a fundamental domain of . To find a fundamental domain for , denote the stabilizer of fi in by i ; then z i . Let i be a fundamental domain of i in z i. i acts on Assume that there are only finitely many -orbits in ffi j i 2 I g. Let fi1 ; : : : ; fik be a set of representatives. Then it can be shown that the union i1 [ [ ik is a fundamental domain for the -action on X . If we take fi to be the distance function dx0 above such that x0 is not fixed by any nontrivial element of , then the stabilizer of each function fi is trivial, and this construction specializes to the previous case of Dirichlet fundamental domains. Note that in this case, there is only one -orbit in the collection of functions dx0 . If the Q-rank of G is equal to 1, take P to be the set of proper Q-parabolic subgroups P of G . For each Q-parabolic subgroup P, let ˛ 2 ˆ.AP ; P / be the unique short root. Choose a Q-Langlands decomposition P D NP AP MP , and hence an associated horopherical decomposition of X: X Š NP AP XP ;

x 7! .nP .x/; aP .x/; zP .x//:

Define a height function on X associated with the parabolic subgroup P by hP .x/ D ˛.log aP /: Note that the Langlands decomposition and horospherical decomposition of P depend on the choice of a basepoint in X (See Equation 2). A different choice of the basepoint will lead to a shift of the height function hP . It turns out that there are choices such that the family of height functions hP is stable under . The basic idea is as follows. By the reduction theory (Proposition 4.34), there are only finitely many conjugacy classes of Q-parabolic subgroups of G . Fix some representatives of these conjugacy classes and choose arbitrary basepoints for them. Then there are choices

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of basepoints, or equivalently height functions, for all other parabolic subgroups such that the family of height functions hP is stable under (see [428]). For each height function hP .x/, its stabilizer is equal to \ P which is a uniform lattice in NP MP . (Recall that P D NP AP MP is the Q-Langlands decomposition of P .) Since fundamental domains for the \ P -action on NP XP and X can be described relatively easily in terms of the horospherical decomposition of X with respect to P, the above approach can be used to obtain fundamental domains for when the Q-rank of G is equal to 1. For each P 2 P and for every x 2 X, by the reduction theory, supP2P hP .x/ is z P for each P as in the Q-rank 1 case above by realized. Then we define a domain z P D fx 2 X j hP .x/ hP 0 .x/; P 0 2 Pg: In the horospherical decomposition X D NP AP XP Š NP XP AP , the stabilizer of hP in is equal to \ P D \ NP MP . It acts cocompactly on NP XP and leaves the component AP fixed. Let \NP MP be a compact fundamental domain in NP XP for the stabilizer. Define z P \ .\N M AP /: P D P P Let P1 ; : : : ; Pm be a set of representatives of -conjugacy classes of Q-parabolic subgroups. Then the union P1 [ [ Pm is a fundamental domain for the action on X. It can be shown that each of the domains P1 ; : : : ; Pm is contractible. Hence the topology of the fundamental domain is relatively simple. The Hilbert modular groups are some of the most important examples of Q-rank 1 arithmetic groups. There has been a lot of work on their fundamental domains. See [393] (Chapter III, §2), [96], [97], [408] (p. 8–11). Intrinsic fundamental domain for higher Q-rank arithmetic subgroups. A natural problem is to obtain a generalization of the Minkowski reduction theory to a general arithmetic subgroup by picking out points of X that are minimal (or rather maximal) with respect to a family of height functions. Given the result discussed above for the Q-rank 1 case, it is natural to define height functions hP for all maximal Q-parabolic subgroups P of G and use them to define a reduced domain analogous to the Minkowski reduction domain. Let Pmax be the set of all maximal Q-parabolic subgroups. For every P 2 Pmax , we can also define z P as above. But the stabilizer of hP in is \ P D \ NP MP , a domain which does not act cocompactly on NP XP and involves a non-compact locally symmetric space MP nXP . On the other hand, if we do this by induction and find MP -fundamental domains in XP and hence fundamental domains of \ NP MP , then we can follow the steps for the Q-rank 1 case and define \NP MP and P etc. Here is another approach that avoids inductive difficulties. In some sense, it is a generalization of the Minkowski reduction theory. The idea is as follows. For every x 2 X , define an ordered set of maximal parabolic subgroups P1 ; : : : ; Pr such that P1 \ \ Pr is a minimal Q-parabolic subgroup of G , where r is the Q-rank of G .


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The important point is that the stabilizer in of the ordered set of height functions hP1 ; : : : ; hPr is equal to \ P , where P D P1 \ \ Pr . Since P is a minimal Q-parabolic subgroup, \ P D \ NP MP acts cocompactly on NP XP and it is easier to find its fundamental domains in NP XP . For every point x, choose P1 2 Pmax such that hP1 .x/ hP .x/ for all P 2 Pmax . Consider all maximal parabolic subgroups P such that P1 \ P is a Q-parabolic subgroup of G and pick one, denoted by P2 , such that hP2 .x/ has a maximum value. Suppose that P1 ; : : : ; Pi have been picked and P1 \ \ Pi is a non-minimal parabolic subgroup. Consider all maximal Q-parabolic subgroups P such that P1 \ \ Pi \ P is a Q-parabolic subgroup of G and pick one PiC1 such that hPi C1 .x/ takes a maximum value. The ordered sequence P1 ; : : : ; Pr is called a reduced sequence of maximal Qparabolic subgroups for the point x. In the above procedure, the choices of P1 ; : : : ; Pr are not unique for points x when several height functions take the maximum value. On the other hand, for a generic point x, there is a unique maximum height function, and the ordered groups P1 ; : : : ; Pr are unique for x. For every ordered sequence of P1 ; : : : ; Pr as above such that P1 \ \ Pr is a minimal Q-parabolic subgroup, we define a region z P1 ;:::;Pr D fx 2 X j P1 ; : : : ; Pr is the reduced sequence for xg: Let P D P1 \ \ Pr . Let ;P be a fundamental domain for \ NP MP acting on NP XP . Define z P1 ;:::;Pr \ ;P : P1 ;:::;Pr D By the reduction theory, there are only finitely many conjugacy classes of such ordered r-tuples P1 ; : : : ; Pr . Pick and fix representatives of these classes. Then the union of their domains P1 ;:::;Pr is a fundamental domain for the -action on X . It can also be shown that each domain P1 ;:::;Pr is contractible, by deforming along the orbits of the geodesic action of AP on X, where P D P1 \ \ Pr as above.

4.11 Combinatorial properties of arithmetic groups: finite presentation and bounded generation An immediate application of the reduction theory (Proposition 4.34), in particular, the Siegel finiteness, is the finite generation of arithmetic groups. This follows from the following general result. Lemma 4.39. If X is a connected topological space and a -action on X admits a rough open (or closed) fundamental domain , then the set S D f 2 \ ¤ ;g generates . If this set S is finite, i.e., if satisfies the global finiteness condition, then is finitely generated.

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The idea of the proof is that if the subgroup 0 generated in S is S S by these elements not equal to , then the two unions of translates of , 2 0 and 2 0 , give a disjoint decomposition of X into two open subsets. This contradicts the assumption that X is connected. See [390], [29] and [348], Lemma 4.9, p. 196. Since we can take Siegel sets that are open to construct a rough fundamental domain for arithmetic groups in Proposition 4.34, the Siegel finiteness condition implies the following result. Corollary 4.40. Arithmetic subgroups are finitely generated. For special arithmetic subgroups such as SL.n; Z/, more explicit generators are also known. See [386] and [395] for references. To prove finite presentation of arithmetic groups, we need another general fact. Proposition 4.41. Assume X is a connected and simply connected locally path connected topological space (for example a simply connected manifold or a simply connected locally finite CW-complex), and that some -action on X admits a rough open fundamental domain that contains only finitely many connected components. If the set S D f j \ ¤ ;g is finite, then is finitely presented. In fact, relations between generators are given by local ones in the following sense: given any three elements 1 ; 2 ; 3 2 S, the relation 1 2 D 3 holds if and only if 1 2 and 3 induce the same action on , and these are all the relations needed to present . See [348], p. 196–198, for a proof. Since the symmetric space X D G=K is simply connected and we can pick Siegel sets to be open and connected, we obtain the following result. Corollary 4.42. Every arithmetic subgroup is finitely presented. There is also a related important notion of bounded generation. A group is called boundedly generated if there is a finite generating set S D f1 ; : : : ; k g such that m every element is of the form 1m1 : : : k k , where m1 ; : : : ; mk 2 Z. The arithmetic group SL.n; Z/, n 3, and more general integral subgroups of Chevalley groups of higher rank are boundedly generated. See [401]. On the other hand, if the R-rank of G is equal to 1, then is not boundedly generated. See [141]. We note that under the R-rank 1 assumption, if is a uniform arithmetic subgroup, then is a hyperbolic group. It is also known that any nonelementary hyperbolic group is not boundedly generated. Bounded generation is closely related to the congruence subgroup problem. For example, a special case of a theorem states that if G GL.n; C/ is an absolutely simple simply connected algebraic group over the rational number field Q and if normal subgroups of G .Q/ have the standard description,13 then bounded generation of G .Z/ 13 For a semisimple simply connected algebraic group G defined over Q, we say that normal subgroups of G .Q/ have the standard description if there exists a finite set S of places of Q such that any Zariski-dense normal subgroup of G .Q/ is open in G .Q/ in the S -adic topology [348], p. 537.


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implies that the congruence subgroup problem for G .Z/ has a positive solution, i.e., the congruence subgroup kernel is finite. See [354], §6, [265], Theorem D, [349] for the general result. See also [326] for a survey and more references in [197], p. 76.

4.12 Subgroups and overgroups Given any group , a natural problem is to understand its subgroups. Two natural classes of groups are finite subgroups and subgroups of finite index. Another immediate corollary of the reduction theory for arithmetic groups is the following finiteness result. Proposition 4.43. Let be an arithmetic subgroup as in the previous subsection. Then there are only finitely many conjugacy classes of finite subgroups of . Proof. Since every finite subgroup of fixes a point in X , it has a conjugate that fixes a point in a fundamental set, which is the union of finitely many Siegel sets (Proposition 4.34). By the Siegel finiteness property, the fundamental set meets only finitely many translates of itself, and it follows that there are only finitely many conjugacy classes of finite subgroups. For comparison, there are infinitely many subgroups of finite index of , and also infinitely many finite quotient subgroups. A lot of work has been done on counting of subgroups of finite index. The following result holds. Proposition 4.44. Every arithmetic subgroup is residually finite, i.e., for every nontrivial element 2 , there exists a homomorphism to a finite group F , ' W ! F , such that './ ¤ e. In particular, contains infinitely many subgroups of infinite index. For every fixed n 2 N, there are only finitely many subgroups of index at most n. To make the notion of residual finiteness of quantitative, for every nontrivial element 2 , consider all finite quotients of such that the image of in them are nontrivial. Then the minimal cardinality of such finite quotients as a function of the word length of with respect to any fixed word metric on gives an invariant of the residual finiteness property of . See [63] for the precise definition and some results for arithmetic subgroups of Chevalley groups. Congruence subgroups provide a large number of subgroups of finite index. Specifically, for any positive integer N , the kernel of the homomorphism GL.n; Z/ ! GL.n; Z=N Z/ is clearly an arithmetic subgroup and called a principal congruence subgroup of level N of GL.n; Z/. Any arithmetic subgroup of GL.n; Z/ containing a principal congruence subgroup is called a congruence subgroup. Congruence subgroups of Q-linear algebraic groups G can also be defined similarly.

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It is known that for any finitely generated group, there are only finitely many subgroups of any fixed index. The growth of the number of subgroups of index at most n (or equal to n) has been actively studied. See the book [268] for an introduction and summary. See also [253], [17] for related questions on growth of finite-dimensional representations of arithmetic groups. Since arithmetic groups are linear, as a consequence of a famous Selberg Lemma [383], we have the following result. Proposition 4.45. Every arithmetic subgroup is virtually torsion-free, i.e., it admits torsion-free subgroups of finite index. In §16 of [57], a notion of neat arithmetic subgroups was introduced and it was proved there that every neat arithmetic subgroup is torsion-free and every arithmetic subgroup admits neat subgroups of finite index. One important difference is that many groups induced from neat arithmetic groups are also neat and hence torsion-free. After discussing subgroups, a natural question is about groups that contain an arithmetic subgroup , or overgroups of . The following result holds [227]. Proposition 4.46. Assume that G is a semisimple linear algebraic group, then every arithmetic subgroup is contained in only finitely many discrete subgroups of G, in particular, in finitely many arithmetic subgroups of G .Q/. This result is related to uniform lower bounds for the volume of locally symmetric spaces nX for every fixed symmetric space X . It is also related to maximal arithmetic subgroups. See [36] and the references there. It is a fact that if a group contains a proper subgroupTof finite index 0 , then it also contains a normal subgroup of finite index. In fact, 2 0 1 is clearly a normal subgroup of and contained in 0 . To show that it is also of finite index, consider the action of 0 on the finite coset = 0 . The kernel of this action, or of the homomorphism 0 ! Sym.= 0 /, is equal to the above intersection and hence is of finite index. The famous normal subgroup theorem of Margulis [278] states. Proposition 4.47. Assume that is an irreducible arithmetic subgroup of a semisimple linear algebraic group G of R-rank at least 2. Then every normal subgroup of is either finite or of finite index. This result says roughly that such an irreducible higher rank lattice is, as an abstract group, an almost simple group. The rank 1 assumption is necessary. See [109].

4.13 Borel density theorem As mentioned in the introduction, the realization of an arithmetic subgroup as a discrete subgroup of the Lie group G D G .R/ is important for many questions about


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. A natural problem is to understand relations between and G. If G is semisimple, then is a lattice in G, i.e., the quotient nG has finite volume with respect to any Haar measure of G. If is a cocompact lattice, then with any word metric is quasi-isometric to G. The former statement means that in terms of measure theory, is not too small, and the second means that in terms of large-scale geometry, is not too small. Since G is the real locus of an algebraic group, it also admits the Zariski topology. Since the Zariski topology is much coarser than the regular topology of G, it is naturally expected that might not be a discrete subgroup in the Zariski topology. The Borel density theorem shows that this is indeed true. Proposition 4.48. Assume that G is a connected semisimple linear algebraic group over Q, and G D G .R/ has no compact factor. Then any arithmetic subgroup G .Q/ is Zariski dense in G. One corollary of this result is the following result, which also shows one way in which the Borel density can be used. Corollary 4.49. Under the assumption of the above proposition, the normalizer of in G is a discrete subgroup and hence contains as a subgroup of finite index. Proof. Let N./ be the normalizer of in G, and M be the closure of N./ in the regular topology. Then M is the real locus of an algebraic subgroup of G . The identity component M 0 of M centralizes elements of , and the Borel density theorem implies that it also centralizes G. Since G is semisimple, M 0 consists of the identity element. Hence M D N./ is a discrete subgroup. The Borel density theorem has many applications in rigidity theory of discrete subgroups of Lie groups. One basic reason is that in dealing with actions that are algebraic, the Borel density theorem allows one to pass from an arithmetic subgroup to the whole algebraic group, as the proof of the above corollary shows. See [432] for applications in rigidity properties of lattices. There are also some results on discrete subgroups of G that are Zariski dense but not lattices. See [353], [38].

4.14 The Tits alternative and exponential growth Besides finite subgroups and finite index subgroups, it is also a natural question to understand other subgroups. The famous Tits alternative is the following result ([407], Corollary 1). Proposition 4.50. Every finitely generated linear group either contains a non-abelian free subgroup or a solvable subgroup of finite index.

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A special case of another result in [407], p. 250, is the following. Proposition 4.51. If G is a semisimple linear algebraic group defined over Q and if a subgroup G .Q/ is Zariski dense, then contains a free non-abelian subgroup. As a corollary of this and the Borel Density Theorem 4.48, we obtain the exponential growth of arithmetic groups. Proposition 4.52. Assume that G is a semisimple linear algebraic group defined over Q, and G .Q/ is an arithmetic group, then grows exponentially. We recall that for any finitely generated group , there is a word metric dS associated with every finite generating set S . For arbitrary R, let B.R; e/ D f 2 j dS .; e/ Rg be the ball of radius R with center e, and let jB.R; e/j be the number of elements in the ball. We say that grows exponentially if jB.R; e/j grows exponentially in R. Polynomial growth can be defined similarly. Though the word metric dS and jB.R; e/j depends on the choice of the generating set S, the growth type of does not depend on the choice of S . The growth type often reflects algebraic properties of the group. For example, a famous theorem of Gromov says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth. For any finitely generated subgroup 0 , if a generating set S 0 is contained in a generating set S of , then it is clear from the definition that the restriction of the word metric dS to 0 is bounded from above by dS 0 . This implies that if 0 has exponential growth, then has at least exponential growth. Proof of Proposition 4.52. It can be checked easily that a non-abelian free group has exponential growth. The Borel density Theorem and Proposition 4.51 implies that the arithmetic subgroup has at least exponential growth. By some general results [105], p. 181, Remark 53 (iii), it grows exponentially. Remark 4.53. Another way to show that the arithmetic subgroup in Proposition 4.52 has at most exponential growth is to use the growth of the symmetric space X. We note that if is torsion-free and identified with an orbit x in X, then the induced distance on x from the invariant metric on X is bounded from above by a multiple of the word metric on . Since the volume of balls in X grows exponentially, it follows that x with the induced metric grows exponentially and hence also grows exponentially.

4.15 Ends of groups and locally symmetric spaces If is a cocompact subgroup of G, i.e., if the quotient nG (or equivalently nX , where X D G=K) is compact, then the number of ends of the group is the same as the number of ends of X, which is equal to 1. On the other hand, if is not a cocompact subgroup, then the situation is different.


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It is known that every infinite group has either 1, 2 or infinitely many ends. (See [380].) Since a group has two ends if and only if it is infinite and virtually cyclic, an arithmetic subgroup of a semisimple Lie group has either 1 or infinitely many ends. Proposition 4.54. If is an irreducible lattice of a semisimple Lie group G and the rank of the associated symmetric space X D G=K is at least 2, then has one end. Proof. Since is irreducible and the rank of X is at least 2, has Property T and hence also Serre’s Property FA (see [416], and also [279]). If has infinitely many ends, then a theorem Stalling [394] implies that is an amalgam (i.e., a free product with amalgamation over finite groups) and hence by Bass–Serre theory, acts on a tree without a fixed point. This contradicts Property FA. It is clear that the symmetric space X has one end. On the other hand, the number of ends of nX depends on the Q-rank. For example, the following result is true. Proposition 4.55. If the Q-rank of G (or rather nX , ) is greater than or equal to 2, then nX has one end, i.e., it is connected at infinity. If the Q-rank of is equal to one, then the ends of nX are parametrized by the set of -conjugacy classes of Q-parabolic subgroups. By passing to smaller subgroups of finite index, there exist such that nX has as many ends as desired. The basic reason for which this proposition is true is that the Q-Tits building Q .G / is connected if and only if the Q-rank of G is greater than or equal to 2. This arises from the following two facts: (1) any two simplices in the Tits building are contained in a common apartment, (2) an apartment is connected if and only if the Q-rank is greater than or equal to 2. Given this fact, Proposition 4.55 can be proved roughly as follows. By reduction theory (Proposition 4.34), the neighborhoods of infinity of nX are described by Siegel sets. We can choose Siegel sets to be connected. For two Q-parabolic subgroups P1 , P2 , the following facts can be proved: If P1 P2 , then a Siegel set of P1 contains a Siegel set of P2 . Suppose that P1 \ P2 is a Q-parabolic subgroup. Then the intersection of Siegel sets of P1 and P2 is contained in a Siegel set of P1 \ P2 . The above intersection pattern of the Siegel sets and the connectedness of the Q-Tits building Q .G / imply that nX is connected at infinity. On the other hand, if the Q-rank of G is equal to 1, it can be shown that if P1 ¤ P2 , then sufficiently small Siegel sets of P1 and P2 are disjoint. If P1 and P2 are not conjugate under , then the image of suitable small Siegel sets of P1 and P2 in nX are disjoint but are neighborhoods of the ends of nX. The above argument is basically clear and convincing. To make it rigorous, it is easier to use the Borel–Serre compactification of nX defined in §4.18. See [204] for a complete proof of Proposition 4.55.

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4.16 Compactifications and boundaries of symmetric spaces As mentioned earlier, if G is semisimple, then X D G=K is a symmetric space of noncompact type and consequently is noncompact. For many applications, it is important to compactify X in such a way that the G-action on X extends continuously to the compactification. There are many different compactifications with different boundary structures that are suitable for various applications. See the books [162] and [61] for a detailed discussion about compactifications of symmetric spaces and references. We recall several basic facts and use them to motivate the following facts: (1) Points at infinity of the symmetric space X are naturally described by parabolic subgroups P of G. For example, the Furstenberg boundaries G=P appear in several different compactifications of X . (2) Structures at infinity of X are related to an infinite simplicial complex, the Tits building .G/ of G. We hope that this discussion will help explain similarities between Tits buildings and the curve complex C.Sg;n / of surfaces Sg;n introduced later, which has played a foundational role in the study of mapping class groups and Teichmüller spaces (see [307], and [210] for summaries of recent results on applications of the curve complexes and references). For applications of Tits buildings in geometry and topology, see [208] and the extensive references there. It is known that the symmetric space X is simply connected and nonpositively curved, i.e., it is a Hadamard manifold. Therefore, X admits a geodesic compactification X [ X.1/, where X.1/ is the set of equivalence classes of directed geodesics of X and is called the sphere at infinity. Since any parametrization of a geodesic is of constant speed and can be scaled to have unit speed, we assume that geodesics are directed and of unit speed. Recall that two unit speed directed geodesics 1 .t /, 2 .t / are called equivalent if14 lim sup d.1 .t/; 2 .t // < C1: t!C1

For any basepoint x0 2 X, let Tx0 X be the tangent space of X at x0 . Then X.1/ can be canonically identified with the unit speed sphere in Tx0 X , since each equivalence class of geodesics contains exactly one unit directed geodesic through x0 . For each unit vector v 2 Tx0 X, denote the corresponding geodesic passing through x0 with direction v by v . Then the topology of the compactification X [ X.1/ is described as follows: a sequence of points xj 2 X going to infinity converges to the equivalence class of v if and only if the direction of the geodesic segment x0 xj converges to v. 14 The assumption of the unit speed of geodesics is convenient in defining this equivalence relation. Otherwise, we need to use d.1 .t /; 2 / D inf d.1 .t /; 2 .s/ j s 2 Rg, since for two equivalent geodesics 1 .t /; 2 .t / of different constant speeds, d.1 .t /; 2 .t // ! C1 as t ! C1.


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It can be shown that this topology does not depend on the choice of the basepoint x0 and the natural action of G on geodesics of X and hence on X.1/ induces a continuous action on the geodesic compactification X [ X.1/. Proposition 4.56. For every boundary point z 2 X.1/, its stabilizer Gz D fg 2 G j gz D zg is a proper parabolic subgroup of G. Furthermore, every proper parabolic subgroup of G fixes some boundary point in X.1/; in fact, the subgroup is equal to the stabilizer of some boundary point. When X D SL.2; R/=SO.2/ is identified with the Poincaré upper half plane H2 , then the standard parabolic subgroup P1 of upper triangular matrices in SL.2; R/ is the stabilizer of the boundary point i1, and the parabolic subgroup P0 of lower triangular matrices in SL.2; R/ is the stabilizer of the boundary point 0 2 R. In this example, X.1/ D H2 .1/ D R [ fi 1g D S 1 , and SL.2; R/ acts transitively on X.1/, which can be written as G=P1 . Proposition 4.57. The group G acts transitively on the sphere at infinity X.1/ if and only if the rank of X is equal to one. If the rank of X is at least 2, then there are infinitely many G-orbits in the boundary X.1/, and each orbit is of the form G=P , where P is a proper parabolic subgroup of G. Probably the simplest example of higher rank symmetric spaces is the product H2 H2 . A maximal flat subspace of X D H2 H2 can be identified with R2 , and the decomposition into four coordinate quadrants corresponds to the Weyl chamber decomposition. The set of unit vectors in a positive closed Weyl chamber, say, the first quadrant, is a 1-simplex, and it parametrizes the set of G D SL.2; R/SL.2; R/-orbits in X.1/. Proposition 4.58. For a general symmetric space X D G=K, the set of G-orbits in X.1/ is parametrized by the set of unit vectors in a positive closed Weyl chamber of a maximal flat subspace of X, which is an .r 1/-simplex, where r is the rank of X . The homogeneous spaces G=P in Proposition 4.57 are called Furstenberg boundaries. When P is a minimal parabolic subgroup, G=P is called the maximal Furstenberg boundary and it has played a fundamental role in the rigidity theory of lattices of G. See [432] and [279] for details. For every parabolic subgroup P of G, let P be the set of points of X.1/ that are fixed by P . Let P0 be the set of points of X.1/ whose stabilizers are exactly equal to P . Proposition 4.59. For every parabolic subgroup P , the closure of P0 in X.1/ is equal to P , and P is a simplex. Furthermore, P0 is the interior of P when all its boundary faces are removed. When P runs over all proper parabolic subgroups of G, the subsets P0 give a disjoint decomposition of X.1/. The simplices P give

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the geodesic sphere X.1/ the structure of an infinite simplicial complex, which is a geometric realization of the Tits building .G/. Recall that the Tits building .G/ is an infinite simplicial complex whose simplices are parametrized by proper parabolic subgroups of G satisfying the following conditions: (1) For every parabolic subgroup P of G, denote its simplex by P . Then P1 P2 if and only if P1 contains P2 as a face. (2) †P is a 0-simplex (i.e., a point) if and only if P is a maximal proper parabolic subgroup of G. Since G acts on the set of parabolic subgroups by conjugation, it acts on .G/ simplicially. The quotient Gn.G/ can be identified with P for a minimal parabolic subgroup P of G. This result is consistent with Propositions 4.59 and 4.58. Besides the geodesic compactification X [ X.1/, another important compactifix Smax . cation is the maximal Satake compactification X For every real parabolic subgroup P , there is an R-Langlands decomposition P D NP AP MP Š NP AP MP ; with respect to any basepoint x0 . The dependence on the basepoint x0 is that AP and MP are stable under the Cartan involution of G associated with K. Remark 4.60. When P is the real locus of a Q-parabolic subgroup P, we have introduced a Q-Langlands decomposition of P in §4.9. The difference between these two decompositions is that in the Q-Langlands decomposition, AP is a maximal Qsplit component of P , but in the R-decomposition here, AP is a maximal R-split component of P . In general AP AP . Define a boundary symmetric space XP associated with the real parabolic subgroup P by XP D MP =.MP \ K/: Unlike the boundary symmetric space XP for a Q-parabolic subgroup P (or rather its real locus P defined in §4.9), XP is always a symmetric space of noncompact type. On the other hand, XP is equal to XP times a possible Euclidean factor. x Smax admits a disjoint deProposition 4.61. The maximal Satake compactification X composition a x Smax D X [ X XP : P

It is a compact Hausdorff space on which G acts continuously. x Smax D H2 H2 , where H2 D H2 [H2 .1/ is the geodesic If X D H2 H2 , then X compactification. Its boundary symmetric spaces consist of either points or H2 .


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x Smax X , Proposition 4.62. There are finitely many G-orbits in the boundary X which are parametrized by G-conjugacy classes of parabolic subgroups, or standard parabolic subgroups containing a minimal parabolic subgroup P0 of G. There is a unique closed orbit, which can be identified with the maximal Furstenberg boundary G=P0 mentioned in the previous subsection.

4.17 Baily–Borel compactification of locally symmetric spaces Compactifications of locally symmetric spaces are closely related to compactifications of symmetric spaces. The basic ideas and steps can be seen in compactifications of nH2 briefly discussed in §4.7. When is a lattice of SL.2; R/, i.e., a Fuchsian group of the first kind, we picked out -rational points in the boundary H2 .1/ and added them to H2 to form a partial compactification with the Satake topology. Then the quotient of this partial compactification by gives a compactification of nH2 . A natural generalization to compactify arithmetic locally symmetric spaces nX initiated by Satake [374] is as follows: x and decompose its boundary X x X into boundary (1) Start with a compactification X components, which are usually parametrized by some real parabolic subgroups of G. (2) Pick out rational boundary components, which are usually characterized by nonempty intersection with the closure of suitable fundamental sets (or Siegel sets of Q-parabolic subgroups) and hence are associated with Q-parabolic subgroups of G . (3) Attach the rational boundary components to X to form a partial compactification xS Q X of X with a suitable topology, called the Satake topology. x S with a com(4) Show that acts continuously on the partial compactification Q X S x pact Hausdorff quotient nQ X , which is a desired compactification of nX. x Smax , its boundary components are For the maximal Satake compactification X boundary symmetric spaces XP . The rational boundary components are exactly XP when P is the real locus of Q-parabolic subgroups of G . This procedure leads to the maximal Satake compactification of nX. In this construction, the reduction theory, in particular, the Siegel finiteness property is used crucially. In other cases, there are complications with Steps 2 and 4. The reason is that once rational boundary components are chosen, it is not obvious whether the extended -action is continuous and the quotient is Hausdorff. A general arithmetic locally symmetric space nX admits finitely many nonisomorphic Satake compactifications, and they are partially ordered, where one compactification nX 1 is greater than or dominates another compactification nX 2 if the identity map on nX extends to a continuous map nX 1 ! nX 2 , which is automatically surjective. Besides the maximal Satake compactification, some minimal Satake compactifications are important for applications as well.

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In order to motivate a new construction of the Deligne–Mumford compactification of Mg using the Bers compactification of the Teichmüller space Tg in §5.12, we outline the topological aspect of the Baily–Borel compactification [20] of Hermitian locally symmetric spaces. The Baily–Borel compactification is a minimal Satake compactification. See [61] for details and references. Assume that X D G=K is a Hermitian symmetric space of noncompact type, i.e., a symmetric space of noncompact type with a G-invariant complex structure. Then by a theorem of Harish-Chandra, X can be embedded into C n as a bounded symmetric x of X in C n is called domain, where n is the complex dimension of X . The closure X the Baily–Borel (or Baily–Borel–Satake) compactification. x As a subspace of C n , we can define analytic arc components of the boundary @X. They are Hermitian symmetric spaces of smaller dimension. Unlike the maximal Satake compactification of X, they are not of the form XP . Instead, they are the Hermitian part of the boundary symmetric spaces XP for maximal parabolic subgroups P of G. (Note that the boundary symmetric space XP splits into a product of a Hermitian symmetric space and a linear symmetric space.) In the extended action x the stabilizer in G of such a boundary component is equal to a maximal of G on X, parabolic subgroup P . Then the Baily–Borel compactification of a Hermitian locally symmetric space nX can be constructed as follows: x C n into (1) Decompose the boundary of the Baily–Borel compactification X analytic arc components. (2) A boundary component is called rational if it has nonempty intersection with the closure of a Siegel set of a minimal Q-parabolic subgroup. x BB of X by adding the rational boundary (3) Form a partial compactification Q X components at infinity and impose the Satake topology on it. x BB with a (4) Show that the -action on X extends to a continuous action on Q X compact Hausdorff quotient. x BB admits the structure of a (5) Show that the topological compactification nQ X normal complex space by constructing a sheaf of holomorphic functions on it. x BB is a normal projective space by (6) Show that the normal complex space nQ X n embedding it into some CP using Poincaré–Eisenstein series. An important new feature in this case is that the boundary components have an intrinsic interpretation in terms of the analytic structure.

4.18 Borel–Serre compactification of locally symmetric spaces and cohomological properties of arithmetic groups For applications to understand topological properties of , the Borel–Serre compactification of nX [62] is sometimes more useful than the Satake compactifications.


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For example, for any Satake compactification nX S , the inclusion nX ! nX S is not a homotopy equivalence. The fundamental group of nX S is not equal to . For example, when is irreducible and the rank of X is at least 2, then the fundamental group is finite. It is also trivial in some cases. (See [211] and references there.) The importance of the Borel–Serre compactifcation is that it preserves the homotopy type of the locally symmetric space nX. The basic idea is to avoid Step 1 in the procedure of Satake compactifications in §4.17 since there may not exist a compactification of X whose rational boundary components give rise to the desired partial compactification of X. The procedure of the Borel–Serre compactification, as slightly reformulated in [61], is as follows: (1) For every Q-parabolic subgroup P, define its boundary component to be e.P/ D NP XP . (2) Attach e.P/ to the infinity of X using the horospherical decomposition of X x BS . The with respect to P to obtain a partial Borel–Serre compactification Q X topology of the partial compactifcation is naturally determined by such a gluing procedure and the inductive step that if P1 P2 , then e.P1 / is contained in the closure of e.P2 /. x BS (3) Show that the -action on X extends to a continuous and proper action on Q X with a compact quotient, which is the Borel–Serre compactification of nX and denoted by nX BS . In the case of X D SL.2; R/=SO.2/, for every Q-parabolic subgroup P, its BS

boundary component e.P/ D NP Š R, and the partial compactification Q H2 is obtained by blowing up every rational boundary point (or -rational point in the sense for Fuchsian groups) into R, and the resulting Borel–Serre compactification of nH2 has a boundary circle for every cusp end of nH2 , as explained earlier. We would like to point out that the reduction theory in Proposition 4.34 is used crucially in the above construction. For example, the Siegel finiteness condition is x BS is proper, and the fact that a union of used to show that the action of on Q X x BS is finitely many Siegel sets form a fundamental set implies that the quotient nQ X BS x compact, since the closure of each Siegel set in Q X is compact. We now recall several basic facts about the Borel–Serre compactification and applications. x BS is a real analytic Proposition 4.63. The partial Borel–Serre compactification Q X manifold with corners whose interior is equal to X . Consequently, it is contractible. The extended -action on it is real analytic. Corollary 4.64. If is torsion-free, then nX BS is a compact real analytic manifold with corners and hence gives a finite model of B, i.e., a model given by a finite CW-complex.

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For the last statement, we use the general fact that a smooth compact manifold with corners admits a finite triangulation. For the application to the virtual duality properties of , we need the following theorem of Solomon–Tits. Proposition 4.65. Let rQ be the Q-rank of G . Then the Tits building Q .G / is homotopy equivalent to a bouquet of infinitely many spheres S rQ 1 of dimension rQ 1. x BS is homotopy equivalent to the Q-Tits buildProposition 4.66. The boundary of Q X ing Q .G / and hence is homotopy equivalent to a bouquet of infinitely many spheres S rQ 1 . Recall that a group is called a Poincaré duality group of dimension d if for every Z-module A, there exists an isomorphism H i .; A/ Š Hd i .; A/; for all i . This is motivated by the Poincaré duality for closed manifolds. In fact, if admits a model of B by a closed manifold, which is necessarily an aspherical manifold, then is a Poincaré duality group. More generally, a group is called a duality group (or a generalized Poincaré duality group) of dimension d if there exists a Z-module D such that for every Z-module A, there exists an isomorphism H i .; A/ Š Hd i .; D ˝ A/; for all i . In this case, D is called the dualizing module of , and the cohomological dimension of is equal to d . See [79], Chapter IIIV, §10, for the history and various results on duality groups. It follows from the general theory that a duality group is torsion-free. A group is called a virtual duality group if it admits a finite index torsion-free subgroup that is a duality group. Virtual Poincaré duality groups can be defined similarly. Corollary 4.67. Assume that is a torsion-free arithmetic subgroup of G .Q/. Then is a duality group of dimension dim X rQ , and the dualizing module is equal to HrQ 1 .Q .G /; Z/. Consequently, is a Poincaré duality group if and only if rQ is equal to 0, i.e., the quotient nX is compact. x BS is contractible, it is a model of The idea of the proof is as follows. Since Q X cofinite E-space. By the general theory of cohomology of groups, it suffices to x BS ; Z/ is not equal to zero in only one degree. Then show that H i .; Z/ D Hci .Q X this degree is the cohomological dimension of and this Z-module is the dualizing module. By the Poincaré–Lefschetz duality for noncompact manifolds with corners, we have the following equalities: x BS ; Z/ Š Hni .Q X x BS ; @Q X x BS ; Z/ D Hni1 .@Q X x BS ; Z/; Hci .Q X


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where n D dim X. By the Solomon–Tits Theorem (Proposition 4.65), the last group is zero if and only if i ¤ n rQ , and furthermore, if rQ is positive, then x BS ; Z/ is an infinitely generated abelian group. This implies that is a HrQ 1 .@Q X duality group of dimension dim X rQ , and is a Poincaré duality group if and only if rQ D 0.

4.19 The universal spaces E and E via the Borel–Serre partial compactification x BS is a finite model of B and hence Q X x BS is a cofinite When is torsion-free, nQ X E-space, which is also an E-space. But most natural arithmetic subgroups such as SL.n; Z/ and Sp.n; Z/ are not torsion-free. As explained in the introduction, a natural question is whether arithmetic groups that contain torsion-elements admit cofinite E-spaces. First we note the following result. Proposition 4.68. For any arithmetic subgroup , the symmetric space X is a model of E. Proof. Since acts properly on X, we only need to check that for any finite subgroup F of , the set of fixed points X F is nonempty and contractible. By the Cartan fixed point theorem (Proposition 4.18), X F ¤ ;. Since F acts by isometries, X F is a totally geodesic submanifold. Since X is simply connected and nonpositively curved, X F is also simply connected and nonpositively curved and hence contractible. If the quotient nX is compact, then X is a cofinite E-space. On the other hand, x BS is compact, if nX is noncompact, then X is not a cofinite E-space. Since nQ X BS x is a cofinite E-space. It is indeed true [206]. a natural guess is that Q X Proposition 4.69. For any arithmetic subgroup , the partial Borel–Serre compactx BS is a cofinite E-space. ification Q X Since any finite subgroup F of has some fixed point in X, we only need to show x BS /F is contractible. This can be proved by using the fact that the fixed point set .Q X x BS are contained in the corresponding that the stabilizers in of boundary points of Q X parabolic subgroups. In the above approach, to get a cofinite E-space, we enlarge the space X by adding x BS so that the quotient some boundary points to get a partial compactification Q X BS x nQ X becomes compact. One important requirement on the compactification is x BS is a -equivariant homotopy equivalence. that the inclusion X ,! Q X Another way to overcome the noncompactness is to take a subspace S of X such that (1) S admits the structure of a -CW complex such that nS is compact.

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(2) S is a -equivariant deformation retract of X. Then it can be checked easily that S is also an E-space and hence is a cofinite model of E. This is the problem of existence of a good equivariant spine in X. If X is the upper half plane H2 and is commensurable with SL.2; Z/, by removing small horodisks around the rational boundary points in a -equivariant way, we obtain a -stable truncated subspace X."/, which is a submanifold with boundary, where " measures the sizes of the horodisks. It can be shown that the above two conditions are satisfied by X."/. It turns out that if we push these horodisks further until they meet and flatten out, the subspace X."/ becomes a tree T which is stable under . (See [384].) What is important is that dim T is equal to the virtual cohomological dimension of , which is the optimal dimension. The above truncation procedure of removing horoballs equivariantly from H2 can be generalized to a general symmetric space X by removing horoballs in X associated with Q-parabolic subgroups which are equivariant with respect to an arithmetic subgroup . Then the remaining subspace is a manifold with corners and is stable under . Denote it by XT . Then the quotient nXT is a compact manifold with corners. (It might be worthwhile to point out that if the rank of X is greater than or equal to 2, then the equivariant horoballs above intersect each other no matter how small the horoballs are.) This is related to -equivariant tilings of X mentioned before in §4.10. The central tile in [371] is a cocompact deformation retract of X and is equal to XT , which corresponds to X."/ in the case of the upper half plane H2 above. This gives another cofinite model of E different from the one in Proposition 4.69. As in the case of the upper half plane H2 , a natural and important question is whether for a non-uniform arithmetic subgroup , the symmetric space X admits a -equivariant deformation retraction S such that nS is compact and dim S is equal to the virtual cohomological dimension of . This seems to be a difficult problem and the answer is not known in general. The following is a list of cases where the answer is positive. (1) When X is a linear symmetric space. (2) When the Q-rank of nX is equal to 1. (3) When the R-rank of X is equal to 1. (4) When D Sp.4; Z/ and X is the Siegel upper space of degree 2. See the papers [14], [281], [428] and references there. In a work in progress, we are able to find such an equivariant spine when the Q-rank of nX is less than or equal to 2.

5 Mapping class groups Modg;n In this section, we introduce some definitions and results about Teichmüller spaces Tg;n and mapping class groups Modg;n by emphasizing their similarities to symmetric


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spaces and arithmetic subgroups. Since some results and proofs are motivated by and related to results for symmetric and locally symmetric spaces in the previous section, they will be rather brief. For a systematic discussion about Teichmüller spaces and mapping class groups, see the first two volumes of this Handbook [334], [335]. See also the book [126] and the survey papers [188], [123], [169].

5.1 Definitions and examples Let Sg;n be an oriented surface of genus g with n punctures. When n D 0, we denote it by Sg . A natural procedure to construct new topological spaces and groups from Sg;n is to consider the group HomeoC .Sg;n / of orientation preserving homeomorphisms of Sg;n . Denote its identity component by Homeo0 .Sg;n /. Then this identity component Homeo0 .Sg;n / is a normal subgroup of HomeoC .Sg;n /, and the quotient HomeoC .Sg;n /=Homeo0 .Sg;n / is called the mapping class group of Sg;n and denoted by Modg;n . Since the set of connected components of a topological space has a natural discrete topology, it seems reasonable to give Modg;n the discrete topology. Equivalently, if DiffC .Sg;n / denotes the group of orientation preserving diffeomorphisms of Sg;n , its identity component Diff0 .Sg;n / is a normal subgroup of DiffC .Sg;n /. Then the quotient DiffC .Sg;n /=Diff0 .Sg;n / is also equal to Modg;n . When g D 1 and n D 0, it can be shown that Modg;n Š SL.2; Z/. For any smooth manifold M , in a similar way we can also define its mapping class group Mod.M / D DiffC .M /=Diff0 .M /. It can be shown that when M D Rn =Zn , the n-dimensional torus, Mod.M / Š SL.n; Z/.

5.2 Teichmüller spaces It is known that the surface Sg;n admits complex structures so that each puncture admits neighborhoods that are biholomorphic to the punctured disc D D fz 2 C j 0 < jzj < 1g. The moduli space of all such complex structures on Sg;n is denoted by Mg;n . It was first introduced by Riemann and has been intensively studied since then. If 2g 2 C n > 0, then by the uniformization theorem for Riemann surfaces, each complex structure on Sg;n is biholomorphic to nH2 , where P SL.2; R/ is a torsion-free lattice subgroup (or rather a Fuchsian group of the first kind.) Therefore, for every complex structure on Sg;n as above, there exists a unique complete hyperbolic metric on Sg;n of finite total area that is conformal to the complex structure. Then Mg;n is also the moduli space of all complete hyperbolic metrics of finite total area of Sg;n . If we identify each complex structure on Sg;n with a projective curve over C with n marked points, then Mg;n is also the moduli space of projective curves over C with n marked points. This is one of the most important moduli spaces in algebraic geometry.

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According to a general philosophy, the moduli space of objects with certain structures (smooth structures, complex structures, algebraic structures etc.) should inherit structures similar to those of the objects. Based on this philosophy, we expect to have and should be able to establish: (1) As the moduli space of complex structures on Sg;n , the moduli space Mg;n should admit a complex structure, i.e., it should be a complex space; (2) as the moduli space of Riemann metrics of constant curvature, Mg;n should admit (Riemannian) metrics; (3) as the moduli space of algebraic curves over C, Mg;n should admit the structure of an algebraic variety, i.e., it should be an algebraic variety. It turns out that all these statements are true, and hence the moduli space Mg;n is the best example to show the above philosophy. In some sense, the rich structures on Mg;n make it one of the most interesting and important spaces in mathematics. Remark 5.1. There is a theorem of Wielandt (see [366] for example) saying that for any finite group G with trivial center, its automorphism tower G Aut.G/ Aut2 .G/ D Aut.Aut.G// terminates after finitely many steps. One way to interpret this is to view Aut.G/ as a genuinely new group constructed from G. The above result says that this process will terminate after finitely many steps. We can also consider the related outer automorphism groups Out.G/, Out2 .G/; : : : . The natural generalization of Wielandt’s theorem to infinite groups does not hold in general. On the other hand, for the following three classes of groups related to this chapter: the abelian free groups Zn , the non-abelian free groups Fn , and the surface groups 1 .Sg /, their automorphism towers do terminate after finitely many steps. Indeed, Aut.Zn / D GL.n; Z/, Out.Fn /, Out.1 .Sg // D Modg are rigid. The automorphism groups of GL.n; Z/ and more generally arithmetic groups, Modg;n and Out.Fn / are related to Mostow strong rigidity and have been discussed in the earlier sections. A natural question is how to construct new spaces starting from some new spaces. As discussed in the previous subsection, one way to construct new spaces is to consider the space of homeomorphisms (or diffeomorphisms). Another natural way is to consider moduli spaces of certain structures on the spaces we started with. If we mimic the automorphism tower and construct the moduli spaces inductively, a natural guess is that this process should terminate after finitely many steps, i.e., the moduli space will eventually become stationary. This turns out to be true for Riemann surfaces (hyperbolic surfaces, or algebraic curves over C), since Mg;n is rigid. If we start with the torus Zn nRn D .S 1 /n and consider the moduli space of flat metrics of total volume 1, then we get SL.n; Z/nSL.n; R/=SO.n/, which is also rigid. If we consider other objects such as K-3 surfaces etc, their moduli spaces are given by arithmetic locally symmetric spaces and are often rigid. See the book [185] and the article [207] for additional references.


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In order to study Mg and put a complex structure on it, Teichmüller initiated the systematic study of the Teichmüller space Tg via quasi-conformal maps and hence introduced a metric on it. He also considered the question of complex structures on Tg . One reason why it is easier to study Teichmüller space than moduli space is that in general it is easier to study moduli spaces of more rigid objects, i.e., those that do not admit self-automorphisms, and to put structures of smooth manifolds on these rigid moduli spaces. Let Sg;n be a compact oriented surface of genus g with n points removed. A Riemann surface †g;n of genus g with n punctures together with a homotopy class of orientation preserving diffeomorphisms ' W †g;n ! Sg;n is called a marked Riemann surface and denoted by .†g;n ; Œ'g;n /. Two marked Riemann surfaces .†g;n ; Œ'g;n / and .†0g;n ; Œ' 0 g;n / are defined to be equivalent if there exists a biholomorphic map h W †g;n ! †0g;n such that ' 0 B h is homotopy equivalent to '. Then the Teichmüller space Tg;n is defined to be the set of equivalence classes of marked Riemann surfaces .†g;n ; Œ'g;n /: Tg;n D f.†g;n ; Œ'g;n /g= :

(1)

Remark 5.2. A marking on a Riemann surface †g;n above is equivalent to a choice of a set of generators of the fundamental group 1 .†g;n /, but not equivalent to a choice of a basis of H1 .†g;n /, which is more common in Hodge theory. In this chapter, we assume (except if we specify the contrary) that 2g 2 C n > 0. Then each Riemann surface †g;n admits a unique complete hyperbolic metric of finite area that is conformal to the complex structure. Under this assumption, the Teichmüller space Tg;n can also be defined to be the moduli space of marked complete metrics of finite volume on Sg;n . As defined earlier, DiffC .Sg;n / is the group of orientation preserving diffeomorphisms of Sg;n , and Diff0 .Sg;n / its identity component. Then the quotient group DiffC .Sg;n /=Diff0 .Sg;n / is the mapping class group Modg;n . Modg;n acts on Tg;n by changing the markings: for any 2 DiffC .Sg;n / and a marked Riemann surface .†g;n ; Œ'/, .†g;n ; Œ'/ D .†g;n ; Œ

B '/:

Clearly, the quotient Modg;n nTg;n is equal to the moduli spaces Mg;n of Riemann surfaces †g;n of genus g with n punctures. Remark 5.3. The idea of Teichmüller spaces has also been used for other spaces and groups. For example, the outer spaces Xn of marked metric graphs are defined in an almost identical way. As the earlier discussions indicated, the similarity between the action of Out.Fn / on Xn with the action of Modg;n on Tg;n has inspired a lot of exciting work. See [412], [414], [45] and references there. For general manifolds, the theory of Teichmüller spaces is often different. See [133].

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5.3 Properties of Teichmüller spaces In comparison with symmetric spaces of noncompact type, the first basic result is the following. Proposition 5.4. The Teichmüller space Tg;n is a real analytic manifold diffeomorphic to R6g6C2n , and Modg;n acts properly on Tg;n . It was first shown by Fricke that Tg;n has the structure of a real analytic manifold and is homeomorphic to R6g6C2n . An explicit homeomorphism (or rather diffeomorphism) was given by Teichmüller via quasiconformal maps of smallest distortion. Probably the easiest way to see the diffeomorphism is to use the Fenchel–Nielsen coordinates. For details, see [327], [40], p. 269. See also [365] for a historical summary of Teichmüller spaces. For the purpose of constructing the Deligne–Mumford compactification of Mg;n using the Bers compactification of Tg;n , the following result is important as well. Proposition 5.5. Tg;n is a complex manifold of dimension 3g 3 C n and can be realized as a bounded domain in C 3g3n under the Bers embedding. Modg;n acts on Tg;n by biholomorphic maps. One way to view the complex structure of Tg;n is to take a base Riemann surface .†g;n ; Œ'/ and consider the complex Banach space B of all Beltrami differentials on †g;n , and realize Tg;n as a quotient of B. See [327]. The tangent space of Tg;n at .†g;n ; Œ'/ can be identified with the space of harmonic Beltrami differentials on †g;n , and the cotangent space can be identified with the space of holomorphic quadratic differential forms on †g;n . For the action of Modg;n on Tg;n , the following Nielsen realization result [233], [424] is important. Proposition 5.6. For every finite subgroup F Modg;n , there exists a Riemann surface †g;n such that F is contained in Aut.†g;n / and hence the set of fixed points F Tg;n is nonempty.

5.4 Metrics on Teichmüller spaces For the symmetric space X D G=K discussed earlier, there is a G-invariant Riemannian metric which is unique up to scaling. This invariant metric enjoys many good properties. It has been used effectively in many contexts and is suitable for different applications. On the other hand, the Teichmüller space Tg;n admits many different metrics introduced for various applications. They are all natural. In some sense, the presence of different metrics on Tg;n makes it a more interesting space. But the lack of homogeneity of these metrics makes it more difficult to understand their properties.


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We will mention two such metrics: the Teichmüller metric and the Weil–Petersson metric. Given any two marked Riemann surfaces p1 D .†g;n ; Œ'/, p2 D .†0g;n ; Œ' 0 /, for any quasi-conformal map f in the homotopy class Œ.' 0 /1 B ' from †g;n to †0g;n , let K.f / be the dilation of f . Then the Teichmüller distance between p1 ; p2 is dT .p1 ; p2 / D inf log K.f /: f

It is a complete Finsler metric and has the property that any two distinct points are connected by a unique geodesic. But it is not a CAT.0/-metric. Another important metric is the Weil–Petersson metric on Tg;n . At any point .†g;n ; Œ'/ 2 Tg;n , the cotangent bundle of Tg;n is equal to the space Q.†g;n / of holomorphic quadratic differential forms on †g . Let ds 2 be the hyperbolic metric of †g;n . Then the inner product on the cotangent bundle is given by: for '1 ; '2 2 Q.†g;n /, Z h'1 ; '2 i D

†g;n

'1 '2 .ds 2 /1 :

It is a Kähler metric but is incomplete. On the other hand, it has the following important property [425], [426]. Proposition 5.7. The Weil–Petersson metric on Tg;n is negatively curved and geodesically convex in the sense that every two points are connected by a unique geodesic. Though the Weil–Petersson metric is not complete, the result in this proposition is a good replacement. For example, the basic Cartan fixed point theorem (Proposition 4.18) for actions of compact isometry groups on complete Riemannian manifolds of nonpositive curvature holds in this case.

5.5 Compactifications and boundaries of Teichmüller spaces Since Tg;n is noncompact, a natural problem is to construct and study some natural compactifications. The following are a few compactifications among all compactifications of Tg;n : (1) The Teichmüller ray compactification of Tg;n obtained by identifying Tg;n with R6g6C2n using Teichmüller rays from a fixed basepoint and adding the sphere S 6g6C2n1 at infinity [235]. (2) The Bers compactification of Tg;n obtained by taking the closure of the image of Tg;n under the Bers embedding [327]. (3) The Thurston compactification [403]. (4) The harmonic map compactification [423]. (5) The extremal length compactification [144].

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(6) The horofunction compactification with respect to the asymmetric Thurston metric [415]. (7) The real spectrum compactification of Tg [82]. (8) The compactification via actions on ƒ-trees [316]. Among these eight compactifications, the harmonic map compactification, the horofunction compactification and the compactification via actions on R-trees (i.e., when ƒ D R in [316]) are isomorphic to the Thurston compactification. It is also known that the action of Modg;n on Tg;n extends to a continuous action to the Thurston compactification and hence also to the harmonic map compactification and the horofunction compactification. It also extends continuously to the real spectrum compactification of Tg . On the other hand, it does not extend continuously to the Teichmüller ray compactification or the Bers compactification [234], [235]. The continuous extension of the action of Modg;n to the Thurston compactification was used crucially in the classification of elements of Modg;n . The action of Modg;n on the Thurston boundary is also important for various rigidity results [165], [166], [236], [237]. It seems that the other compactifications have not been used for similar applications. On the other hand, there is a closely related partial compactification of Tg;n , which can be obtained from the Bers compactification. It can be described in several different ways. Recall that a Riemann surface † is called stable if its group of biholomorphic automorphisms is finite. This is equivalent to the condition that each connected component of † has negative Euler characteristic and hence it is also equivalent to the condition that † admits a complete hyperbolic metric of finite area. (We note that the isometry group of any hyperbolic manifold of finite volume is finite.) For example, a compact Riemann surface †g of genus g is stable if and only if g > 1. More generally, †g;n is stable if and only if 2g 2 C n > 0. In [1], the augmented Teichmüller space Tyg;n was introduced. As a set, it is the union of Tg;n and the set of stable Riemann surfaces which are obtained from †g;n by pinching along some simple closed geodesics. These boundary stable Riemann surfaces are also marked in some sense. More specifically, they correspond to the socalled regular b-groups in the boundary of the Bers compactification [42], [284], [1]. Three equivalent topologies were introduced on Tyg;n in [1]. They basically correspond to the intuitive idea that a sequence of marked Riemann surfaces †g;n converges to a boundary stable curve if and only if a collection of disjoint simple closed curves are pinched. Its relation with the induced subspace topology from the Bers compactification was not clear or discussed in [1]. We will address this later by using the method of Satake compactifications of locally symmetric spaces. One application of the augmented Teichmüller space Tyg;n is the following result [1].


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Proposition 5.8. The action of Modg;n on Tg;n extends to a continuous, but not proper, action on Tyg;n , and the quotient Modg;n nTyg;n is a compact Hausdorff space. The compact quotient Modg;n nTyg;n is a compactification of the moduli space Mg;n and equal to the Deligne–Mumford compactification. As mentioned before, the Weil–Petersson metric is not complete. An important result is the following [291], [425], [427]. Proposition 5.9. The completion of Tg;n with respect to the Weil–Petersson metric is canonically homeomorphic to the augmented Teichmüller space Tyg;n . Furthermore, with the extended distance function, Tyg;n is a CAT.0/-space. This result is satisfying since it realizes the augmented Teichmüller space Tyg;n completely in terms of an intrinsic metric of Tg;n . One consequence is the following realization of the Deligne–Mumford compactification Mg;n

DM

.

Corollary 5.10. The completion of Mg;n with respect to the Weil–Petersson metric is DM equal to the Deligne–Mumford compactification Mg;n . One consequence of this together with the results on the automorphism group of the curve complex C.Sg;n / [188], [271] is the following corollary [293], [425]. Corollary 5.11. With a few exceptions, the isometry group of the Weil–Petersson metric of Tg;n is equal to Modg;n .

5.6 Curve complexes and boundaries of partial compactifications The boundary of the augmented Teichmüller space Tyg;n consists of Teichmüller spaces of Riemann surfaces of lower genus with more punctures and of the same Euler characteristic. The inclusion relations between these boundary components of Tyg;n can be described in terms of the curve complex C .Sg;n / of the surface Sg;n . This is an infinite simplicial complex and plays an important role for the Teichmüller space Tg;n as does the spherical Tits building Q .G / for the symmetric space X . Specifically, consider the collection of the homotopy classes Œc of all essential simple closed curves in Sg;n , i.e., simple closed curves that are not trivial or homotopic to a loop around a puncture. They parametrize the vertices of C .Sg;n /. The vertices Œc1 ; : : : ; ŒckC1 form the vertices of a k-simplex if and only if they admit disjoint representatives. It is known that C .Sg;n / is an infinite simplicial complex of infinite diameter [188]. It clear that Modg;n acts simplicially on C.Sg;n /.

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Proposition 5.12. The quotient Modg;n nC .Sg;n / is a finite complex. Proof. Since a simplex of C.Sg;n / of maximal dimension corresponds to a maximal collection of disjoint non-homotopic simple closed curves on Sg;n , i.e., a pants decomposition, and since there are only finitely many homeomorphism classes of pants decompositions, the proposition follows. An important result due to Harer [169] is the following. Proposition 5.13. The curve complex C .Sg;n / is homotopy equivalent to a bouquet of spheres. This is the analogue of the Solomon–Tits theorem for Tits buildings (Proposition 4.65). From the proof, it was not clear if the bouquet contains at least one sphere, i.e., if C.Sg;n / has trivial topology. This was answered in [192]. Proposition 5.14. The curve complex C.Sg;n / is homotopy equivalent to a bouquet of infinitely many spheres. The dimension d of the spheres is d D 2g 2 when n D 0 and g 2, and d D 2g 3 C n when g > 0 and n > 0, and d D n 4 when g D 0 and n 4. Due to the collar theorem, two (or more) simple closed geodesics on a hyperbolic surface †g;n can be pinched simultaneously if and only if they are disjoint. Then it is easy to imagine that each boundary Teichmüller space of Tyg;n corresponds to a simplex of C .Sg;n /. Proposition 5.15. For each simplex of C .Sg;n /, there is a boundary Teichmüller space T of Tyg;n , and for any two simplices 1 , 2 in C .Sg;n /, T1 is contained in T2 as a face if and only if 1 contains 2 as a face. Since each boundary Teichmüller space T is contractible, we get the following result. Corollary 5.16. The boundary of the augmented Teichmüller space Tyg;n is connected and has the homotopy type of a bouquet of infinitely many spheres. Remark 5.17. One way to decompose intrinsically the boundary of the augmented Teichmüller space Tyg;n into boundary Teichmüller spaces is to consider the maximal totally geodesic subspaces when Tyg;n is considered as the completion of the Weil– Petersson metric and as a CAT.0/-space. For a detailed discussion of the CAT.0/geometry of the augmented Teichmüller space Tyg;n , see [425], [427]. See also [426], [427] for a detailed description of the geometry of the boundary Teichmüller spaces in Tyg;n .


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5.7 Universal spaces for proper actions As mentioned before, given any discrete group , a natural problem is to construct good models of universal spaces E for proper actions of , in particular, cofinite models of E spaces. Proposition 5.18. The Teichmüller space Tg;n is a model of the universal spaces EModg;n for proper actions of Modg;n . Proof. For any finite subgroup F Modg;n , by the Nielsen realization result (PropoF is nonempty. To show that it is contractible, sition 5.6), the set of fixed points Tg;n F take any two points p; q 2 Tg;n . Consider the Weil–Petersson geodesic connecting them. Since such a geodesic is unique and p; q are fixed by F , the geodesic is also F is contractible. fixed by F . This implies that Tg;n Remark 5.19. We can also use the fact that with respect to the Teichmüller metric, every two points are connected by a unique geodesic to prove the second statement in the proof. The negative curvature of the Weil–Petersson metric can also be used to F . prove nonemptiness of the fixed point set Tg;n Since Modg;n nTg;n is noncompact, Tg;n is not a cofinite space of Modg;n . To overcome this difficulty, a natural method is to construct an analogue of the Borel– Serre partial compactification of symmetric spaces. Such a construction was outlined by Harvey [172] and carried out by Ivanov (see [188] and references). On the other hand, it is not obvious that it satisfies the property that the fixed point set of any finite subgroup on the partial compactifcation is contractible. Another way is to remove suitable neighborhoods of the infinity of Tg;n so that its quotient by Modg;n is compact. For any small constant " > 0, define the thick part of Tg;n by Tg;n ."/ D f.†g;n ; Œ'/ 2 Tg;n j †g;n has no closed geodesic of length < "g: (2) Proposition 5.20. For " sufficiently small, the thick part Tg;n ."/ is a real analytic manifold with corners and invariant under Modg;n , and the quotient Modg;n nTg;n ."/ is a compact real analytic manifold with corners and hence has the structure of a finite CW-complex. The key statement that Modg;n nTg;n ."/ is compact follows from the Mumford compactness criterion for subsets of Mg;n [324]. Remark 5.21. The Mahler compactness criterion for subsets of the locally symmetric space SL.n; Z/nGLC .n; R/=SO.n/, the space of lattices of Rn , is a foundational result in the reduction theory of arithmetic groups. The Mumford compactness criterion was motivated by that, and this is another instance of fruitful interaction between two spaces discussed in this chapter.

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The expected fact that Tg;n ."/ is a cofinite model of EModg;n was proved in [216]. Proposition 5.22. There exists a Modg;n -equivariant deformation retraction of Tg;n to the thick part Tg;n ."/. In particular, for any finite subgroup F Modg;n , the fixed point set Tg;n ."/F is nonempty and contractible, and hence Tg;n ."/ is a cofinite model of EModg;n . It is clear that the thick part Tg;n ."/ is similar to the truncated subspace of the symmetric space X mentioned in §4.19. Both spaces give cofinite models for universal spaces of proper actions.

5.8 Cohomological properties of Modg;n We now discuss some consequences of the existence of a cofinite EModg;n -space in the previous subsection. Proposition 5.23. Modg;n is of type WFL, which means that for any torsion-free subgroup 0 of finite index there is a free resolution of Z over Z 0 of finite length, and Modg;n is also of type FP1 . In particular, in every degree i , Hi .Modg;n ; Z/ and H i .Modg;n ; Z/ are finitely generated. Determining the cohomology groups H i .Modg;n ; Z/ is important and complicated. The stable cohomology groups with rational coefficients H i .Modg;n ; Q/ can be computed (see [169], [188], [276], [275]). Another result is the following [169], [192]. Proposition 5.24. Modg;n is a virtual duality group and its dualizing module is equal to Hi .C .Sg;n /; Z/, where i is the only positive degree in which the homology of the curve complex C .Sg;n / is not equal to 0. On the other hand, Modg;n is not a virtual Poincaré duality group. The similarities with duality results for arithmetic subgroups in Proposition 4.67 are striking. The proof of these results are also similar, by using the fact that the boundary of Tg;n ."/ has the same homotopy type as the curve complex C .Sg;n / and the analogue of the Solomon–Tits theorem. This is one instance showing similar roles played by the Tits building Q .G / and the curve complex C .Sg;n /. For more results showing similarities between the Tits building Q .G / and the curve complex C .Sg;n / and references, see [210].

5.9 Pants decompositions and the Bers constant An important technique in studying hyperbolic metrics on surfaces Sg;n is to understand pants decompositions. The reason is that hyperbolic surfaces can be built up


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from the basic pieces, pairs of pants. For example, the Fenchel–Nielsen coordinates can be defined for every pants decomposition. Recall that for any hyperbolic surface †g;n , a pants decomposition consists of a collection of simple closed geodesics c1 ; : : : ; c3g3Cn such that every connected component of the complement in Sg;n is biholomorphic to the unit disk with two disjoint disks or points removed, i.e., a pair of pants. Pants decompositions are not unique. Since every essential simple closed curve in †g;n , i.e., a curve not homotopic to a point or a loop around a puncture, contains a unique simple closed geodesic in its homotopy class, any collection of disjoint essential simple closed curves 1 ; : : : ; 3g3Cn in †g;n that are pairwise nonhomotopic induces a pants decomposition. Then any element 2 DiffC .†g;n /, . 1 /; : : : ; . 3g3Cn / also induces a pants decomposition, which is in general different from the previous one. By the above argument, we can see that any collection of disjoint essential simple closed curves 1 ; : : : ; 3g3Cn in Sg;n that are pairwise nonhomotopic induces a pants decomposition for every marked Riemann surface .†g;n ; Œ'/ in Tg;n . A natural question is, for a given hyperbolic surface †g;n , what kind of pants decompositions are optimal in some sense. This is answered by the following basic result (see Chapter 5 in [90]). Proposition 5.25. There exists a constant ı D ı.g; n/ such that every hyperbolic surface †g;n admits a pants decomposition such that the lengths of the geodesics in the pants decomposition are bounded from above by ı. The constant ı.g; n/ in the proposition is called a Bers constant. The minimal value it can take is not known. It is not clear how many pants decompositions satisfying the above conditions exist.

5.10 Fundamental domains and rough fundamental domains Motivated by the reduction theory for arithmetic groups and its applications to understanding the structure of arithmetic groups and associated locally symmetric spaces, a natural and important problem for the action of Modg;n on Tg;n is to find fundamental domains and rough fundamental domains that reflect properties of Modg;n and Tg;n . Besides their importance in understanding structures of mapping class groups, finding good fundamental domains is an interesting problem in itself. The earlier discussion about the reduction theory of arithmetic subgroups indicates that it is not easy to construct fundamental domains. For many applications, rough fundamental domains with properties adapted to structures at infinity of Teichmüller spaces might be equally or even more useful than complicated fundamental domains. In this subsection, we construct rough fundamental domains using the so-called Bers sets by using pants decompositions that satisfy the conditions in Proposition 5.25. In the next subsection, we generalize Minkowski reduction to the action of Modg on

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Tg to obtain an intrinsically defined fundamental domain. This is closely related to a folklore open problem on constructing intrinsic Modg -equivariant cell decompositions of Tg (see Problems 5.30 and 5.31 in the next subsection). Before we define Bers sets, we summarize some earlier results on fundamental domains, rough fundamental domains for mapping class groups Modg;n and related Modg;n -equivariant tilings of Tg;n : (1) A fundamental domain for Modg in Tg was constructed and defined by finitely many equations involving lengths of (non-separating) simple closed geodesics in [285], [286]. In some lower genus cases, defining equations are worked out in [287], [288] and [151], [152] more explicitly. The topology of the fundamental domain is not clear; for example, it is not known whether it is a cell. (2) The Dirichlet fundamental domain for Modg;n with respect to the Teichmüller metric of Tg;n was studied in [297]. It is star-shaped with respect to the center. It is intrinsically defined in terms of the geometry of all Riemann surfaces in Tg;n once the center is chosen, but it is not defined in terms of the intrinsic geometry of each Riemann surface alone. It is not clear whether the closure in the augmented Teichmüller space Tyg;n is a cell. (3) A fundamental domain for Modg when g D 2 was given in [229] explicitly in terms of nonlinear polynomials in some special coordinates of Tg . The topology of the fundamental domain is not clear. There are also related results in [230]. (4) Rough (or approximate) fundamental domains for Modg;n were first introduced in [228] to solve a conjecture of Bers. Later in [90] and also implicitly in [3], different rough fundamental domains for Modg;n were introduced using Bers sets and Fenchel–Nielsen coordinates. This is analogous to the reduction theory for arithmetic groups acting on symmetric spaces of noncompact type (see [57], [61]). (5) Equivariant cell decompositions of Tg;n for pairs .g; n/ with n > 0 for small values g and n or for some subgroups of Modg;n were given in [375], [376]. Though equivariant cell decompositions of Tg;n are known in [169], [170], [65], [345], [346], [347] (see the next subsection for more detail), the point of the papers [375], [376] is to use systoles (minimal length of geodesics) to obtain such cell decompositions so that they might be generalized to the case Tg . (6) Generalizing the precise reduction theory of arithmetic groups acting on symmetric spaces of noncompact type [371], an equivariant tiling of Tg was given in [257]. (A tiling of a symmetric space means here an equivariant decomposition of the symmetric space. But each piece could have large stabilizers, and hence it is not an equivariant cell decomposition. See §4.10 for more detail.) To get an fundamental domain from this tiling, one needs to get a fundamental domain for each tile with respect to the stabilizer in Modg of the tile. The central tile is invariant under Modg with a compact quotient, and how to get such a fundamental domain of Modg for the central tile is not automatic or obvious. Getting


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a fundamental domain for other tiles depends on the central tile of Teichmüller spaces of smaller dimensions. (7) In the case of genus 2, an equivariant cell decomposition of Tg was obtained in [367] using Weierstrass points. Now we define the rough fundamental domains introduced in [228], [3], [90]. For every pants decomposition P D fc1 ; : : : ; c3g3Cn g of Sg;n , there is a Fenchel–Nielsen coordinate system: FN P W Tg;n ! R3g3Cn R3g3Cn ; C .†g;n ; Œ'/ 7! .`1 ; : : : ; `3g3Cn ; 1 ; : : : ; 3g3Cn /; where ì is the length of the simple closed geodesic i in the homotopy class ' 1 .ci /, and i is the twisting angle along the geodesic i . The twisting angles are not canonically defined and depend on an additional choice, for example, some extra combinatorial data. For each pants decomposition P , for any positive constants b1 ; b2 , define a Bers region BP ;b1 ;b2 D FN 1 P ..0; b1 .0; b1 Œb2 ; b2 Œb2 ; b2 /: It is known that up to the action of Modg;n , there are only finitely many equivalence classes of pants decompositions of Sg;n . Let P1 ; : : : ; Pm be representatives of these equivalence classes of pants decompositions. Let ı D ı.g; n/ be the Bers constant in Proposition 5.25. For each Pi , let Bi be the Bers region BPi ;ı; . Proposition 5.26. The union B D B1 [ [ Bm is a rough fundamental domain for the Modg;n -action on Tg;n . It satisfies both the local finiteness and the global finiteness conditions. The fact that the Modg;n -translates of B cover Tg;n follows from Proposition 5.25. For the proof that it is a rough fundamental domain satisfying the finiteness conditions, see [90], §6.6. Remark 5.27. As mentioned before, the curve complex C .Sg;n / is similar to the spherical Tits building Q .G /, and hence minimal Q-parabolic subgroups of G correspond to pants decompositions P of Sg;n . The above discussion give a concrete example of such a comparison. The Fenchel–Nielsen coordinate system of Tg;n associated with P is similar to the horospherical decomposition of the symmetric space X associated the minimal Q-parabolic subgroup P. Then the Bers subsets of Tg;n associated with P correspond to the Siegel subsets of X associated with P in Definition 4.33. The horospherical decomposition of the symmetric space X associated to non-minimal Q-parabolic subgroups are important in the reduction theory of arithmetic groups and compactifications of the locally symmetric space nX. Similarly, there is also a generalization of the Fenchel–Nielsen coordinate system of Tg;n for any sub-collection of simple closed curves contained in any pants decomposition P .

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Proposition 5.26 is the analogue of the reduction theory for arithmetic groups in Proposition 4.34. As mentioned in the summary earlier in this subsection, an analogue of the reduction theory in [371], i.e., the -equivariant tiling of X recalled in §4.10, also holds for Modg;n [257].

5.11 Generalized Minkowski reduction and fundamental domains After obtaining rough fundamental domains for the Modg;n -action on Tg;n in the previous subsection, a natural problem is to construct a fundamental domain for Modg;n . In the case of arithmetic groups , there are two cases depending on whether the symmetric space X is linear or not. If X is linear, a fundamental domain for the -action can be described and is given by the union of finitely many polyhedral cones in the associated symmetric cone. In the general case, the situation is more difficult and is not understood. For the action of Modg;n on Tg;n , if n > 0, a stronger result than constructing fundamental domains is known. Specifically, there is the following important result. Proposition 5.28. Assume n > 0. Then Tg;n admits an intrinsic Modg;n -equivariant cell decomposition, and there are only finitely many Modg;n -orbits of cells. It is due to many people including Mumford, Thurston, Harer [169], Penner [345], Bowditch–Epstein [65]. This result is similar to the -equivariant polyhedral cone decompositions of linear symmetric spaces. An immediate corollary is Corollary 5.29. The Modg;n -action on Tg;n admits a fundamental domain consisting of finite cells in the equivariant cell decomposition in Proposition 5.28. Proposition 5.28 has several important applications. (1) A proof of the Witten conjecture on the intersection theory of the moduli space Mg;n by Kontsevich [244]. (See also Chapter 5 of volume II of this Handbook [314] for a survey of Witten’s conjecture and its various proofs.) (2) Evaluation of the Euler characteristic of Modg;n by Harer–Zagier [171]. The method of proof of Proposition 5.28 depends crucially on the presence of punctures, i.e., n > 0. Partially motivated by the above results, a longstanding folklore problem is the following. Problem 5.30. Construct an intrinsic cell decomposition of Tg such that the following conditions are satisfied: (1) It is equivariant with respect to Modg and there are only finitely many Modg equivalence classes of cells. (Naturally some cells are not closed since Modg nTg is noncompact.)


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(2) It descends to a finite cell decomposition of Mg . (3) The closure of each cell in the augmented Teichmüller space Tyg is a closed cell so that the cell decomposition of Mg extends to a finite cell decomposition of DM the Deligne–Mumford compactification Mg Š Modg nTyg . It should be emphasized that the key point is that the cell decomposition should be intrinsic, i.e., depending on the intrinsic hyperbolic geometry or complex structure of the marked Riemann surfaces in Tg . Otherwise, the existence of such an equivariant cell decomposition follows easily from general facts on the existence of triangulations of analytic spaces and the fact that Mg admits a compactification which is a projective variety (See [202]). The existence of an equivariant cell decompositions of Tg implies the existence of fundamental domains for the action of Modg on Tg , which consist of a union of finitely many representatives of the cells. As one step towards solving Problem 5.30, a weaker problem is to find an intrinsic fundamental domain for Modg which is the union of finitely many cells such that they have no overlap in the interior, and the closure of each cell in Tyg is also a cell. Then the -translates of these cells give an equivariant decomposition of Tg with disjoint interior, and their closures in Tyg also give an equivariant decomposition of Tyg into cells. Therefore, a weaker version of Problem 5.30 is the following: Problem 5.31. Construct a fundamental domain of the Modg -action on Tg which consists of a finite union of cells such that these cells are defined intrinsically and their interiors are disjoint and their closures in the augmented Teichmüller space Tyg are also cells. Remark 5.32. For a public statement of Problem 5.30 on equivariant intrinsic cell decompositions of Tg with an extension to the augmented space Tyg , see Problems 1 and 2 by D. Sullivan of the CTQM problem list. In these problems, Sullivan proposed to use Weierstrass points of compact Riemann surfaces to replace the punctures to solve this problem. This list of open problem was created in 2006 at the opening symposium of Center for the Topology and Quantization of Moduli Spaces, University of Aarhus. It is posted as the website http://www.ctqm.au.dk/PL/. It was also raised at a workshop on the moduli space of curves and is posted at http://www.aimath.org/WWN/modspacecurves/open-problems/index.html In this subsection, we discuss a generalization of Minkowski reduction for the action of SL.n; Z/ on the space of positive definite quadratic forms to the action of Modg on Tg , and hence give a solution to Problem 5.31. For more details, see [202]. The key concept is the notion of reduced ordered pants decomposition of a marked hyperbolic Riemann surface. Let P D fc1 ; : : : ; c3g3 g be an ordered collection of simple closed geodesics of a hyperbolic surface †g such that they form an ordered pants decomposition of †g .

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It is called a reduced ordered pants decomposition of †g if the following conditions are satisfied: (1) The geodesic c1 has shortest length among all simple closed geodesics in †g . (2) The geodesic c2 has shortest length among all simple closed geodesics in †g that are disjoint from c1 . (3) More generally, for any i 2, ci has shortest length among all simple closed geodesics in †g that are disjoint from c1 ; : : : ; ci1 . It is clear that such a reduced ordered pants decomposition always exists, but is not necessarily unique. It is not unique if and only if there are more than one simple closed geodesics of minimal length at some stages in the above definition. On the other hand, if c1 is a unique simple closed geodesic of shortest length, i.e., a unique systole, and for every i 2, ci is a unique simple closed geodesic of shortest length that is disjoint from c1 ; : : : ; ci1 , then fc1 ; : : : ; c3g3 g is a unique reduced ordered pants decomposition. It is clear that a generic hyperbolic surface †g has a unique reduced ordered pants decomposition. For any ordered pants decomposition P D fc1 ; : : : ; c3g3 g of Sg , define a domain z P of Tg as follows: z P D f.†g ; Œ'/ 2 Tg j Œ' 1 .P / is a reduced ordered pants decomposition of †g g; (3) where Œ' 1 .P / represents the ordered pants decomposition of †g consisting of the unique geodesics in the homotopy classes Œ' 1 .ci /, i D 1; : : : ; 3g 3. z P is invariant under the stabilizer of P in Modg , denoted by StabP . The domain The reason is that if P is a reduced ordered pants decomposition for a marked Riemann surface .†g ; Œ'/, then for any element Œ 2 Modg , .P / is also a reduced ordered pants decomposition of the new marked Riemann surface Œ .†g ; Œ'/. To construct a fundamental domain for the Modg -action on Tg , we need to find z P . For this purpose, we need to identify StabP . fundamental domains of StabP in It is clear that StabP contains the subgroup generated by the Dehn twists along curves in P . But it could also contain some half Dehn twists. To explain this, we call a curve ci 2 P hyper-elliptic if ci separates off a one-holed torus, i.e., one connected component of †g ci is biholomorphic to a Riemann surface of genus 1 with a small disk removed. Lemma 5.33. For every hyper-elliptic curve ci 2 P , the half Dehn twist along ci is contained in Modg and also in the stabilizer StabP . The idea of the proof is as follows. Each compact Riemann surface of genus 1 with one distinguished point admits an involution that fixes the distinguished point. Remove a small disk around this point that is stable under the involution. Then this involution corresponds to a half Dehn twist along the boundary circle. Let †0 ; †00 be the two connected components of †g ci . Suppose that †0 is a one-holed torus. Then the involution on the pointed elliptic curve defines an involution on †0 . Extend it to


261

†00 such that it is the identity map outside a small tubular neighborhood of ci , and the extended map on †g is a half Dehn twist of ci . Given an ordered pants decomposition P , let be the subgroup of Modg generated by the half Dehn twists on all hyper-elliptic curves in P and full Dehn twists on the other curves in P . Then we have the following result. z P is Proposition 5.34. The stabilizer StabP is equal to P , and hence the domain invariant under P . We note that in the definition of a reduced ordered pants decomposition, we have only imposed conditions on the lengths of the geodesics ci in Œ' 1 .P /. Since the Dehn twists along these geodesics do not change the fact that P is an ordered reduced pants decomposition and the lengths of these geodesics, it is natural to find conditions on the twisting parameters 1 ; : : : ; 3g3 . For each curve ci 2 P , define mi D 12 if ci is hyper-elliptic, and mi D 1 otherwise. The choice of the value of mi is determined by the minimal Dehn twist along ci that is contained in StabP . z P by Define a subdomain P of z P j i 2 Œ0; 2 mi ; i D 1; : : : ; 3g 3g: (4) P Df.`1 ; 1 I : : : I `3g3 ; 3g3 / 2 From the description of the stabilizer StabP , it is clear that the subdomain P is zP. a fundamental domain of the StabP -action on For any pants decomposition P , the twisting angles i are not uniquely defined and depend on various choices. In [48], some particularly nice ones are chosen so that the length functions associated with simple closed geodesics are convex functions in the associated Fenchel–Nielsen coordinates. A crucial property is the following result [202], Proposition 5.3. Proposition 5.35. With respect to a suitable choice of Fenchel–Nielsen coordinates of Tg for each pants decomposition P in [48], P is contractible. The basic idea is to deform along the anti-stretch lines in the Thurston metric of Tg [404] so that in the deformation process, P is kept as a reduced ordered pants decomposition and the twisting coordinates remain invariant. Let P1 ; : : : ; Pn0 be representatives of Modg -equivalence classes of pants decompositions of Sg as above. Let P1 ; : : : ; Pn0 be the domains associated with them as defined in Equation (4). Define D P1 [ [ Pn0 :

(5)

Then one of the main results of [202] is the following, which gives a solution to Problem 5.31. Theorem 5.36. The domain is an intrinsically defined fundamental domain for the Modg -action on Tg satisfying the following properties:

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(1) It satisfies both the local finiteness and global finiteness conditions. (2) Each domain Pi in Tg is a cell and its closure Pi in Tyg is also a cell. By the same argument, a fundamental domain for the action of Modg;n on Tg;n can be constructed and enjoys the same properties.

5.12 Compactifications of moduli spaces and a conjecture of Bers Suppose that a discrete group acts properly on X with a noncompact quotient nX. A natural and important problem is to understand relations between compactifications x of X by is of X and nX. If is infinite, the quotient of a compactification X non-Hausdorff in general, since the -action on the boundary is not proper since any infinite group cannot act properly on a compact space. This problem has been discussed earlier in the setup of actions of arithmetic groups on symmetric spaces of noncompact type and compactifications of symmetric and locally symmetric spaces. Though this problem was known for a long time for the compactification of the upper half plane and its quotients, it was Satake [374] who formulated it for general symmetric spaces and their arithmetic quotients. In this section, we follow the method of Satake compactifications of locally symmetric spaces to construct the Deligne–Mumford compactification of Mg;n from the Bers compactification of Tg;n . We believe that this might be the motivation for a conjecture of Bers ([42], Conjecture IV, p. 599). Near the end of this subsection, we also explain how to apply the same procedure to construct a new compactification of Mg;n whose boundary is equal to Modg;n nC.Sg;n /, a finite simplicial complex. Recall that for every fixed base point .†g;n ; Œ'/ in Tg;n , there is a Bers embedding iB W Tg;n ,! C 3g3Cn Š Q.†g;n /; where Q.†g;n / is the space of holomorphic quadratic differentials on †g;n . It is a holomorphic embedding and the image is a bounded star-shaped domain. The closure B iB .Tg;n / is the Bers compactification and it is denoted by Tg;n . The geometry of the Bers boundary @Tg;n analytic arc components in @Tg;n

B

B

is complicated. We can also define

as for bounded symmetric domains. Though we B

cannot determine all the analytic arc components, it is known that the boundary @Tg;n contains some natural complex submanifolds. In fact, it is known [2], §5 (see also [299], Theorem 1.2, Corollary 1.3) that every stable Riemann surface of Euler characteristic 2 2g n appears in the boundary B @Tg;n . By [2], Corollary 1, p. 230, we have the following result.


263

Proposition 5.37. For every marked stable Riemann surface †0 that is contained B in the boundary @Tg;n , the Teichmüller space of †0 is contained in the boundary B

@Tg;n as well. In [42], Bers stated the following conjecture (Conjecture IV, p. 599). Conjecture 5.38. There exists a fundamental domain for the Modg;n -action on B

Tg;n such that the intersection of the closure of in Tg;n consists of cusps. The notion of cusp is defined as follows. Let †g;n be the fixed base Riemann surface that defines the Bers embedding. Write it as a quotient † nH2 , where † is a discrete subgroup of P SL.2; R/. Then there is an embedding Tg;n ! Hom.† ; PSL.2; C//=conjugation: B

The closure of this embedding can be identified with the Bers embedding Tg;n [299], p. 218. Under this identification, and according to the definition in [42], p. 571, (see B also Theorem 10 in [42]), a boundary point in @Tg;n given by a discrete faithful representation ' W † ! PSL.2; C/ is called a cusp if a hyperbolic element in † is mapped to a parabolic element. A coarse fundamental domain for the Modg;n -action on Tg;n was constructed in [228] and it was shown that the intersection of the closure of the rough fundamental domain with the boundary @Tg;n consists of cusps. An immediate corollary of [228] and of the above discussion of fundamental domains is the following result. Proposition 5.39. The fundamental domain for the Modg -action on Tg in Theorem 5.36 satisfies the Bers conjecture. More generally, a similarly defined fundamental domain for the Modg;n -action on Tg;n also satisfies the Bers conjecture. Proof. By construction, the fundamental domain in Theorem 5.36 is contained in the rough fundamental domains of [228], and the proposition follows immediately. Alternatively, we can see directly that for any unbounded sequence of marked hyperbolic surfaces in each domain Pi in Theorem 5.36, some geodesics in the pants decomposition Pi are pinched, i.e., their lengths go to 0. This implies that every x \ @Tg B is a cusp. boundary point in Denote by g;n the fundamental domain for the action of Modg;n on Tg;n . When n D 0, it is reduced to in Theorem 5.36. It is clear that any regular b-group, i.e., a stable Riemann surface that appears B in the boundary of @Tg;n , is a cusp. But the converse is not true in general. For example, assume g 3. We can pinch one separating simple closed geodesic in †g , and deform the connected component of genus at least 2 to a degenerate boundary

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point in the sense of [42], [1]. Then the corresponding boundary point of Tg is a cusp but not a b-regular group. The following slightly stronger result holds and is important for the discussion in this subsection. B

Proposition 5.40. The intersection of g;n \ @Tg;n consists of stable Riemann surfaces, and every marked stable Riemann surface belongs to a translate g;n for some 2 Modg;n . Proof. For simplicity, we discuss the case of Tg . For a pants decomposition Pi in Theorem 5.36 and its associated domain Pi , if a sequence of marked Riemann surB

faces .†j ; Œ'j / 2 Pi converges to a boundary point in Pi \ @Tg , then by passing to a subsequence if necessary, we can assume that a subset of geodesics of .†j ; Œ'j / contained in Pi is pinched, i.e., their lengths go to 0, and for the other geodesics in Pi , their lengths converge to positive numbers, and their Fenchel–Nielsen twisting parameters also converge. Such a sequence .†j ; Œ'j / determines a marked stable Riemann surface .†1 ; Œ'1 /. By [299], Theorem 1.2, Corollary 1.3, the sequence B .†j ; Œ'j / also converges to .†1 ; Œ'1 / in the Bers compactification Tg . (Note that this is a crucial point. Of course, .†j ; Œ'j / converges to .†1 ; Œ'1 / in the augmented Teichmüller space Tyg with respect to the three equivalent topologies in [1]. B

But we need convergence with respect to the Bers compactification Tg .) In the above proof, we have used the fact that the marked stable Riemann surface .†1 ; Œ'1 / is B contained in the Bers compactification Tg . This proves that the limit point of the B

sequence .†j ; Œ'j / 2 Pi in Tg is a stable Riemann surface, and the first statement is proved. For the second statement, we note that for any marked stable Riemann surface .†1 ; Œ'1 / of Euler characteristic 2 2g n, by opening up the nodes, i.e., pairs of cusps, we obtain marked Riemann surfaces .†j ; Œ'j / in Tg . By passing to a suitable subsequence and under the action of some elements of the subgroup of Modg generated by the Dehn twists of the opened up geodesics, we can assume that .†j ; Œ'j / is contained in Pi for some pants decomposition Pi and in the stabilizer StabPi of Pi in Modg . Then the arguments in the previous paragraph show that .†1 ; Œ'1 / B

is the limit of a subsequence of .†j ; Œ'j / in the Bers compactification Tg , and hence is contained in the closure Pi . In constructing Satake compactifications of locally symmetric spaces nX, a boundary point of a Satake compactification of X is called Siegel rational [61], p. 289, [374] if it meets the closure of a Siegel set of a Q-parabolic subgroup. Recall that we defined the Bers set BP ;b1 ;b2 in §5.10. For simplicity, denote it by BP . Similarly we can introduce the following.


265

B

Definition 5.41. A boundary point in @Tg;n is called rational if it is in the closure of a Bers set BP for some pants decomposition P . B

Then Proposition 5.40 says that the set of rational boundary points of Tg;n consists of exactly regular b-groups. The boundary Teichmüller spaces in Proposition 5.37 consist of rational points and hence can be called rational boundary components. Now we can apply the method in defining Satake compactifications of locally symmetric spaces in [374], [61] (§III.3) to construct the Deligne–Mumford compactification of Mg;n and also to recover the topologies on the augmented Teichmüller space in [1]. By the above discussions, we have the following result. Proposition 5.42. The augmented Teichmüller space Tyg;n mentioned in §5.4 is equal B

to the union of Tg;n with all rational boundary points of Tg;n . It is clear that Modg;n acts on Tyg;n . For every Bers set BP , the closure BP in B Tg;n is contained in Tyg;n . Endow BP with the subspace topology induced from the B

Bers compactification Tg;n . Proposition 5.43. There is a natural topology on Tyg;n that is induced from the topology B of the Bers compactification Tg;n such that the action of Modg;n on Tyg;n satisfies the following properties: (1) It induces the topology on Tg;n and the closure of every Bers set BP . (2) The Modg;n -action on Tyg;n is continuous. (3) For every point p 2 Tyg;n , there exists a fundamental system of neighborhoods fU g of p such that for in Modg;n that fixes p, U D U , and for the other , U \ U D ;. (4) If p; p 0 2 Tyg;n are not in one Modg;n -orbit, then there exist neighborhoods U of p and U 0 of p 0 such that Modg;n U \ U 0 D ;. Furthermore, any topology on Tyg;n satisfying the above conditions is equal to the natural one defined above. The basic idea is that for any boundary point p contained in the closure BP of a Bers set, take a neighborhood V of p in BP . Then the union of translates of V by elements of the stabilizer of p in Modg;n gives a neighborhood in the Satake topology of Tyg;n . This can be seen clearly in the context of SL.2; Z/ acting on H2 where horodisk neighborhoods of rational boundary points give the Satake topology. The same proof of [374] works by noticing the fact that the induced action of B Modg;n on each boundary Teichmüller space in Tg;n is the action of corresponding mapping class groups and hence is proper.

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It should be stressed that this Satake topology on Tyg;n is definitely different from (i.e., strictly finer than) the subspace topology on Tyg;n when it is considered as a B

subspace of Tg;n . Remark 5.44. The proof of Proposition 5.40 shows that the topology of BP is the same as the topologies induced from the three equivalent topologies on Tyg;n defined in [1]. Therefore, the Satake topology on the augmented Teichmüller space Tyg;n is equivalent to the three topologies in [1]. One important fact might be that the Satake topology here B is defined using the topology of the Bers compactification Tg;n . Therefore, we have constructed the augmented Teichmüller space Tyg;n (both the underlying space and the topology) purely in terms of the Bers compactification. This is in some sense similar to the fact that the Weil–Petersson completion of Tg;n gives an intrinsic construction of Tyg;n in Proposition 5.9. An immediate corollary of Proposition 5.43 is the following. Proposition 5.45. The quotient Modg;n nTyg;n of Tyg;n with the Satake topology is a compact Hausdorff space, which is equal to the Deligne–Mumford compactification DM Mg;n . Proof. The first statement follows from the properties of the Satake topology. The second statement follows from the construction of the Deligne–Mumford compactDM ification Mg;n that is the moduli space of all stable Riemann surfaces of Euler characteristic 2 2g n. Remark 5.46. In his papers [40], [41], [42], Bers did not explain his motivations for making Conjecture IV of [42] (see also [40], p. 296), i.e., Conjecture 5.38 above. It DM seems that the above construction of the Deligne–Mumford compactification Mg;n B

from the Bers compactification Tg;n following the method of compactifications of locally symmetric spaces should be one of the motivations. In some of his earlier works, Siegel had considered compactifications of fundamental domains of special arithmetic groups. Bers might have been motivated by some work of Siegel. The comments in [40], p. 296, might also justify the claim in this remark. On the other hand, it is important to note that compactifications of fundamental domains are related to, but different from, compactifications of locally symmetric spaces. As discussed earlier in this chapter, besides the Bers compactification, the Teichmüller space Tg;n admits several compactifications. Among them, the Thurston comTh pactification Tg;n is probably the most interesting. A natural problem is to apply the Th above procedure to the Thurston compactification Tg;n and to construct the corresponding compactification of Mg;n . It turns out to be a new compactification of Mg;n whose boundary is equal to Modg;n nC.Sg;n /, a natural finite simplicial complex [205].


267 Th

As in Definition 5.41, a boundary point of the Thurston compactification Tg;n is called rational if it is contained in the closure of a Bers set BP of a pants decomposition P . It is known that the curve complex C.Sg;n / can be canonically embedded into the Th boundary of Tg;n . Then the following result can be proved [198]. Proposition 5.47. For any Bers set BP , the intersection of the closure of BP with the Thurston boundary @BP is equal to the simplex corresponding to the pants decomposition P . A corollary is the following result. Corollary 5.48. The set of rational boundary points of Tg;n complex C.Sg;n /.

Th

is equal to the curve

Consequently, the partial compactification of Tg;n corresponding to the Thurston Th

compactification Tg;n is equal to Tg;n [C.Sg;n /, and the associated compactification of Mg;n is Mg;n [ Modg;n nC.Sg;n /. This compactification is similar to the Tits compactification of an arithmetic locally symmetric space nX in [215], whose boundary is nQ .G /, the quotient by of the Tits building Q .G /. Besides this formal similarity, the construction is also similar. Remark 5.49. Naturally, we will also get a different compactification of Mg;n from the Teichmüller compactification of Tg;n by the above procedure. It would be interesting to identify this compactification.

5.13 Geometric analysis on moduli spaces The moduli space Mg;n has been extensively studied from the points of view of algebraic topology, complex geometry, algebraic geometry and mathematical physics etc. In this section, we would like to raise several questions about Mg;n and emphasize the point of view of geometric analysis. The basic point is that Mg;n is also an important Riemannian orbifold and its geometry and analysis should be studied and better understood. We believe that this is an important direction to be explored. The spectral theory of arithmetic locally symmetric spaces nX has played a fundamental role in the theory of automorphism forms for . A natural problem is to study the spectral theory of Mg;n . As mentioned before, Tg;m admits several Modg;n -invariant Riemannian metrics, for example, the Weil–Petersson metric, the Bergman metric, the Ricci metric, the McMullen metric, etc. They induce Riemannian metrics on Mg;n . Though Mg;n is an orbifold, many concepts and techniques for Riemannian manifolds can be generalized

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to orbifolds and hence to Mg;n . In particular, each Riemannian metric on Mg;n induces a Laplace operator. Since the Weil–Petersson metric is incomplete, the first question is whether the Laplace operator with domain C01 .Mg;n / is essentially self-adjoint. The answer seems to be positive. In a joint work in progress with R. Mazzeo, W. Müller, and A. Vasy, we expect to prove the following result. Theorem 5.50. The Laplace operator of Mg;n acting on functions with respect to the Weil–Petersson metric is essentially self-adjoint and hence has a unique self-adjoint extension. Its spectrum is discrete and its counting function satisfies the Weyl law for the counting function of eigenvalues of compact Riemannian manifolds. For other complete metrics such as the Bergman metric and the Ricci metric, it is known that the Laplace operator is essentially self-adjoint and has a unique selfadjoint extension. Using the asymptotic behaviors of these metrics near the infinity of Mg;n , it can be shown that the spectrum of the Laplace operator is not discrete. On the other hand, it is not clear whether the non-discrete part of the spectrum is absolutely continuous, i.e., whether the spectrum measure is absolutely continuous. It is also desirable to understand structures of generalized eigenfunctions. For Hermitian arithmetic locally symmetric spaces nX, an important result is the validity of the Zucker conjecture, which says that the L2 -cohomology group of nX is canonically isomorphic to the intersection cohomology group of the Baily–Borel compactification of nX. The Lp -cohomology groups of nX were also studied in [433]. See [217] in this volume for a more detailed discussion. A natural problem is to relate the L2 -cohomology group of Mg;n to some cohomology groups of compactifications of Mg;n . Probably the most natural and important compactification of Mg;n is the Deligne– DM Mumford compactification Mg;n . It is a compact orbifold and hence its intersection cohomology group is equal to the usual cohomology group. Of course, the Lp -cohomolgy group of a Riemannian manifold (or orbifold) depends only on the quasi-isometry class of the metric. The following two results are proved in [217]. Proposition 5.51. For any p with 4=3 p < C1, the Lp -cohomology group of Mg;n with respect to the Weil–Petersson metric is canonically isomorphic to the DM cohomology group of the Deligne–Mumford compactification Mg;n . For 1 p < 4=3, the Lp -cohomology group of Mg;n with respect to the Weil–Petersson metric is canonically isomorphic to the cohomology group of Mg;n . The paper [372] proves only the case Mg and p D 2, and the same proof works for the more general case Mg;n and p D 2. Proposition 5.52. With respect to any Riemannian metric that is quasi-isometric to the Teichmüller metric, for any 1 < p < 1, the Lp -cohomology group of Mg;n


269

is isomorphic to the cohomology group of the Deligne–Mumford compactification DM Mg;n . In the case when the metric is the Bergman metric and p D 2, this result was proved in [431] (Theorem 4).

6 Interactions between locally symmetric spaces and moduli spaces of Riemann surfaces The most basic example of a symmetric space is the upper half plane H2 D fx C iy j x 2 R; y > 0g. It admits three important generalizations depending on different interpretations. First, H2 is the moduli space of marked elliptic curves (or Abelian varieties of dimension 1) and the quotient SL.2; Z/nH2 is the moduli space of elliptic curves. Second, by writing H2 D SL.2; R/=SO.2/, we can identify it with the space of positive definite binary quadratic forms of determinant 1. Third, H2 is the Teichmüller space Tg when g D 1. The generalization based on the first interpretation is the Siegel upper half space hg D fX C iY j X; Y are real g g matrices; Y > 0g. The symplectic group Sp.2g; R/ acts transitively and holomorphically on hg , and the stabilizer of the point iIg is equal to U.g/, and hence we have the identification hg D Sp.2g; R/=U.g/: This is a Hermitian symmetric space of noncompact type. The Siegel modular group Sp.2g; Z/ acts properly on hg and the quotient Sp.2g; Z/nhg is called the Siegel modular variety. It can be identified with the moduli space of principally polarized abelian varieties of dimension g and usually denoted by Ag . Clearly, when g D 1, hg is equal to H2 . The generalization based on the second interpretation is the symmetric space SL.n; R/=SO.n/. This space and its quotient SL.n; Z/nSL.n; R/=SO.n/ have been discussed before. The generalization based on the third interpretation is the Teichmüller space Tg for g 2, and Modg corresponds to SL.2; Z/. The quotient space Modg nTg is the moduli space Mg . Of course, Tg;n and Modg;n are also natural generalizations.

6.1 The Jacobian map and the Schottky problem It turns out that there is an important map between the two generalizations Mg and Ag of the space SL.2; Z/nH2 in the previous paragraph, i.e., the Jacobian (or period) map J W Mg ! Ag :

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Clearly, Ag D Sp.2g; Z/nhg is an important locally symmetric space. In the previous sections, we were mainly interested in analogies between locally symmetric spaces and the moduli spaces Mg;n . This Jacobian map establishes a direct connection between them. We briefly recall its definition. Let †g be a compact Riemann surface of genus g, and let Ai ; Bi , i D 1; : : : ; g, be a symplectic basis of H1 .†g ; Z/, i.e., a basis satisfying the conditions: for i; j D 1; : : : ; g, Ai Aj D 0;

Bi Bj D 0;

Ai Bj D ıij :

Associated to this basis is a normalized basis f!1 ; : : : ; !g g of the complexR vector space H 0 .†g ; 1 / of holomorphic 1-forms on †g satisfying the condition Ai !j D ıij . The corresponding period matrix … D .…ij / of †g is the complex g g matrix with entries defined by Z …ij D

Bi

!j :

Riemann’s bilinear relations [154], p. 232, imply that … D .…ij / belongs to the Siegel upper half space hg . The choice of a different homology basis Ai ; Bi of H1 .†g ; Z/ yields a new period …0 D … for some 2 Sp.2g; Z/. We thus have the well-defined period map J W Mg ! Ag D Sp.2g; Z/nhg which associates to (the isomorphism class of) a Riemann surface †g the Sp.2g; Z/orbit through the period … above. To explain the name of Jacobian map, we note that L D Zg ˚ …Zg is a lattice in g C and the Jacobian variety J.†g / of the Riemann surface †g is the torus LnC g , which turns out to be an abelian variety, i.e., it admits the structure of a projective variety. Moreover, the intersection pairing on homology H1 .†g ; Z/ determines a Hermitian bilinear form on C g with respect to which the torus C g =L is principally polarized [154], p. 359. Similarly, different choices of symplectic bases give rise to an isomorphism class of abelian varieties Zg ˚ …Zg , i.e., its Jacobian variety J.†g /. This gives the Jacobian map J W Mg ! Ag : Intrinsically, without using the period …, the Jacobian variety J.†g / is equal to H1 .†g ; Z/n.H 0 .†g ; 1 // , and the inclusion of H1 .†g ; Z/ in the dual space .H 0 .†g ; 1 // is obtained by integrating 1-forms along cycles in H1 .†g ; Z/ [154], p. 307. Remark 6.1. For another way to define the Jacobian map and an application of the Jacobian map to construct 2-forms on the moduli space Mg , see [226]. By Torelli’s Theorem (see [154], p. 359), the Jacobian map J is injective. When g D 1, Mg D Ag , and J is an isomorphism. For g 2, dimC Mg D 3g 3


271

and dimC hg D .gC1/g . It can be shown that when g D 2; 3, the image J.Mg / is 2 a Zariski dense subvariety of Ag , and when g 4, J.Mg / is a lower dimensional subvariety of Ag . The classical Schottky problem is to characterize the Jacobian locus J.Mg / inside the moduli space Ag of all principally polarized abelian varieties. A lot of work has been devoted on this important problem since 1882 or even earlier. Basically there are two approaches: (1) the analytic one is to find polynomials that “cut out” the locus J.Mg / inside Ag ; (2) the geometric approach is to find geometric properties of principally polarized abelian varieties that are satisfied only by Jacobians. It was finally proved in [388], [389] that a Jacobian variety is characterized by the condition that its Riemann theta function satisfies a nonlinear partial differential equation. See [325] for a history of the Schottky problem and [28] and [106] for more recent surveys of the status of the Schottky problem.

6.2 The coarse Schottky problem It is difficult to check whether a given abelian variety is a Jacobian variety using the characterization in [388], [389]. In order to construct explicit examples of abelian varieties that are not Jacobian varieties, Buser and Sarnak [91] studied the position of the Jacobian locus J.Mg / in Ag for large genera g from the point of view of differential geometry. We note that hg has an invariant metric as a Riemannian symmetric space, and this metric induces a metric on Ag . Buser and Sarnak consider a certain systolic function m W Ag ! R, i.e., the length of shortest geodesics of the abelian variety with a suitable normalized flat metric, which can be thought of as giving a “distance” to the boundary of Ag . Then they prove that J.Mg / Vg WD fx 2 Ag j m.x/

3

log.4g C 3/g:

Moreover, as g ! C1, Vol.Vg /=Vol.Ag / D O.g g / for any < 1. The volumes are computed with respect to the volume form on Ag induced from the invariant metric. This means that for large genus g the entire Jacobian locus lies in a “very small” neighborhood Vg of the boundary of Ag . Motivated by this work of Buser and Sarnak, B. Farb proposed in [123], Problem 4.11, to study the Schottky problem from the point of view of large-scale geometry, called the “Coarse Schottky Problem”: How does J.Mg / look “from far away”, or how “dense” is J.Mg / inside Ag in the sense of coarse geometry? This question can be made precise by using the concept of an asymptotic cone (or tangent cone at infinity) introduced by Gromov. Recall that a sequence .Xn ; pn ; dn / of unbounded, pointed metric spaces converges in the sense of Gromov–Hausdorff to a pointed metric space .X; p; d / if for every r > 0; the Hausdorff distance between the balls Br .pn / in .Xn ; dn / and the ball Br .p/ in .X; d / goes to zero as n ! 1.

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Let x0 be an (arbitrary) point of Ag . The asymptotic cone of Ag endowed with the locally symmetric metric dAg is defined as the Gromov–Hausdorff limit of rescaled pointed spaces: Cone1 .Ag / WD GH -limn!1 .Ag ; x0 ; n1 dAg /: Note that Cone1 .Ag / is independent of the choice of the base point x0 . For some spaces asymptotic cones are easy to describe. For example, the asymptotic cone of the Euclidean space Rn is isometric to Rn . Similarly, if C is a cone in Rn , then Cone1 .C / is isometric to C . The asymptotic cone of the Poincaré half plane H2 is more complicated and turns out to be an R-tree, i.e., a tree which branches everywhere. (Note that a usual simplicial tree branches at points that do not have any accumulation points.) For a hyperbolic Riemann surface †g;n with n > 0, its Cone1 .†g;n / is a “cone” over n points, i.e., n rays with a common origin. For Siegel’s modular variety Ag with respect to the metric induced from the invariant metric of hg , Cone1 .Ag / is known to be isometric to a g-dimensional metric cone over a simplex, which is equal to Sp.2g; Z/nQ .Sp.2g; C// (see [215] for example). Farb’s question can now be stated as follows ([123], Problem 4.11): Coarse Schottky problem: Describe, as a subset of a g-dimensional Euclidean cone, the subset of Cone1 .Ag / determined by the Jacobian locus Jg .Mg / in Ag . One of the results of [212] gives a solution to the coarse Schottky problem. It asserts that the locus J.Mg / is coarsely dense. Theorem 6.2. Let Cone1 .Ag / be the asymptotic cone of Siegel’s modular variety. Then the subset of Cone1 .Ag / determined by the Jacobian locus J.Mg / Ag is equal to the entire Cone1 .Ag /. More precisely, J.Mg / is coarsely dense in Ag , i.e., there exists a constant ıg depending only on g such that Ag is contained in a ıg -neighbourhood of J.Mg /. The basic idea of the proof is to degenerate a general compact Riemann surface †g to a stable Riemann surface such that each of its component is of genus 1, and then apply the fact that the Jacobian map J is an isomorphism when g D 1. It might be worthwhile to emphasize that in the theorem of [91] mentioned, the genus g ! C1, while g is fixed here and hence there is no contradiction between these two seemingly opposite conclusions. The result of [91] implies that the constant ıg in the above theorem goes to infinity as g ! C1. A natural problem is to estimate how fast ıg goes to infinity. Remark 6.3. The Jacobian map J W Mg ! Ag has played an important role in the study of Mg . For example, it was used in [242] to show that the moduli space of stable Riemann surfaces, the natural compactification of Mg which is equal to the later Deligne–Mumford compactification, is a projective variety. For some related results on maps between locally symmetric spaces and Mg , see [163].


273

References

[1] [2] [3] [4] [5] [6] [7]

[8]

[9] [10] [11] [12] [13] [14]

[15] [16] [17] [18] [19] [20]

W. Abikoff, Degenerating families of Riemann surfaces. Ann. of Math. (2) 105 (1977), no. 1, 29–44. 184, 250, 264, 265, 266 W. Abikoff, On boundaries of Teichmüller spaces and on Kleinian groups. III. Acta Math. 134 (1975), 211–237. 262 W. Abikoff, The real analytic theory of Teichmüller space. Lecture Notes in Math. 820, Springer-Verlag, Berlin 1980. 182, 187, 256, 257 S. Adyan and V. Durnev, Algorithmic problems for groups and semigroups. Russian Math. Surveys 55 (2000), no. 2, 207–296. 168 P. Albuquerque, Patterson–Sullivan theory in higher rank symmetric spaces. Geom. Funct. Anal. 9 (1999), 1–28. 207 R. Alperin, Locally compact groups acting on trees and property T . Monatsh. Math. 93 (1982), no. 4, 261–265. 172 J. Amorós, M. Burger, K. Corlette, D. Kotschick, and D. Toledo, Fundamental groups of compact Kähler manifolds. Math. Surveys Monogr. 44, Amer. Math. Soc., Providence, R.I., 1996. 196 A. Ancona, Théorie du potentiel sur les graphes et les variétés. In École d’été de Probabilités de Saint-Flour XVIII–1988, Lecture Notes in Math. 1427, Springer-Verlag, Berlin 1990, 1–112. 193 J. Andersen, Mapping class groups do not have Kazhdan’s property (T). Preprint, arXiv:0706.2184v1 [math.QA]. 181 B. Apanasov, Discrete groups in space and uniformization problems. Math. Appl. (Soviet Ser.) 40, Kluwer Academic Publishers Group, Dordrecht 1991. 173 B. Apanasov, Conformal geometry of discrete groups and manifolds. De Gruyter Exp. Math. 32, Walter de Gruyter, Berlin 2000. 174 J. Arthur, The trace formula for noncompact quotient. In Proceedings of the International Congress of Mathematicians (Warsaw, 1983), Vol. 2, PWN, Warsaw 1984, 849–859. 178 G. Arzhantseva and I. Lysenok, A lower bound on the growth of word hyperbolic groups. J. London Math. Soc. (2) 73 (2006), no. 1, 109–125 193 A. Ash, Deformation retracts with lowest possible dimension of arithmetic quotients of self-adjoint homogeneous cones. Math. Ann. 225 (1977), no. 1, 69–76. 172, 224, 225, 244 A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties. Math. Sci. Press, Brookline, Mass., 1975. 173, 224, 225 J. Athreya, A. Bufetov, A. Eskin, and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space. Preprint, arXiv:math/0610715v5 [math.DS] 183 N. Avni, Arithmetic groups have rational representation growth. Ann. of Math. (2) 174 (2011), no. 2, 1009–1056. 232 P. Bahls, The isomorphism problem in Coxeter groups. Imperial College Press, London 2005. 191 P. Bahls, Automorphisms of Coxeter groups. Trans. Amer. Math. Soc. 358 (2006), no. 4, 1781–1796. W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) 84 (1966), 442–528. 169, 240

274

Lizhen Ji

[21]

W. Ballmann, Nonpositively curved manifolds of higher rank. Ann. of Math. (2) 122 (1985), no. 3, 597–609. 176

[22]

W. Ballmann and P. Eberlein, Fundamental groups of manifolds of nonpositive curvature, J. Differential Geom. 25 (1987), no. 1, 1–22. 169, 192

[23]

W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature. Progr. Math. 61, Birkhäuser, Boston, Mass., 1985. 176

[24]

A. Bartels and W. Lück, The Borel Conjecture for hyperbolic and CAT(0)-groups. Ann. of Math. 175 (2012), 631–689. 193

[25]

A. Bartels, W. Lück, and H. Reich, The K-theoretic Farrell–Jones conjecture for hyperbolic groups. Invent. Math. 172 (2008), no. 1, 29–70. 193

[26]

H. Bass and A. Lubotzky, Automorphisms of groups and of schemes of finite type. Israel J. Math. 44 (1983), 1–22. 187

[27]

G. Baumslag, Automorphism groups of residually finite groups. J. London Math. Soc. 38 (1963), 117–118. 187

[28]

A. Beauville, Le problème de Schottky et la conjecture de Novikov. Séminaire Bourbaki 1986/87, Astérisque 152–153 (1988), 101–112. 271

[29]

A. Beardon, The geometry of discrete groups. Grad. Texts in Math. 91, Springer-Verlag, New York 1983. 173, 212, 213, 214, 217, 230

[30]

J. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space. Geom. Topol. 10 (2006), 1523–1578.

[31]

J. Behrstock, C. Drutu, and L. Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Math. Ann. 344 (2009), no. 3, 543–595. 170

[32]

J. Behrstock, B. Kleiner, Y. Minsky, and L. Mosher, Geometry and rigidity of mapping class groups. Preprint, arXiv:0801.2006v4 [math.GT]. 181, 187

[33]

J. Behrstock andY. Minsky, Dimension and rank for mapping class groups. Ann. of Math. (2) 167 (2008), no. 3, 1055–1077. 180, 181

[34]

M. Bekka, M. Cowling, and P. de la Harpe, Some groups whose reduced C -algebra is simple. Inst. Hautes Études Sci. Publ. Math. 80 (1994), 117–134. 171

[35]

B. Bekka, P., de la Harpe, and A. Valette, Kazhdan’s property (T). New Math. Monogr. 11, Cambridge University Press, Cambridge 2008. 172

[36]

M. Belolipetsky, Counting maximal arithmetic subgroups. With an appendix by Jordan Ellenberg and Akshay Venkatesh. Duke Math. J. 140 (2007), no. 1, 1–33. 232

[37]

R. Benedetti and C. Petronio, Lectures on hyperbolic geometry. Universitext, SpringerVerlag, Berlin 1992. 173

[38]

Y. Benoist, Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7 (1997), no. 1, 1–47. 233

[39]

Y. Benoist, Five lectures on lattices in semisimiple Lie groups. In Géométries à courbure négative ou nulle, groupes discrets et rigidités, Sémin. Congr. 18, Soc. Math. France, Paris 2009, 117–176. 166

[40]

L. Bers, Uniformization, moduli, and Kleinian groups. Bull. London Math. Soc. 4 (1972), 257–300. 248, 266


275

[41]

L. Bers, Finite-dimensional Teichmüller spaces and generalizations. Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 131–172. 266

[42]

L. Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I. Ann. of Math. (2) 91 (1970), 570–600. 166, 250, 262, 263, 264, 266

[43]

G. Besson, G. Courtois, and S. Gallot, Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5 (1995), no. 5, 731– 799. 176

[44]

G. Besson, G. Courtois, and S. Gallot, Volume et entropie minimale des espaces localement symétriques. Invent. Math. 103 (1991), no. 2, 417–445.

[45]

M. Bestvina, The topology of Out.Fn /. In Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. II, Higher Education Press, Beijing 2002, 373–384. 189, 190, 247

[46]

M. Bestvina and N. Brady, Morse theory and finiteness properties of groups. Invent. Math. 129 (1997), no. 3, 445–470. 198

[47]

M. Bestvina, K. Bromberg, and K. Fujiwara, The asymptotic dimension of mapping class groups is finite. Preprint, arXiv:1006.1939v2 [math.GR]. 181

[48]

M. Bestvina, K. Bromberg, K. Fujiwara, and J. Souto, Shearing coordinates and convexity of length functions on Teichmüller space. Preprint, arXiv:0902.0829v1 [math.GT]. 261

[49]

M. Bestvina and M. Feighn, The topology at infinity of Out.Fn /. Invent. Math. 140 (2000), no. 3, 651–692. 188, 190

[50]

M. Bestvina and M. Feighn, Proper actions of lattices on contractible manifolds, Invent. Math. 150 (2002), no. 2, 237–256. 177

[51]

M. Bestvina, M. Feighn, and M. Handel, The Tits alternative for Out.Fn /. I. Dynamics of exponentially-growing automorphisms. Ann. of Math. (2) 151 (2000), no. 2, 517–623. 187

[52]

M. Bestvina and K. Fujiwara, A characterization of higher rank symmetric spaces via bounded cohomology. Geom. Funct. Anal. 19 (2009), no. 1, 11–40. 176

[53]

M. Bestvina and G. Mess, The boundary of negatively curved groups. J. Amer. Math. Soc. 4 (1991), no. 3, 469–481. 193

[54]

J. Birman, Braids, links, and mapping class groups. Ann. of Math. Stud. 82, Princeton University Press, Princeton, N.J., 1974. 160, 179

[55]

J. Birman, A. Lubotzky, and J. McCarthy, Abelian and solvable subgroups of the mapping class groups. Duke Math. J. 50 (1983), no. 4, 1107–1120. 180

[56]

F. Bonahon, Low-dimensional geometry. From Euclidean surfaces to hyperbolic knots. Stud. Math. Libr. 49, IAS/Park City Mathematical Subseries, Amer. Math. Soc., Providence, R.I., Institute for Advanced Study (IAS), Princeton, N.J., 2009. 174

[57]

A. Borel, Introduction aux groupes arithmétiques. Hermann, Paris 1969. 173, 203, 213, 217, 222, 224, 232, 256

[58]

A. Borel, Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963), 111–122. 203

[59]

A. Borel, Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272. 170, 223

276

Lizhen Ji

[60]

A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geometry 6 (1972), 543–560. 177, 223

[61]

A. Borel and L. Ji, Compactifications of symmetric and locally symmetric spaces. Math. Theory Appl., Birkhäuser, Boston, Mass., 2006. 177, 226, 236, 240, 241, 256, 264, 265

[62]

A. Borel and J. P. Serre, Corners and arithmetic groups. Comment. Math. Helv. 48 (1973), 436–491. 177, 240

[63]

K. Bou-Rabee and T. Kaletha, Quantifying residual finiteness of arithmetic groups. Compositio Math., to appear; preprint arXiv:1008.3579v1 [math.GR]. 231

[64]

B. Bowditch, A course on geometric group theory. MSJ Memoirs 16, Math. Soc. Japan, Tokyo 2006.

[65]

B. Bowditch and D. Epstein, Natural triangulations associated to a surface. Topology 27 (1988), 91–117. 256, 258

[66]

M. Bozejko, T. Januszkiewicz, and R. Spatzier, Infinite Coxeter groups do not have Kazhdan’s property. J. Operator Theory 19 (1988), no. 1, 63–67. 191

[67]

T. Brendle and B. Farb, Every mapping class group is generated by 6 involutions. J. Algebra 278 (2004), no. 1, 187–198. 179

[68]

M. Bridson, Non-positive curvature and complexity for finitely presented groups. Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. II, Eur. Math. Soc., Zurich 2006, 961–987. 192

[69]

M. Bridson, Geometric and combinatorial group theory. In The Princeton companion to mathematics (T. Gower, ed.), Princeton University Press, Princeton, N.J., 2008, 431–448. 192

[70]

M. Bridson, Semisimple actions of mapping class groups on CAT(0) spaces. In Geometry of Riemann surfaces, London Math. Soc. Lecture Note Ser. 368, Cambridge University Press, Cambridge 2010, 1–14.

[71]

M. Bridson andA. Haefliger, Metric spaces of non-positive curvature. Grundlehren Math. Wiss. 319, Springer-Verlag, Berlin 1999. 168, 191, 192, 195

[72]

M. Bridson and P. de la Harpe, Mapping class groups and outer automorphism groups of free groups are C -simple. J. Funct. Anal. 212 (2004), 195–205. 182, 188, 193

[73]

M. Bridson and K. Vogtmann, On the geometry of the automorphism group of a free group. Bull. London Math. Soc. 27 (1995), no. 6, 544–552. 160, 187

[74]

M. Bridson and K. Vogtmann, Automorphisms of automorphism groups of free groups. J. Algebra 229 (2000), no. 2, 785–792. 188

[75]

M. Bridson and K. Vogtmann, Automorphism groups of free groups, surface groups and free abelian groups. In Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, R.I., 2006, 301–316. 166

[76]

J. Brock, The Weil–Petersson visual sphere. Geom. Dedicata 115 (2005), 1–18. 184

[77]

J. Brock and B. Farb, Curvature and rank of Teichmüller space. Amer. J. Math. 128 (2006), no. 1, 1–22. 192

[78]

H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, and H. Zassenhaus, Crystallographic groups of four-dimensional space. Wiley Monogr. Crystallography, WileyInterscience, New York 1978. 173


277

[79]

K. Brown, Cohomology of groups. Corrected reprint of the 1982 original, Grad. Texts in Math. 87, Springer-Verlag, New York 1994. 170, 242

[80]

K. Brown, Finiteness properties of groups. J. Pure Appl. Algebra 44 (1987), no. 1–3, 45–75. 198

[81]

K. Brown and R. Geoghegan, An infinite-dimensional torsion-free FP1 group. Invent. Math. 77 (1984), no. 2, 367–381. 198

[82]

G. Brumfiel, The real spectrum compactification of Teichmüller space. In Geometry of group representations, Contemp. Math. 74, Amer. Math. Soc., Providence, R.I., (1988), 51–75. 166, 184, 185, 250

[83]

M. Bucher-Karlsson, Simplicial volume of locally symmetric spaces covered by SL3 R=SO.3/. Geom. Dedicata 125 (2007), 203–224. 175

[84]

M. Bucher-Karlsson, The simplicial volume of closed manifolds covered by H2 H2 . J. Topol. 1 (2008), no. 3, 584–602. 175

[85]

I. Bumagina, The Hopf property for subgroups of hyperbolic groups. Geom. Dedicata 106 (2004), 211–230. 193

[86]

U. Bunke and M. Olbrich, Scattering theory for geometrically finite groups. In Geometry, analysis and topology of discrete groups, Adv. Lect. Math. (ALM) 6, International Press, Somerville, Mass., 2008, 40–136. 207

[87]

M. Burger, A. Iozzi, Alessandra, and A. Wienhard, Surface group representations with maximal Toledo invariant. C. R. Math. Acad. Sci. Paris 336 (2003), no. 5, 387–390. 207

[88]

M. Burger and N. Monod, Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. 1 (1999), 199–235. 176

[89]

K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings. Inst. Hautes Études Sci. Publ. Math. 65 (1987), 35–59. 176

[90]

P. Buser, Geometry and spectra of compact Riemann surfaces. Progr. Math., 106, Birkhäuser, Boston, Mass., 1992. 255, 256, 257

[91]

P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus (with an appendix by J. H. Conway and N. J. A. Sloane). Invent. Math. 117 (1994), 27–56. 271, 272

[92]

J. Cassels, An introduction to the geometry of numbers. Corrected reprint of the 1971 edition, Classics Math., Springer-Verlag, Berlin 1997. 176

[93]

J. Cassels, Rational quadratic forms. London Math. Soc. Monogr. 13, Academic Press, New York 1978.

[94]

R. Charney, J. Crisp, and K. Vogtmann, Automorphisms of 2-dimensional right-angled Artin groups. Geom. Topol. 11 (2007), 2227–2264. 198

[95]

I. Chatterji and K. Ruane, Some geometric groups with rapid decay. Geom. Funct. Anal. 15 (2005), no. 2, 311–339. 171

[96]

H. Cohn, On the shape of the fundamental domain of the Hilbert modular group. Proc. Sympos. Pure Math. VIII, Amer. Math. Soc., Providence, R.I., 1965, 190–202. 228

[97]

H. Cohn, A numerical survey of the floors of various Hilbert fundamental domains. Math. Comp. 19 (1965), 594–605. 228

278

Lizhen Ji

[98]

M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96 (1989), 507–549. 171, 193

[99]

H. Coxeter and W. Moser, Generators and relations for discrete groups. Fourth edition, Ergeb. Math. Grenzgeb. 14, Springer-Verlag, Berlin 1980. 198

[100] M. Culler and J. Morgan, Group actions on R-trees. Proc. London Math. Soc. (3) 55 (1987), no. 3, 571–604. 190 [101] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups. Invent. Math. 84 (1986), no. 1, 91–119. 188, 189, 190 [102] M. Culler and K. Vogtmann, A group-theoretic criterion for Property FA. Proc. Amer. Math. Soc. 124 (1996) 677–683. 181, 188 [103] G. Daskalopoulos and R. Wentworth, Harmonic maps and Teichmüller theory. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, EMS Publishing House, Zurich 2007, 33–109. [104] M. Davis, The geometry and topology of Coxeter groups. London Math. Soc. Monogr. 32, Princeton University Press, Princeton, N.J., 2008. 190, 191, 198 [105] P. de la Harpe, Topics in geometric group theory. Chicago Lect. Math., University of Chicago Press, Chicago, Ill., 2000. 181, 188, 234 [106] O. Debarre, The Schottky problem: an update. In Current topics in complex algebraic geometry, Math. Sci. Res. Inst. Publ. 28, Cambridge University Press, Cambridge 1995, 57–64. 271 [107] P. Deligne and G. Mostow, Monodromy of hypergeometric functions and non- lattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986) 5–90. 196 [108] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. 186 [109] T. Delzant, Sous-groupes distingués et quotients de groupes hyperboliques. Duke Math. J. 83 (1996), no. 3, 661–682. 169, 232 [110] A. Dranishnikov and T. Januszkiewicz, Every Coxeter group acts amenably on a compact space. Topology Proc. 24 (1999), Spring, 135–141. 191 [111] C. Drutu and M. Kapovich, Lectures on geometric group theory, preprint of a book, 2010, http://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf. [112] C. Drutu, S. Mozes, and M. Sapir, Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362 (2010), no. 5, 2451–2505. 170 [113] M. Duchin and K. Rafi, Divergence of geodesics in Teichmüller space and the mapping class group. Geom. Funct. Anal. 19 (2009), no. 3, 722–742. 183 [114] J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1975), no. 1, 39–79. 178 [115] J. Dymara and T. Januszkiewicz, Cohomology of buildings and their automorphism groups. Invent. Math. 150 (2002), no. 3, 579–627. 172 [116] P. Ehrlich and H. Im Hof, Dirichlet regions in manifolds without conjugate points. Comment. Math. Helv. 54 (1979), no. 4, 642–658. 214 [117] J. Elstrodt, F. Grunewald, and J. Mennicke, Groups acting on hyperbolic space. Springer Monogr. Math., Springer-Verlag, Berlin 1998. 173


279

[118] M. Ershov and A. Jaikin-Zapirain, Property (T) for noncommutative universal lattices, Invent. Math. 179 (2010), no. 2, 303–347. 172 [119] A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space. J. Mod. Dyn. 5 (2011), no. 1, 71–105. 182 [120] E. Falbel and J. Parker, The geometry of the Eisenstein-Picard modular group. Duke Math. J. 131 (2006), no. 2, 249–289. 226 [121] B. Farb, The quasi-isometry classification of lattices in semisimple Lie groups. Math. Res. Lett. 4 (1997), no. 5, 705–717. 171 [122] B. Farb, Some problems on mapping class groups and moduli space. In Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, R.I., 2006, 11–55. 179, 180 [123] B. Farb (ed.), Problems on mapping class groups and related topics. Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, R.I., 2006. 160, 166, 180, 181, 187, 245, 271, 272 [124] B. Farb and M. Handel, Commensurations of Out.Fn /. Inst. Hautes Études Sci. Publ. Math. 105 (2007), 1–48. 188 [125] B. Farb, A. Lubotzky, andY. Minsky, Rank-1 phenomena for mapping class groups. Duke Math. J. 106 (2001), no. 3, 581–597. 179, 182, 187, 192 [126] B. Farb and D. Magalit, A primer on mapping class groups. Princeton Math. Ser. 49, Princeton University Press, Princeton, N.J., 2012. 179, 197, 245 [127] B. Farb and H. Masur, Teichmüller geometry of moduli space, I: Distance minimizing rays and the Deligne–Mumford compactification. J. Differential Geom. 85 (2010), no. 2, 187–227. 186 [128] B. Farb and H. Masur, Teichmüller geometry of moduli space, II: M.S/ seen from far away. In In the tradition of Ahlfors-Bers, V (M. Bonk, J. Gilman, H. Masur, Y. Minsky, and M. Wolf, eds.), Contemp. Math. 510, Amer. Math. Soc., Providence, R.I., 2010, 71–79. 185 [129] B. Farb and H. Masur, Geometry of the complex of curves. I. Hyperbolicity. Invent. Math. 138 (1999), no. 1, 103–149. 186 [130] B. Farb and S. Weinberger, The intrinsic asymmetry and inhomogeneity of Teichmüller space. Duke Math. J. 155 (2010), no. 1, 91–103. 183 [131] F. Farrell, Lectures on surgical methods in rigidity. Tata Institute of Fundamental Research, Springer-Verlag, Berlin 1996. 165 [132] F. Farrell and L. Jones, Rigidity for aspherical manifolds with 1 GLm .R/. Asian J. Math. 2 (1998), no. 2, 215–262. 171, 191 [133] F. Farrell and P. Ontaneda, The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible. Ann. of Math. (2) 170 (2009), no. 1, 45–65. 247 [134] W. Fenchel and J. Nielsen, Discontinuous groups of isometries in the hyperbolic plane. Edited and with a preface by Asmus L. Schmidt, biography of the authors by Bent Fuglede, de Gruyter Stud. Math. 29, Walter de Gruyter, Berlin 2003. 174 [135] S. Ferry, A. Ranicki, and J. Rosenberg, Novikov conjectures, index theorems and rigidity. Vol. 1. London Math. Soc. Lecture Note Ser. 226, Cambridge University Press, Cam-

280

Lizhen Ji bridge, 1995. 165, 182

[136] E. Formanek and C. Procesi, The automorphism group of a free group is not linear. J. Algebra (2) 149 (1992), 494–499. 187 [137] G. Francsics and P. Lax, Analysis of a Picard modular group. Proc. Natl. Acad. Sci. USA 103 (2006), no. 30, 11103–11105. 226 [138] G. Francsics and P. Lax, A semi-explicit fundamental domain for a Picard modular group in complex hyperbolic space. In Geometric analysis of PDE and several complex variables, Contemp. Math. 368, Amer. Math. Soc., Providence, R.I., 2005, 211–226. 226 [139] L. Funar, C. Kapoudjian, and V. Sergiescu, Asymptotically rigid mapping class groups and Thompson’s groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2012, 595–664. 198 [140] R. Fricke and F. Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band I: Die gruppentheoretischen Grundlagen, Band II: Die funktionentheoretischen Ausführungen und die Anwendungen, Bibliotheca Mathematica Teubneriana, Bände 3, 4 Johnson Reprint Corp., New York; B. G. Teubner Verlagsgesellschaft, Stuttgart 1965. 172 [141] K. Fujiwara, On non-bounded generation of discrete subgroups in rank-1 Lie group. In Geometry, spectral theory, groups, and dynamics, Contemp. Math. 387, Amer. Math. Soc., Providence, R.I., 2005, 153–156. 230 [142] K. Fujiwara, Quasi-homomorphisms on mapping class groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. II, EMS Publishing House, Zurich 2009, 241–269. [143] S. Galatius, Stable homology of automorphism groups of free groups. Ann. of Math. (2) 173 (2011), no. 2, 705–768. 188 [144] F. Gardiner and H. Masur, Extremal length geometry of Teichmüller space. Complex Variables Theory Appl. 16 (1991), no. 2–3, 209–237. 184, 249 [145] H. Garland and M. Raghunathan, Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups. Ann. of Math. (2) 92 (1970), 279–326. 205 [146] I. M. Gel’fand, M. Graev, and I. Pyatetskii-Shapiro, Representation theory and automorphic functions. Generalized Funct.6, Academic Press, Boston, Mass., 1990. 173, 195 [147] R. Geoghegan, Topological methods in group theory, Grad. Texts in Math. 243, SpringerVerlag, New York 2008. 198 [148] W. Goldman, Complex hyperbolic geometry. Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York 1999. 174, 225 [149] W. Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. II, EMS Publishing House, Zurich 2009, 611–684. [150] A. Gorodnik and H. Oh, Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary. Duke Math. J. 139 (2007), no. 3, 483–525. 175 [151] D. Griffiths, The side-pairing elements of Maskit’s fundamental domain for the modular group in genus two. Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 1, 3–50. 256 [152] D. Griffiths, At most 27 length inequalities define Maskit’s fundamental domain for the modular group in genus 2. In The Epstein birthday schrift, Geom. Topol. Monogr. 1,


281

Geometry & Topology Publications, Coventry 1998, 167–180. 256 [153] D. Griffiths, Length inequalities for systems of geodesic loops on a surface of genus two. I. Bull. London Math. Soc. 28 (1996), no. 5, 505–508. [154] P. Griffiths and J. Harris, Principles of algebraic geometry. Wiley Classics Lib., John Wiley & Sons, New York 1994. 270 [155] M. Gromov, Hyperbolic groups. In Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer-Verlag, New York 1987, 75–263. 192, 195 [156] M. Gromov, Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), 1–295, London Math. Soc. Lecture Note Ser. 182, Cambridge University Press, Cambridge 1993. 195 [157] M. Gromov and I. Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces. Inst. Hautes Études Sci. Publ. Math. 66 (1988), 93–103. 204 [158] P. Gruber and C. Lekkerkerker, Geometry of numbers. Second edition, North-Holland Math. Library 37, North-Holland Publishing Co., Amsterdam 1987. 220 [159] F. Grunewald and D. Segal, The solubility of certain decision problems in arithmetic and algebra. Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 915–918. 168 [160] F. Grunewald and D. Segal, Some general algorithms. I. Arithmetic groups. Ann. of Math. (2) 112 (1980), no. 3, 531–583. 168 [161] F. Grunewald and D. Segal, Some general algorithms. II. Nilpotent groups. Ann. of Math. (2) 112 (1980), no. 3, 585–617. 168 [162] Y. Guivarc’h, L. Ji, and J. C. Taylor, Compactifications of symmetric spaces. Progr. Math. 156, Birkhäuser, Boston, Mass., 1998. 236 [163] R. Hain, Locally symmetric families of curves and Jacobians. In Moduli of curves and abelian varieties, Aspects Math., E33, Vieweg, Braunschweig 1999, 91–108. 272 [164] R. Hain and E. Looijenga, Mapping class groups and moduli spaces of curves. In Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, R.I., 1997 97–142,. 160 [165] U. Hamenstädt, Geometry of the mapping class groups. I. Boundary amenability. Invent. Math. 175 (2009), no. 3, 545–609. 185, 250 [166] U. Hamenstädt, Geometry of the mapping class groups III: Quasi-isometric rigidity. Preprint, arXiv:math/0512429v2 [math.GT] 181, 187, 250 [167] U. Hamenstädt, Geometric properties of the mapping class group. In Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, R.I., 2006, 215–232. 179 [168] G. Harder, A Gauss–Bonnet formula for discrete arithmetically defined groups. Ann. Sci. École Norm. Sup. (4) 4 (1971), 409–455. 170, 176 [169] J. Harer, The cohomology of the moduli space of curves. In Theory of moduli, Lecture Notes in Math. 1337, Springer-Verlag, Berlin 1988, 138–221. 160, 166, 179, 180, 181, 185, 245, 252, 254, 256, 258 [170] J. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84 (1986), 157–176. 256

282

Lizhen Ji

[171] J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves. Invent. Math. 85 (1986), no. 3, 457–485. 180, 258 [172] W. Harvey, Boundary structure of the modular group. In Riemann surfaces and related topics, Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Stud. 97, Princeton University Press, Princeton, N.J., 1981, 245–251. 253 [173] W. Harvey, Geometric structure of surface mapping class groups. In Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser. 36, Cambridge University Press, Cambridge 1979, 255–269. [174] W. Harvey, Teichmüller spaces, triangle groups and Grothendieck dessins. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, EMS Publishing House, Zurich 2007, 249–292. 204 [175] A. Hatcher and K. Vogtmann, Homology stability for outer automorphism groups of free groups. Algebr. Geom. Topol. 4 (2004), 1253–1272. 188 [176] A. Hatcher and K. Vogtmann, The complex of free factors of a free group. Quart. J. Math. Oxford Ser. (2) 49 (1998), no. 196, 459–468. 188 [177] T. Hattori, Asymptotic geometry of arithmetic quotients of symmetric spaces. Math. Z. 222 (1996), no. 2, 247–277. 177 [178] D. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups. Mem. Amer. Math. Soc. 97 (1992), no. 469. 204, 205 [179] G. Hemion, On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds. Acta Math. 142 (1979), no. 1-2, 123–155. 179 [180] F. Herrlich and G. Schmithüsen, On the boundary of Teichmüller disks in Teichmüller and in Schottky space. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, EMS Publishing House, Zurich 2007, 293–349. [181] R. Holzapfel, Geometry and arithmetic around Euler partial differential equations. Math. Appl. (East European Ser.) 11, D. Reidel Publishing Co., Dordrecht 1986. 196 [182] R. Holzapfel, The ball and some Hilbert problems. Lectures Math. ETH Zurich, Birkhäuser, Basel 1995. 196 [183] P. Humbert, Théorie de la réduction des formes quadratiques définies positives dans un corps algébrique K fini. Comment. Math. Helv. 12 (1940), 263–306. 225 [184] J. Humphreys, Arithmetic groups. Lecture Notes in Math. 789, Springer-Verlag, Berlin 1980. 173 [185] B. Hunt, The geometry of some special arithmetic quotients. Lecture Notes in Math. 1637, Springer-Verlag, Berlin 1996. 246 [186] Y. Imayoshi, A construction of holomorphic families of Riemann surfaces over the punctured disk with given monodromy, in Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. II, EMS Publishing House, Zurich 2009, 93–130. 186 [187] Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces. Springer-Verlag, Tokyo 1992. 182 [188] N. Ivanov, Mapping class groups. In Handbook of geometric topology, North-Holland, Amsterdam 2002, 523–633. 160, 166, 179, 180, 181, 182, 183, 186, 187, 197, 245, 251, 253, 254 [189] N. Ivanov, Automorphism of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices 14 (1997), 651–666.


283

[190] N. Ivanov, Subgroups of Teichmüller modular groups. Transl. Math. Monogr. 115, Amer. Math. Soc., Providence, R.I., 1992. 160, 180 [191] N. Ivanov, Fifteen problems about the mapping class groups. In Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, R.I., 2006, 71–80,. 180, 181 [192] N. Ivanov and L. Ji, Infinite topology of curve complexes and non-Poincaré duality of Teichmüller modular groups. Enseign. Math. (2) 54 (2008), no. 3–4, 381–395. 180, 252, 254 [193] N. Ivanov and J. McCarthy, On injective homomorphisms between Teichmüller modular groups. I. Invent. Math. 135 (1999), no. 2, 425–486. 181 [194] B. Iversen, Hyperbolic geometry. London Math. Soc. Student Texts 25, Cambridge University Press, Cambridge 1992. 173 [195] L. Ji, Mostow strong rigidity and nonisomorphism for outer automorphism groups of free groups and mapping class groups. In New developments in Lie theory and geometry, Contemp. Math. 491, Amer. Math. Soc., Providence, R.I., 2009, 57–81. 169, 179, 181, 187, 188 [196] L. Ji, Steinberg representations and duality properties of arithmetic groups, mapping class groups, and outer automorphism groups of free groups, in Representation theory, pp. 71–98, Contemp. Math. 478, Amer. Math. Soc., Providence, R.I., 2009. 188 [197] L. Ji, Arithmetic groups and their generalizations. What, why, and how. AMS/IP Stud. Adv. Math. 43, Amer. Math. Soc., Providence, R.I.; International Press, Cambridge, Mass., 2008. 168, 169, 170, 171, 174, 203, 231 [198] L. Ji, A summary of the work of Gregory Margulis. Pure Appl. Math. Q. 4 (2008), no. 1 (Special Issue: In honor of Grigory Margulis), Part 2, 1–69. 166, 176, 204, 267 [199] L. Ji, The integral Novikov conjectures for S-arithmetic groups. I. K-Theory 38 (2007), no. 1, 35–47. 206 [200] L. Ji, Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups. J. Differential Geom. 68 (2004), no. 3, 535–544. 170, 171, 181 [201] L. Ji, The integral Novikov conjectures for linear groups containing torsion elements. J. Topol. 1 (2008), no. 2, 306–316. 181 [202] L. Ji, Fundamental domains for mapping class groups and equivariant cell decomposition for Teichmüller spaces via Minkowski reduction. Preprint, 2009. 259, 261 [203] L. Ji, Well-rounded equivariant deformation retractions of Teichmüller spaces. Preprint, 2011. 185 [204] L. Ji, From symmetric spaces to buildings, curve complexes and outer spaces. Innov. Incidence Geom. 10 (2009), 33–80. 188, 190, 224, 235 [205] L. Ji, Arithmetic groups, mapping class groups, related groups, and their associated spaces. In Fourth International Congress of Chinese Mathematicians, AMS/IP Stud. Adv. Math. 48, Amer. Math. Soc., Providence, R.I., 2010, 127–175. 176, 266 [206] L. Ji, Integral Novikov conjectures and arithmetic groups containing torsion elements. Comm. Anal. Geom. 15 (2007), no. 3, 509–533. 177, 243 [207] L. Ji, Lectures on locally symmetric spaces and arithmetic groups. In Lie groups and automorphic forms, AMS/IP Stud. Adv. Math. 37, Amer. Math. Soc., Providence, R.I., 2006, 87–146. 246

284

Lizhen Ji

[208] L. Ji, Buildings and their applications in geometry and topology. Asian J. Math. 10 (2006), no. 1, 11–80. 236 [209] L. Ji, Simplicial volume of moduli spaces of Riemann surfaces. J. Differential Geom., to appear. 183 [210] L. Ji, Curve complexes versus Tits buildings: structures and applications. Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zurich, to appear. 183, 186, 236, 254 [211] L. Ji, V. Kumar Murty, L. Saper, and J. Scherk, The congruence subgroup kernel and the fundamental group of the reductive Borel-Serre compactification. Preprint, 2010. 241 [212] L. Ji and E. Leuzinger, The asymptotic Schottky problem. Geom. Funct. Anal. 19 (2010), 1693–1712. 272 [213] L. Ji and Z. Lin, Finite generation properties of cohomology rings for infinite groups. J. Pure Appl. Algebra 216, no. 5, 1118–1133. 169, 180, 188, 191, 192 [214] L. Ji, K. Liu, X. Sun, and S. T. Yau, in preparation. 186 [215] L. Ji and R. MacPherson, Geometry of compactifications of locally symmetric spaces. Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 457–559. 177, 267, 272 [216] L. Ji and S. Wolpert, A cofinite universal space for proper actions for mappingclass groups. In In the Tradition of Ahlfors-Bers, V (M. Bonk, J. Gilman, H. Masur, Y. Minsky, and M. Wolf, eds.), Contemp. Math. 510, Amer. Math. Soc., Providence, R.I., 2010, 151–163. 180, 181, 185, 254 [217] L. Ji and S. Zucker, On Lp -cohomology and the geometry of metrics of moduli space of curves. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2012, 747–775. 178, 186, 268 [218] L. Ji and M. Zworski, Scattering matrices and scattering geodesics of locally symmetric spaces. Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 3, 441–469. 178 [219] P. Jolissaint, Rapidly decreasing functions in reduced C -algebras of groups. Trans. Amer. Math. Soc. 317 (1990), no. 1, 167–196. 171 [220] I. Kapovich and M. Lustig, Intersection form, laminations and currents on free groups. Geom. Funct. Anal. 19 (2010), no. 5, 1426–1467. 189, 190 [221] I. Kapovich and M. Lustig, Geometric intersection number and analogues of the curve complex for free groups. Geom. Topol. 13 (2009), no. 3, 1805–1833. 188 [222] I. Kapovich and N. Benakli, Boundaries of hyperbolic groups. In Combinatorial and geometric group theory, Contemp. Math. 296, Amer. Math. Soc., Providence, R.I., 2002, 39–93. 193 [223] M. Kapovich, Hyperbolic manifolds and discrete groups. Reprint of the 2001 edition, Mod. Birkhäuser Class., Birkhäuser, Boston, Mass., 2009. 174 [224] M. Kapovich and J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 88 (1998), 5–95 (1999). 196 [225] S. Katok, Fuchsian groups. Chicago Lect. Math., University of Chicago Press, Chicago, Ill., 1992. 173 [226] N. Kawazumi, Canonical 2-forms on the moduli space of Riemann surfaces. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich


285

2009, 217–237. 270 [227] D. Kazdan and G. Margulis, A proof of Selberg’s hypothesis. Mat. Sb. (N.S.) 75 (117) (1968), 163–168. 232 [228] L. Keen, A rough fundamental domain for Teichmüller spaces. Bull. Amer. Math. Soc. 83 (1977), no. 6, 1199–1226. 256, 257, 263 [229] L. Keen, A modular group and Riemann surfaces of genus 2. Math. Z. 142 (1975), 205–219. 256 [230] L. Keen, On fundamental domains and the Teichmüller modular group. In Contributions to analysis, Academic Press, New York 1974, 185–194. 256 [231] L. Keen and N. Lakic, Hyperbolic geometry from a local viewpoint. London Math. Soc. Student Texts 68. Cambridge University Press, Cambridge 2007. 174 [232] R. Kent and C. Leininger, Subgroups of mapping class groups from the geometrical viewpoint. In In the tradition of Ahlfors-Bers, IV (Dick Canary, Jane Gilman, Juha Heinonen, and Howard Masur, eds.), Contemp. Math. 432, Amer. Math. Soc., Providence, R.I., 2007, 119–141. 180 [233] S. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. 180, 248 [234] S. Kerckhoff, The asymptotic geometry of Teichmüller space. Topology 19 (1980), no. 1, 23–41. 183, 184, 250 [235] S. Kerckhoff and W. Thurston, Non-continuity of the action of the modular group at Bers’ boundary of Teichmüller space. Invent. Math. 100 (1990), 25–48. 184, 249, 250 [236] Y. Kida, The mapping class group from the viewpoint of measure equivalence theory. Mem. Amer. Math. Soc. 196 (2008), no. 916. 181, 185, 250 [237] Y. Kida, Introduction to measurable rigidity of mapping class groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 297–367. 181, 250 [238] Y. Kida, Measure equivalence rigidity of the mapping class group. Ann. of Math. (2) 171 (2010), no. 3, 1851–1901. 179 [239] D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999), no. 3, 451–494. 178 [240] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal. 7 (1997), no. 4, 755–782. 166, 210 [241] M. Knopp, On the cuspidal spectrum of the arithmetic Hecke groups, Math. Comp. 61 (1993), no. 203, 269–275. 204, 205 [242] F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand. 39 (1976), no. 1, 19–55. 272 [243] M. Koecher, Beiträge zu einer Reduktionstheorie in Positivitätsbereichen. I. Math. Ann. 141 (1960), 384–432. 225 [244] M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147 (1992), no. 1, 1–23. 258 [245] M. Kontsevich, Formal (non)commutative symplectic geometry. In The Gel’fand Mathematical Seminars, 1990–1992, Birkhäuser, Boston, Mass., 1993, 173–187. 188

286

Lizhen Ji

[246] D. Krammer, The conjugacy problem for Coxeter groups. Groups Geom. Dyn. 3 (2009), 71–171. 191 [247] A. Krieg, Modular forms on half-spaces of quaternions. Lecture Notes in Math. 1143, Springer-Verlag, Berlin 1985. 225 [248] S. Krstić and K. Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups. Comment. Math. Helv. 68 (1993), no. 2, 216–262. 188, 189 [249] S. Krushkal, B. Apanasov, and N. Gusevskii, Kleinian groups and uniformization in examples and problems. Transl. Math. Monogr. 62, Amer. Math. Soc., Providence, R.I., 1986. 173 [250] F. Labourie, Cross ratios, Anosov representations and the energy functional on Teichmüller space. Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 3, 437–469. 207 [251] J. Lafont and B. Schmidt, Simplicial volume of closed locally symmetric spaces of non-compact type. Acta Math. 197 (2006), no. 1, 129–143. 175 [252] R. Langlands, On the functional equations satisfied by Eisenstein series. Lecture Notes in Math. 544, Springer-Verlag, Berlin 1976. 178 [253] M. Larsen and A. Lubotzky, Representation growth of linear groups. J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 351–390. 232 [254] B. Leeb, A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry. Bonner Mathematische Schriften 326, Universität Bonn, Mathematisches Institut, Bonn 2000. 176 [255] J. Lehner, Discontinuous groups and automorphic functions. Math. Surveys VIII, Amer. Math. Soc., Providence, R.I., 1964. 172 [256] E. Leuzinger, Tits geometry, arithmetic groups, and the proof of a conjecture of Siegel. J. Lie Theory 14 (2004), 317–338. 177 [257] E. Leuzinger, Reduction theory for mapping class groups and applications to moduli spaces. J. Reine Angew. Math. 649 (2010), 11–31. 185, 256, 258 [258] G. Levitt, Automorphisms of hyperbolic groups and graphs of groups. Geom. Dedicata 114 (2005), 49–70. [259] G. Link, Ergodicity of generalised Patterson–Sullivan measures in higher rank symmetric spaces. Math. Z. 254 (2006), no. 3, 611–625. 207 [260] K. Liu, X. Sun, and S. T.Yau, Canonical metrics on the moduli space of Riemann surfaces. I. J. Differential Geom. 68 (2004), no. 3, 571–637. 183 [261] K. Liu, X. Sun, and S. T. Yau, New results on the geometry of the moduli space of Riemann surfaces. Sci. China Ser. A 51 (2008), no. 4, 632–651. 183 [262] C. Löh and R. Sauer, Simplicial volume of Hilbert modular varieties. Comment. Math. Helv. 84 (2009), no. 3, 457–470. 175 [263] C. Löh and R. Sauer, Degree theorems and Lipschitz simplicial volume for nonpositively curved manifolds of finite volume. J. Topol. 2 (2009), no. 1, 193–225. 175 [264] E. Looijenga, L2 -cohomology of locally symmetric varieties. Compositio Math. 67 (1988), no. 1, 3–20. 178 [265] A. Lubotzky, Subgroup growth and congruence subgroups. Invent. Math. 119 (1995), no. 2, 267–295. 168, 231


287

[266] A. Lubotzky, Discrete groups, expanding graphs and invariant measures. With an appendix by Jonathan D. Rogawski. Reprint of the 1994 edition, Mod. Birkhäuser Class., Birkhäuser, Basel 2010. 173 [267] A. Lubotzky, S. Mozes, and M. Raghunathan, The word and Riemannian metrics on lattices of semisimple groups. Inst. Hautes Études Sci. Publ. Math. 91 (2000), 5–53. 170, 171, 175 [268] A. Lubotzky and D. Segal, Subgroup growth. Progr. Math. 212, Birkhäuser, Basel 2003. 232 [269] W. Lück, Survey on geometric group theory. Münster J. Math. 1 (2008), 73–108. 193, 200 [270] W. Lück, L2 -invariants: theory and applications to geometry and K-theory. Ergeb. Math. Grenzgeb. (3) 44, Springer-Verlag, Berlin 2002. 163 [271] F. Luo, Automorphisms of the complex of curves. Topology 39 (2000), no. 2, 283–298. 183, 251 [272] R. Lyndon and P. Schupp, Combinatorial group theory. Reprint of the 1977 edition, Classics Math., Springer-Verlag, Berlin 2001. 187 [273] H. Maass, Siegel’s modular forms and Dirichlet series. Dedicated to the last great representative of a passing epoch, Carl Ludwig Siegel on the occasion of his seventy-fifth birthday, Lecture Notes in Math. 216. Springer-Verlag, Berlin 1971. 225 [274] C. Maclachlan and A. Reid, The arithmetic of hyperbolic 3-manifolds. Grad. Texts in Math. 219, Springer-Verlag, New York 2003. 174 [275] I. Madsen, Moduli spaces from a topological viewpoint. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. I, Eur. Math. Soc., Zurich 2007, 385–411. 181, 186, 254 [276] I. Madsen and M. Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math. (2) 165 (2007), no. 3, 843–941. 181, 186, 254 [277] A. Marden, Outer circles. An introduction to hyperbolic 3-manifolds. Cambridge University Press, Cambridge 2007. 174 [278] G. Margulis, Discrete subgroups of semisimple Lie groups. Ergeb. Math. Grenzgeb. (3) 17, Springer-Verlag, Berlin 1991. 169, 171, 172, 173, 175, 204, 232 [279] G. A. Margulis, On the decomposition of discrete subgroups into amalgams. Selecta Math. Soviet. 1 (1981), no. 2, 197–213. 172, 235, 237 [280] G. A. Margulis and G. A. Soifer, Maximal subgroups of infinite index in finitely generated linear groups. J. Algebra 69 (1981), no. 1, 1–23. 169 [281] R. MacPherson, M. McConnell, Explicit reduction theory for Siegel modular threefolds. Invent. Math. 111 (1993), no. 3, 575–625. 244 [282] W. Magnus, Noneuclidean tesselations and their groups. Pure Appl. Math. 61, Academic Press, New York 1974. 173 [283] B. Maskit, Kleinian groups. Grundlehren Math. Wiss. 287, Springer-Verlag, Berlin 1988. 173, 207, 217 [284] B. Maskit, On boundaries of Teiehmüller spaces and on Kleinian groups, II. Ann. of Math. 91 (1970), 607–639. 250

288

Lizhen Ji

[285] B. Maskit, A picture of moduli space. Invent. Math. 126 (1996) 341–390. 256 [286] B. Maskit, On the fundamental domain for the Teichmüller modular group. J. Anal. Math. 75 (1998), 155–172. 256 [287] B. Maskit, Parameters for Fuchsian groups. II. Topological type .1; 1/. Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 2, 265–275. 256 [288] B. Maskit, Parameters for Fuchsian groups. I. Signature .0; 4/. In Holomorphic functions and moduli, Vol. II (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ. 11, Springer-Verlag, New York 1988, 251–265. 256 [289] H. Masur, Logarithmic law for geodesics in moduli space. In Mapping class groups and moduli spaces of Riemann surfaces, Contemp. Math. 150, Amer. Math. Soc., Providence, R.I., 1993, 229–245. 186 [290] H. Masur, Geometry of Teichmüller space with the Teichmüller metric. In Surveys in differential geometry, Vol. XIV, Geometry of Riemann surfaces and their moduli spaces, Surv. Differ. Geom. 14, International Press, Somerville, Mass., 2009, 295–313. 183 [291] H. Masur, Extension of the Weil–Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43 (1976), no. 3, 623–635. 184, 251 [292] H. Masur and M. Wolf, Teichmüller space is not Gromov hyperbolic. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 259–267. 183 [293] H. Masur and M. Wolf, The Weil–Petersson isometry group. Geom. Dedicata 93 (2002), 177–190. 251 [294] K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups. Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York 1998. 174 [295] J. McCarthy, “Tits-alternative” for subgroups of surface mapping class groups. Trans. Amer. Math. Soc. 291 (1985), no. 2, 583–612. 180 [296] J. McCarthy and A. Papadopoulos, The visual sphere of Teichmüller space and a theorem of Masur-Wolf. Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 1, 147–154. 184 [297] J. McCarthy and A. Papadopoulos, Fundamental domains in Teichmüller space. Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 1, 151–166. 256 [298] M. McConnell, Symmetric spaces and arithmetic groups. In Voronoi’s impact on modern science, Book 1, Institute of Mathematics, National Academy of Sciences of Ukraine, Institute of Mathematics, Kyiv 1998, 155–185. 224 [299] C. McMullen, The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. of Math. (2) 151 (2000), no. 1, 327–357. 183, 262, 263, 264 [300] C. McMullen, Cusps are dense. Ann. of Math. (2) 133 (1991), no. 1, 217–247. [301] G. McShane, Simple geodesics and a series constant over Teichmüller space. Invent. Math. 132 (1998), no. 3, 607–632. 163 [302] D. Meintrup and T. Schick, A model for the universal space for proper actions of a hyperbolic group. New York J. Math. 8 (2002), 1–7. 193, 195 [303] J. Milne, Lectures on étale cohomology. http://www.jmilne.org/math/CourseNotes/lec.html 196 [304] I. Mineyev, G. Yu, The Baum–Connes conjecture for hyperbolic groups. Invent. Math. 149 (2002), no. 1, 97–122. 193


289

[305] H. Minkowski, Ausgewählte Arbeiten zur Zahlentheorie und zur Geometrie. Edited and with a foreword and appendices by E. Krätzel and B. Weissbach, Teubner-Arch. Math. 12, BSB B. G. Teubner Verlagsgesellschaft, Leipzig 1989. 218, 220 [306] Y. Minsky, Quasi-projections in Teichmüller space. J. Reine Angew. Math. 473 (1996), 121–136. 183 [307] Y. Minsky, Curve complexes, surfaces and 3-manifolds. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. II, 1001–1033, Eur. Math. Soc., Zurich 2006. 186, 236 [308] M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. of Math. (2) 168 (2008), no. 1, 97–125. [309] M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167 (2007), no. 1, 179–222. [310] M. Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167 (2007), no. 1, 179–222. [311] M. Mirzakhani, Weil–Petersson volumes and intersection theory on the moduli space of curves. J. Amer. Math. Soc. 20 (2007), no. 1, 1–23. [312] H. Miyachi, Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space. Geom. Dedicata 137 (2008), 113–141. 184 [313] E. Moise, Geometric topology in dimensions 2 and 3. Grad. Texts in Math. 47, SpringerVerlag, New York 1977. 179 [314] G. Mondello, Riemann surfaces, ribbon graphs and combinatorial classes. In In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. II, EMS Publishing House, Zurich 2009, 151–215. 258 [315] N. Monod, An invitation to bounded cohomology. In Proceedings of the International Congress of Mathematicians (Madrid, 2006) Vol. II, 1183–1211, Eur. Math. Soc., Zurich 2006. 171 [316] J. Morgan and P. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I. Ann. of Math. (2) 120 (1984), no. 3, 401–476. 184, 250 [317] S. Morita, Introduction to mapping class groups of surfaces and related groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. I, EMS Publishing House, Zurich 2007, 353–386. 160 [318] S. Morita, Structure of the mapping class groups of surfaces: a survey and a prospect. In Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr. 2, Geometry & Topology Publications, Coventry 1999, 349–406. [319] D. Morris, Bounded generation of SL.n; A/ (after D. Carter, G. Keller, and E. Paige). New York J. Math. 13 (2007), 383–421. [320] L. Mosher, Mapping class groups are automatic. Ann. of Math. (2) 142 (1995), no. 2, 303–384. 179 [321] L. Mosher, Geometric survey of subgroups of mapping class groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. I, EMS Publishing House, Zurich 2007, 387–410. 160, 180 [322] G. Mostow, Strong rigidity of locally symmetric spaces. Ann. of Math. Stud. 78, Princeton University Press, Princeton, N.J., 1973. 171, 173, 176

290

Lizhen Ji

[323] G. Moussong and S. Prassidis, Equivariant rigidity theorems. New York J. Math. 10 (2004), 151–167. 191 [324] D. Mumford, A remark on Mahler’s compactness theorem. Proc. Amer. Math. Soc. 28 (1971), 289–294. 185, 253 [325] D. Mumford, Curves and their Jacobians. The University of Michigan Press, Ann Arbor, Mich., 1975. 271 [326] V. K. Murty, Bounded and finite generation of arithmetic groups. In Number theory, CMS Conf. Proc. 15, Amer. Math. Soc., Providence, R.I., 1995, 249–261. 231 [327] S. Nag, The complex analytic theory of Teichmüller spaces. Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, Inc., New York 1988. 182, 248, 249 [328] M. Nathanson, On the fundamental domain of a discrete group. Proc. Amer. Math. Soc. 41 (1973), 629–630. 213 [329] A. Nevo, The spectral theory of amenable actions and invariants of discrete groups. Geom. Dedicata 100 (2003), 187–218. 166, 175 [330] P. Nicholls, The ergodic theory of discrete groups. London Math. Soc. Lecture Note Ser. 143, Cambridge University Press, Cambridge 1989. 207 [331] V. Nikulin and I. Shafarevich, Geometries and groups. Universitext, Springer Ser. in Soviet Math., Springer-Verlag, Berlin 1987. 173 [332] K. Ohshika, Discrete groups. Transl. Math. Monogr. 207, Iwanami Series in Modern Mathematics, Amer. Math. Soc., Providence, R.I., 2002. 174, 195 [333] N. Ozawa, Weak amenability of hyperbolic groups. Groups Geom. Dyn. 2 (2008), no. 2, 271–280. 193 [334] A. Papadopoulos (ed.), Handbook of Teichmüller theory, Vol. I, EMS Publishing House, Zurich 2007. 166, 187, 245 [335] A. Papadopoulos (ed.), Handbook of Teichmüller theory, Vol. II, EMS Publishing House, Zurich 2009. 166, 187, 245 [336] A. Papadopoulos, Introduction to Teichmüller theory, old and new. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. I, EMS Publishing House, Zurich 2007, 1–30. 183 [337] A. Papadopoulos, Metric spaces, convexity and nonpositive curvature. IRMA Lect. Math. Theoret. Phys. 6, EMS Publishing House, Zurich 2005. [338] A. Papadopoulos and G. Théret, On Teichmüller’s metric and Thurston’s asymmetric metric on Teichmüller space. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Vol. I, EMS Publishing House, Zurich 2007, 111–204. 183 [339] L. Paris, Braid groups and Artin groups, Handbook of Teichmüller theory, Vol. II, EMS Publishing House, Zurich 2009, 389–451. 190 [340] L. Paris and D. Rolfsen, Geometric subgroups of mapping class groups. J. Reine Angew. Math. 521 (2000), 47–83. 180 [341] J. Parker, Complex hyperbolic lattices. In Discrete groups and geometric structures, Contemp. Math. 501, Amer. Math. Soc., Providence, R.I., 2009, 1–42. 204, 226 [342] J. Parker, Cone metrics on the sphere and Livné’s lattices. Acta Math. 196 (2006), no. 1, 1–64. 226


291

[343] S. J. Patterson, The limit set of a fuchsian group. Acta mathematica 136 (1976), 241–273. 207 [344] F. Paulin, Sur les automorphismes de groupes libres et de groupes de surface. Version préliminaire, Séminaire Bourbaki 1023 (Juin 2010). 160, 166, 170 [345] R. Penner, The decorated Teichmüller space of puctured surfaces. Comm. Math. Phys. 113 (1987), 299–339. 256, 258 [346] R. Penner, Cell decomposition and compactification of Riemann’s moduli space in decorated Teichmüller theory. In Woods Hole mathematics, Ser. Knots Everything 34, World Scientific Publishing, Hackensack, N.J., 2004, 263–301. 256 [347] R. Penner, Perturbative series and the moduli space of Riemann surfaces. J. Differential Geom. 27 (1988), 35–53. 256 [348] V. Platonov and A. Rapinchuk, Algebraic groups and number theory. Pure Appl. Math. 139, Academic Press, Boston, Mass., 1994. 173, 201, 212, 230 [349] V. Platonov and A. Rapinchuk, Abstract properties of S-arithmetic groups and the congruence problem. Russian Acad. Sci. Izv. Math. 40 (1993), no. 3, 455–476 168, 231 [350] S. Popa, Deformation and rigidity for group actions and von Neumann algebras. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. I, Eur. Math. Soc., Zurich 2007, 445–477. 171 [351] G. Prasad, Discrete subgroups isomorphic to lattices in semisimple Lie groups. Amer. J. Math. 98 (1976), no. 1, 241–261. 171 [352] G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups. Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91–117. 170, 176 [353] G. Prasad and A. Rapinchuk, Zariski-dense subgroups and transcendental number theory. Math. Res. Lett. 12 (2005), no. 2–3, 239–249. 233 [354] G. Prasad and A. Rapinchuk, Developments on the congruence subgroup problem after the work of Bass, Milnor and Serre. Preprint, 2009. 168, 231 [355] S. Prassidis and B. Spieler, Rigidity of Coxeter groups. Trans. Amer. Math. Soc. 352 (2000), no. 6, 2619–2642. 191 [356] M. Puschnigg, The Kadison-Kaplansky conjecture for word-hyperbolic groups. Invent. Math. 149 (2002), no. 1, 153–194. 193 [357] D. Quillen, The spectrum of an equivariant cohomology ring. II. Ann. of Math. (2) 94 (1971), 573–602. 169, 180, 188, 191, 192 [358] J. F. Quint, Mesures de Patterson–Sullivan en rang supérieur. Geom. Funct. Anal. 12 (2002), no. 4, 776–809. 207 [359] K. Rafi,A combinatorial model for the Teichmüller metric. Geom. Funct. Anal. 17 (2007), no. 3, 936–959. 183 [360] M. Raghunathan, Discrete subgroups of Lie groups. Ergeb. Math. Grenzgeb. 68, SpringerVerlag, New York 1972. 173, 202 [361] M. Raghunathan, The congruence subgroup problem. In Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras 1991, 465–494. 169

292

Lizhen Ji

[362] M. Raghunathan, The congruence subgroup problem. Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 4, 299–308; Erratum ibid. 115 (2005), no. 3, 369. 169 [363] J. Ratcliffe, Foundations of hyperbolic manifolds. Second edition, Grad. Texts in Math. 149. Springer-Veralg, New York 2006. 173 [364] D. B. McReynolds, The congruence subgroup problem for braid groups: Thurston’s proof. Preprint, arXiv:0901.4663v2 [math.GT]. 180 [365] H. E. Rauch, A transcendental view of the space of algebraic Riemann surfaces. Bull. Amer. Math. Soc. 71 (1965), 1–39. 248 [366] D. Robinson, A course in the theory of groups. Second edition, Grad. Texts in Math. 80, Springer-Verlag, New York 1996. 246 [367] A. Rodado, Weierstrass points and canonical cell decompositions of the moduli and Teichmüller Spaces of Riemann surfaces of genus two. PhD thesis, Mathematics and Statistics, The University of Melbourne, 2007. 185, 257 [368] J. Roe, Hyperbolic groups have finite asymptotic dimension. Proc. Amer. Math. Soc. 133 (2005), no. 9, 2489–2490. 193 [369] S. Ryshkov, The theory of Hermite-Minkowski reduction of positive definite quadratic forms. J. Soviet Math. 6 (1976), 651–671. 221, 225 [370] S. Ryshkov, Geometry of positive quadratic forms. Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, 501–506. 221, 225 [371] L. Saper, Tilings and finite energy retractions of locally symmetric spaces. Comment. Math. Helv. 72 (1997), no. 2, 167–202. 177, 226, 244, 256, 258 [372] L. Saper, L2 -cohomology of the Weil–Petersson metric. In Mapping class groups and moduli spaces of Riemann surfaces, Contemp. Math. 150, Amer. Math. Soc., Providence, R.I., 1993, 345–360. 186, 268 [373] L. Saper and M. Stern, L2 -cohomology of arithmetic varieties. Ann. of Math. (2) 132 (1990), no. 1, 1–69. 178 [374] I. Satake, On compactifications of the quotient spaces for arithmetically defined discontinuous groups. Ann. of Math. (2) 72 (1960), 555–580. 239, 262, 264, 265 [375] P. Schmutz Schaller, A systolic geometric cell decomposition for the space of once-holed Riemann surfaces of genus 2. Topology 40 (2001), no. 5, 1017–1049. 256 [376] P. Schmutz Schaller, A cell decomposition of Teichmüller space based on geodesic length functions. Geom. Funct. Anal. 11 (2001), no. 1, 142–174. 256 [377] P. Schmutz Schaller, Systoles and topological Morse functions for Riemann surfaces. J. Differential Geom. 52 (1999), no. 3, 407–452. [378] P. Schmutz Schaller, Geometry of Riemann surfaces based on closed geodesics. Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 3, 193–214. [379] J. Schwermer, Geometric cycles, arithmetic groups and their cohomology. Bull. Amer. Math. Soc. 47 (2010), 187–279. 169 [380] P. Scott and T. Wall, Topological methods in group theory. In Homological group theory, London Math. Soc. Lecture Note Ser. 36, Cambridge University Press, Cambridge 1979, 137–203. 235


293

[381] Z. Sela, Endomorphisms of hyperbolic groups. I. The Hopf property. Topology 38 (1999), no. 2, 301–321. 193 [382] Z. Sela, The isomorphism problem for hyperbolic groups. I. Ann. of Math. (2) 141 (1995), 217–283. 192 [383] A. Selberg, Discontinuous groups and harmonic analysis. Proceedings of the International Congress of Mathematicians (Stockholm, 1962), Institut Mittag-Leffler, Djursholm 1963, 177–189. 178, 232 [384] J. P. Serre, Cohomologie des groupes discrets. In Prospects in mathematics, Ann. of Math. Stud. 70, Princeton University Press, Princeton, N.J., 1971, 77–169. 164, 170, 244 [385] J. P. Serre, Arithmetic groups. In Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser. 36, Cambridge University Press, Cambridge 1979, 105–136. 170, 176 [386] R. Sharma and T. Venkataramana, Generations for arithmetic groups. Geom. Dedicata 114 (2005), 103–146. 230 [387] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures 1, Publ. Math. Soc. Japan 11, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. 173 [388] T. Shiota, Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (1986), no. 2, 333–382. 271 [389] T. Shiota, The KP equation and the Schottky problem. Sugaku Expositions 3 (1990), no. 2, 183–211. 271 [390] C. L. Siegel, Discontinuous groups. Ann. of Math. (2) 44 (1943), 674–689. 211, 212, 213, 230 [391] C. L. Siegel, Topics in complex function theory. Vol. I. Elliptic functions and uniformization theory. Reprint of the 1969 edition, Wiley Classics Lib., John Wiley & Sons, New York 1988. 214 [392] C. L. Siegel, Lectures on the geometry of numbers. Springer-Verlag, Berlin 1989. 218, 220, 221 [393] C. L. Siegel, Lectures on advanced analytic number theory. Notes by S. Raghavan, Tata Institute of Fundamental Research Lectures on Mathematics 23, Tata Institute of Fundamental Research, Bombay 1965. 228 [394] J. Stallings, On torsion-free groups with infinitely many ends. Ann. of Math. (2) 88 (1968), 312–334. 170, 181, 235 [395] C. Soulé, An introduction to arithmetic groups. In Frontiers in number theory, physics, and geometry, II, Springer-Verlag, Berlin 2007, 247–276. 230 [396] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149 (1982), 215–237. 178 [397] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. 207 [398] B. Sury, The congruence subgroup problem. An elementary approach aimed at applications. Texts Read. Math. 24, Hindustan Book Agency, New Delhi 2003. 174

294

Lizhen Ji

[399] F. Taherkhani, The Kazhdan property of the mapping class group of closed surfaces and the first cohomology group of its cofinite subgroups. Experiment. Math. 9 (2000), no. 2, 261–274. 181 [400] K. Takeuchi, Arithmetic triangle groups. J. Math. Soc. Japan 29 (1977), no. 1, 91–106. 204, 217 [401] O. Tavgen, Bounded generation of Chevalley groups over rings of algebraic S-integers. Math USSR Izvestiya 36 (1991) 101–128. 230 [402] A. Terras, Harmonic analysis and symmetric spaces and applications. II. SpringerVerlag, Berlin 1988. 218, 220 [403] A. Fathi, F. Laudenbach, and V. Poénaru (eds.), Travaux de Thurston sur les surfaces. Astérisque 66—67 (1979). 182, 184, 187, 249 [404] W. Thurston, Minimal stretch maps between hyperbolic surfaces. Preprint 1998, arXiv:math/9801039v1 [math.GT]. 183, 261 [405] W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. 184 [406] W. Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy, Princeton Math. Ser. 35. Princeton University Press, Princeton, N.J., 1997. 173 [407] J. Tits, Free subgroups in linear groups. J. Algebra 20 (1972), 250–270. 233, 234 [408] G. van der Geer, Hilbert modular surfaces. Ergeb. Math. Grenzgeb.(3) 16, SpringerVerlag, Berlin 1988. 173, 228 [409] M. F. Vignéras, Arithmétique des algèbres de quaternions. Lecture Notes in Math. 800, Springer-Verlag, Berlin 1980. 173 [410] É. Vinberg and O. Shvartsman, Discrete groups of motions of spaces of constant curvature. In Geometry, II, Encyclopaedia Math. Sci. 29, Springer-Verlag, Berlin 1993, 139–248. 173, 204 [411] E. Vinberg, V. Gorbatsevich, and O. Shvartsman, Discrete subgroups of Lie groups. In Lie groups and Lie algebras, II, Encyclopaedia Math. Sci. 21, Springer-Verlag, Berlin 2000, 1–123, 217–223. 166, 174 [412] K. Vogtmann,Automorphisms of free groups and outer space. Geom. Dedicata 94 (2002), 1–31. 160, 187, 188, 189, 190, 247 [413] K. Vogtmann, End invariants of the group of outer automorphisms of a free group. Topology 34 (1995), 533–545. 187, 189 [414] K. Vogtmann, The cohomology of automorphism groups of free groups. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. II, 1101–1117, Eur. Math. Soc., Zurich 2006. 188, 189, 190, 247 [415] C. Walsh, The horoboundary and isometry group of Thurston’s Lipschitz metric. Preprint, arXiv:1006.2158v3 [math.GT]. 166, 184, 250 [416] Y. Watatani, Property T of Kazhdan implies Property FA of Serre. Math. Japon. 27 (1982), no. 1, 97–103. 172, 235 [417] B. Wehrfritz, Infinite linear groups. Ergeb. Math. Grenzgeb. 76, Springer-Verlag, Heidelberg 1973. 171


295

[418] H. Weyl, The theory of reduction for arithmetical equivalence. Trans. Am. Math. Soc 48 (1940), 126–164; 225 [419] H. Weyl, Theory of reduction for arithmetical equivalence. II. Trans. Amer. Math. Soc. 51 (1942), 203–231. 225 [420] T. White, Fixed points of finite groups of free group automorphisms. Proc. Amer. Math. Soc. 118 (1993), no. 3, 681–688. 188, 189 [421] W. Woess, Random walks on infinite graphs and groups. Cambridge Tracts in Math. 138, Cambridge University Press, Cambridge 2000. 193 [422] J. Wolf, Spaces of constant curvature. Fifth edition. Publish or Perish, Inc., Houston, TX, 1984. 172 [423] M. Wolf, The Teichmüller theory of harmonic maps. J. Differential Geom. 29 (1989), 449–479. 184, 249 [424] S. Wolpert, Geodesic length functions and the Nielsen problem. J. Differential Geom. 25 (1987), no. 2, 275–296. 180, 248 [425] S. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space. In Surveys in differential geometry, Vol. VIII, Surv. Differ. Geom. VIII, International Press, Somerville, Mass., 2003, 357–393. 183, 184, 249, 251, 252 [426] S. Wolpert, The Weil–Petersson metric geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 47–64. 183, 249, 252 [427] S. Wolpert, Families of Riemann surfaces and Weil–Petersson geometry. CBMS Regional Conf. Ser. in Math. 113, Amer. Math. Soc., Providence, R.I. 2010. 183, 251, 252 [428] D.Yasaki, On the existence of spines for Q-rank 1 groups. Selecta Math. (N.S.) 12 (2006), no. 3-4, 541–564. 228, 244 [429] M.Yoshida, Hypergeometric functions, my love. Modular interpretations of configuration spaces. Aspects Math. E32, Friedr. Vieweg & Sohn, Braunschweig 1997. 196 [430] M. Yoshida, Fuchsian differential equations. With special emphasis on the Gauss– Schwarz theory. Aspects Math. E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. 196 [431] S. K. Yeung, Bergman metric on Teichmüller spaces and moduli spaces of curves. In Recent progress on some problems in several complex variables and partial differential equations, Contemp. Math. 400, Amer. Math. Soc., Providence, R.I., 2006, 203–217. 269 [432] R. Zimmer, Ergodic theory and semisimple groups. Monogr. Math. 81, Birkhäuser, Basel 1984. 171, 173, 175, 233, 237 [433] S. Zucker, On the reductive Borel-Serre compactification: Lp -cohomology of arithmetic groups (for large p). Amer. J. Math. 123 (2001), no. 5, 951–984. 178, 268

Chapter 5

Simplicial actions of mapping class groups John D. McCarthy and Athanase Papadopoulos

Contents 1 2

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Introduction . . . . . . . . . . . . . . . . . . . . . . . Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Curves . . . . . . . . . . . . . . . . . . . . . . . 2.2 Arcs . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Boundary graphs . . . . . . . . . . . . . . . . . 2.4 Geometric intersection number . . . . . . . . . . 2.5 Orientation . . . . . . . . . . . . . . . . . . . . 2.6 Domains . . . . . . . . . . . . . . . . . . . . . . Simplicial complexes . . . . . . . . . . . . . . . . . . 3.1 Abstract simplicial complexes . . . . . . . . . . 3.2 Exchange automorphisms . . . . . . . . . . . . . 3.3 Quotient complexes . . . . . . . . . . . . . . . . 3.4 Topology . . . . . . . . . . . . . . . . . . . . . Some complexes associated to surfaces . . . . . . . . 4.1 The curve complex C.S/ . . . . . . . . . . . . . 4.2 The arc complex A.S/ . . . . . . . . . . . . . . 4.3 The arc and curve complex AC.S / . . . . . . . . 4.4 The pants decomposition graph P1 .S / . . . . . . 4.5 The ideal triangulation graph T .S / . . . . . . . . 4.6 The Schmutz graph of nonseparating curves G.S/ 4.7 The complex of nonseparating curves N.S / . . . 4.8 The cut system graph HT1 .S/ . . . . . . . . . . 4.9 The complex of separating curves CS.S / . . . . 4.10 The Torelli complex TC.S/ . . . . . . . . . . . The complex of domains and its subcomplexes . . . . 5.1 The complex of domains D.S/ . . . . . . . . . . 5.2 The truncated complex of domains D 2 .S / . . . . 5.3 The complex of boundary graphs B.S / . . . . . 5.4 The complex of peripheral pairs of pants P@ .S / . 5.5 Other subcomplexes of D.S/ . . . . . . . . . . . Surface topology recognized by D.S/ and D 2 .S / . . 6.1 Recognizing elementary vertices in D.S/ . . . . 6.2 Recognizing nonseparating annuli in D.S/ . . .

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6.3 Recognizing elementary vertices in D 2 .S / . . . . . . 6.4 Recognizing annular vertices in D 2 .S / . . . . . . . . 6.5 Recognizing biperipheral edges in D.S/ . . . . . . . . 7 Automorphisms of the truncated complex of domains . . . . 7.1 Distinguishing vertices of D.S/ via their annular links 7.2 Automorphisms of D 2 .S/ are geometric . . . . . . . . 8 Automorphisms of the complex of domains . . . . . . . . . 8.1 Exchange automorphisms of D.S / . . . . . . . . . . . 8.2 Automorphisms of D.S/ . . . . . . . . . . . . . . . . 9 Directions for further work . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction In this chapter, S D Sg;b denotes a connected compact orientable surface of genus g 0 with b 0 boundary components and @S denotes the boundary of S. The mapping class group of S, denoted by D g;b D .S /, is the group of isotopy classes of orientation-preserving self-homeomorphisms of S . The extended mapping class group D .S/, is the group of all isotopy classes of self-homeomorphisms of S , D g;b of S . Thus, is a normal subgroup of index 2 in . The study of these groups has used their action on various abstract simplicial complexes, each of which encodes combinatorial information about the relationship which certain subspaces of S (curves, arcs, cut systems, ideal triangulations, etc.) bear to one another. The aim of this chapter is threefold: (1) to review the definitions and some geometrical properties of the complexes; (2) to review some of the extended mapping class group actions on these complexes; (3) to study in detail the actions on recently defined complexes, namely, the complex of domains and some of its subcomplexes. Let us agree that in what follows, when we talk about an “action of the mapping class group”, we also mean to say the action of the extended mapping class group. The first systematic study of an action of the mapping class group on a combinatorial complex was done by A. Hatcher and W. P. Thurston in a paper published in 1980 [44]. In that paper, Hatcher and Thurston described two actions of the mapping class group, namely, the action on the cut system complex and the one on the pants decomposition complex. We note that these complexes are not simplicial complexes, but CW complexes. The 1-skeleta of these complexes are simplicial graphs, called the cut system graph and the pants decomposition graph, and the actions of mapping class groups on these 1-skeleta contain most of the information of the actions on the corresponding thicker CW complexes. In this chapter, we review the actions of the mapping class group on these complexes and on several others. To be more precise, let us briefly recall the definitions of the cut system and the pants decomposition graphs. In the case where the surface S D Sg is a closed surface

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of genus g 2, the vertices of the cut system graph are systems of g closed curves that cut S into a sphere with 2g holes. The vertices of the pants decomposition graph are systems of 3g 3 closed curves that cut S into 2g 2 pairs of pants, i.e. spheres with three holes. In both cases, two vertices are joined by an edge whenever these vertices are represented by systems of curves on the surface that are related by an elementary move. In each case, an elementary move consists of replacing a curve in a cut system (respectively a pants decomposition) by a new curve which has minimal intersection number with the old curve, and such that the new curve system is again a cut system (respectively a pants decomposition). The elementary moves between cut systems are described in Figure 14 below. The elementary moves between pants decompositions are represented in Figure 10 below. The cut system complex and the pants decomposition complex are CW complexes obtained from these graphs by attaching a certain number of two-cells that make the complexes simply connected. The actions of the mapping class group on these two complexes were used by Hatcher and Thurston to obtain explicit finite presentations of the mapping class group. Both complexes are defined in the paper [44] for closed surfaces of genus g 2. Several years after Hatcher and Thurston introduced these two complexes, it was shown that the actions of the extended mapping class group on these complexes as well as on their 1-skeleta are rigid in the sense that the natural homomorphisms from the mapping class group into the simplicial automorphism group of these complexes are isomorphisms, with the exception of a few number of surfaces which have low genus and small number of boundary components. These rigidity results are due to Irmak and Korkmaz [52] for the cut system complex and to Margalit [86] for the pants decomposition complex. This rigidity phenomenon and its generalization to actions of the mapping class groups on other complexes is one of the main themes of this chapter. The curve complex, C.S/, is a flag complex that was introduced shortly after the work of Hatcher and Thurston by W. Harvey. This complex captures the combinatorics of disjointness vs. intersection in the set of isotopy classes of essential unoriented simple closed curves on S. For all n 0, an n-simplex of C.S / is a set of n C 1 distinct isotopy classes of essential closed curves on S that can be represented by disjoint curves on the surface. Harvey introduced this complex as an analogue for Teichmüller space of Tits buildings which are combinatorial objects that describe the large-scale geometry and the structure at infinity of symmetric spaces and of locally symmetric spaces. After its introduction by Harvey, the curve complex was studied by various people, from different points of view, namely, by Ivanov, Korkmaz, Luo, Masur, Minsky, Bowditch, Klarreich, Hamenstädt, Schleimer and there are others. (We shall mention several works on that complex in §4.1 below.) Ivanov proved the important result (completed by Korkmaz and Luo) stating that the simplicial automorphism group of the curve complex coincides with the natural image of the extended mapping class group in that group. Masur and Minsky studied the curve complex, endowed with its natural simplicial metric, from the point of view of large-scale geometry. They

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showed that this complex is Gromov hyperbolic. Klarreich described the Gromov boundary of the curve complex. We shall review in §4.1 below several of these results as well as other results related to the curve complex. We shall also consider the arc complex, A.S /, defined in analogy with the curve complex, with essential closed curves replaced by essential properly embedded arcs in S . A rigidity theorem for the arc complex, analogous to the theorem by Ivanov– Korkmaz–Luo, was obtained quite recently by Irmak and McCarthy (see Theorem 4.7 below). Besides the cut system graph, the pants decomposition graph, the curve complex and the arc complex, we shall consider the ideal triangulation graph, the arc and curve complex, the Schmutz graph of nonseparating curves, the complex of nonseparating curves (whose 1-skeleton is not the Schmutz graph of nonseparating curves), the complex of separating curves, the Torelli complex and several others. In particular, we shall also consider a newly defined complex D.S/ on which acts simplicially, the complex of domains of S. A domain on S is a nonempty connected compact embedded surface in S which is not equal to S and each of whose boundary components is either contained in @S or is essential on S. The vertex set D0 .S / of D.S/ is the set of isotopy classes of domains on S . For each n 0, an n-simplex of D.S/ is a set of n C 1 distinct vertices of D.S/ that can be represented by disjoint domains on S . One principle guiding the techniques we develop in our study of D.S/ is that its vertices are essentially copies of the curve complex of the subsurfaces they represent, together with the boundary components of these subsurfaces which are contained in the interior of S. A recurrent theme in the theory of mapping class group actions on simplicial complexes associated to surfaces is that in general, these actions are rigid in a sense already mentioned, that is, there are no simplicial actions on these complexes other than those arising from homeomorphisms of surfaces. This fact, i.e., that the natural homomorphism from the extended mapping class group to the simplicial automorphism group of the complex is onto, is naturally expressed by saying that all the automorphisms of the complex are geometric (that is, they are induced by homeomorphisms of the surface). We shall give several examples of rigid actions. Let us note that although the statements of these rigidity results are similar, the proofs are in general different, and they use the properties of the specific complexes that are involved. This makes each of these actions interesting. Another interesting (and may be surprising) fact is that even though all the various complexes that appear in this theory are different in many respects (they all have different dimensions, some of them have all of their maximal simplices of the same dimension, other do not, some of them are locally finite, others are not, etc.) all these complexes have the same automorphism group! We note by the way that there are several other rigidity results of actions of mapping class groups on spaces that are not simplicial complexes. We mention for instance the papers by Royden [118], Luo [83], Masur and Wolf [87], Papadopoulos [108] and [109], Walsh [130] and the recent papers by Ohshika [106], [107], and the paper


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by Charitos, Papadoperakis and Papadopoulos [23]. In each of these papers, the extended mapping class group acts as the automorphism group a specific structure (metric, topological, PL, etc.). There are many applications of the complexes associated to surfaces. Harer used several of the actions of mapping class groups on these complexes in his proof of the stability of the homology of the mapping class group [39]. We shall mention several other applications. Applications of the actions of the mapping class group to curve complexes are also described in detail in the chapter by Lizhen Ji in Volume IV of this Handbook [62], from another point of view, viz., the comparison between mapping class groups and arithmetic groups. Another important theme in the study of the simplicial complexes associated to surfaces is that the combinatorial information at the vertices of these complexes recognizes the topological data on the surface that these vertices represent. This is well illustrated in the detailed study of the complex of domains and the truncated complex of domains that we make in what follows. There are several reasons that led us to study the actions of the mapping class group on the complex of domains. First of all, this complex is the complex which is naturally associated to the Thurston theory of surface diffeomorphisms. Indeed, the various pieces of the Thurston decomposition of a surface diffeomorphism, that we call thick domains and annular (or thin) domains, appear as the vertices of this flag complex. Secondly, the complex of domains contains, as subcomplexes, a certain number of complexes, each of which is induced from a particular subset of the 0-skeleton D0 .S / of D.S /. We mention some of these subcomplexes: The truncated complex of domains, D 2 .S /, whose vertices are the isotopy classes of domains that are not pairs of pants with two boundary components contained in @S . The complex of elementary domains, E.S/, whose vertices are represented by domains that are either annuli or pairs of pants. The complex of annular domains, R.S /, whose vertices are represented by domains that are annuli. This complex is naturally isomorphic to the curve complex C.S /. The complex of pairs of pants, P .S/, whose vertices are represented by domains that are pairs of pants. The complex of peripheral pairs of pants, P@ .S /, whose vertices are represented by domains homeomorphic to pairs of pants having at least one boundary component contained in @S . The complex of thick domains, TD.S /, whose vertices are represented by domains on S that are not annuli. The complex of boundary graphs, B.S /, whose vertices are isotopy classes of graphs on S that are the union of an essential arc on S together with the boundary components of S that contain at least one endpoint of this arc. B.S / is naturally identified with P@ .S/. We shall see that it is also naturally identified with a certain subcomplex of A.S/ with the same vertex set as A.S / but with, in general, fewer

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simplices. (In particular, there is a natural injection of the vertex set of the arc complex into the complex of domains, but this extension does not extend to a simplicial injection of the whole arc complex.) Each of these complexes and sub-complexes has some special combinatorial features, and there are interesting questions that are particular to each of them. Furthermore, besides the study of the individual complexes, there are interesting questions about natural maps between them. Except for a small number of special surfaces (which all belong to the following class: spheres with at most four holes, tori with at most two holes and the closed surface of genus 2), the natural homomorphism of the extended mapping class group to the automorphism group of any one of the complexes that we study is an isomorphism, except in the case of the complex of domains, if the surface has at least two boundary components. Indeed, a special feature of the complex of domains of such a surface is that its automorphism group is uncountable. This automorphism group includes automorphisms that send a biperipheral pair of pants (that is, a pair of pants having two of its boundary components boundary components of S ) to the corresponding biperipheral annulus (that is, the annulus homotopic to the boundary component of the biperipheral pair of pants that is not a boundary component of S) and that fix all the other vertices of D.S/. Such an automorphism is not geometric, since a homeomorphism of the surface cannot send an annulus to a pair of pants. This phenomenon disappears if instead of the complex of domains we consider the truncated complex of domains, and we shall see that the natural homomorphism of the extended mapping class group into the automorphism group of the truncated complex of domains is an isomorphism (except, as usual, for a small number of surfaces that we mentioned above). Valentina Disarlo studied the coarse geometry of the complex of domains and of some of its complexes, see Chapter 6 of the present volume [26]. Finally, we note that all the simplicial complexes that we consider can be seen as induced subcomplexes of a “universal” simplicial complex UC .S / whose simplices are finite collections of isotopy classes of a class C of closed subsets of S which is invariant under the action of the group Homeo.S / of self-homeomorphisms of S on the set of closed subspaces of that surface. This ensures that there is a natural action .S / UC .S/ ! UC .S/ of .S/ on UC .S /. Note that the induced subcomplex UD .S / of UC .S/ corresponding to any subcollection D of C which is also invariant under this action affords a similar simplicial representation W .S / ! Aut.UD .S // of .S /. The plan of this chapter is the following. Section 2 contains some basic principles on surfaces, subsurfaces, curves and geometric intersection numbers that will be used in the later sections. For the convenience of the non-expert in the theory of surfaces, we have included proofs of a few facts that we shall use, although these facts are well known. Of course, a reader who is an expert in these matters will skip these proofs.


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Section 3 contains a short introduction to abstract simplicial complexes. In this section, we define some basic simplicial notions that we use later in the chapter. Some of the notions are standard notions, and others are new. In particular, we introduce the notion of an exchangeable pair of vertices and of an exchange automorphism of a simplicial complex. We use this notion later in the text, in our study of the complex of domains. We give necessary and sufficient conditions for a pair of vertices to be exchangeable. We define certain special subgroups of the automorphism group of a simplicial complex K, which we call Boolean subgroups. Such a group is isomorphic to the Boolean algebra of a collection of subsets of K consisting of exchangeable pairs of vertices. The exchange automorphisms and the Boolean subgroups will be used in an essential way in the section on the automorphism group of the complex of domains, §8. We also develop the theory of quotient complexes. Section 4 is a survey on several simplicial complexes associated to a surface S: the curve complex C.S/, the arc complex A.S /, the arc and curve complex AC.S /, the pants decomposition graph P1 .S/, the ideal triangulation graph T .S /, the Schmutz graph of nonseparating curves G.S/, the complex of nonseparating curves N.S /, the cut system graph HT1 .S/, the complex of separating curves CS.S /, and the Torelli complex TC.S/. We state the rigidity results without proofs, and we sometimes elaborate on some special cases of surfaces that are excluded by the hypotheses of the theorems. It is usually pleasant and instructive to work out the details of these special cases. In Section 5, we introduce the complex of domains, D.S/, and several of its subcomplexes, in particular the truncated complex of domains, D 2 .S /, the complex of boundary graphs B.S/ and the complex of peripheral pairs of pants P@ .S /. There are natural inclusion maps among these complexes and the complexes introduced in the preceding sections, and there is a natural projection map D.S/ ! D 2 .S /. In Section 6, we show on a series of examples that combinatorial information at the vertices of the simplicial complexes D.S / and D 2 .S / can be used to characterize the subsurfaces that are represented by these vertices. To be more specific, we show for instance that a vertex x of D.S/ is elementary (that is, it represents either an annulus or a pair of pants) if and only if we have Lk.Lk.x// D fxg. Some of the characterizations of vertices of D.S/ and of D 2 .S / that are obtained in this section will be used in the following sections, where the rigidity results are proved. In Section 7, we prove that if S is not a sphere with four holes, or a torus with at most two holes, or a closed surface of genus two, then the natural homomorphism W .S / ! Aut.D 2 .S// corresponding to the action of .S / on D 2 .S / is an isomorphism. In particular, every automorphism of D 2 .S / is induced by a homeomorphism S ! S which is uniquely defined up to isotopy on S . In Section 8, we prove the rigidity result for the complex of domains. This involves the notion of an exchange automorphism. We prove that if S is not a sphere with at most four holes, a torus with at most two holes, or a closed surface of genus two, then every automorphism of D.S/ is a composition of an exchange automorphism of D.S/ with a geometric automorphism of D.S/.

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Acknowledgements. The authors thank the Max Planck Institute for Mathematics (Bonn) for the excellent conditions provided during which part of this work was done. They also thank Valentina Disarlo who proof-read an early version of the manuscript.

2 Surfaces In what follows S D Sg;b is a connected compact orientable surface of genus g with b boundary components. We shall say that S is a surface of genus g with b holes. Note that Sg;b is a closed surface of genus g if and only if b D 0. Some surfaces have special names; for instance, S0;1 is a disc, S0;2 is an annulus, S0;3 is a pair of pants, S0;b is a sphere with b holes and S1;b is a torus with b holes. When talking about a torus (respectively sphere) with zero holes, we shall use the terminology closed torus (respectively sphere) in order to avoid ambiguity. Let @S denote the boundary of S. We shall sometimes index the b boundary components of S by @i , 1 i b. For each collection C of subsets of S , the support of C on S is the union in S of the subsets of S in the collection C . We shall usually denote the support of C on S by jC j. Throughout this chapter, all isotopies between subspaces of S will be ambient isotopies. More precisely, if X and Y are subspaces of S , an isotopy from X to Y is a map ' W S Œ0; 1 ! S such that the maps ' t W S ! S , 0 t 1, are homeomorphisms of S , '0 D idS W S ! S , and '1 .X / D Y .

2.1 Curves A curve on S is an embedded connected closed one-dimensional submanifold in the interior of S . (Thus, a curve is homeomorphic to a circle.) Let ˛ be a curve on S. We say that ˛ is k-peripheral if there exists a sphere with k C 1 holes X on S such that ˛ is a boundary component of X and every other boundary component of X is a boundary component of S. We say that ˛ is essential if it is neither 0-peripheral nor 1-peripheral on S. Equivalently, ˛ is essential on S if and only if there does not exist a disc on S whose boundary is equal to ˛ or an annulus on S whose boundary is equal to the union of ˛ with a boundary component of S . If S is a sphere with at most three holes, then there are no essential curves on S . Otherwise, there are infinitely many pairwise nonisotopic essential curves on S. Suppose that ˛ and ˇ are disjoint essential curves on S. Then ˛ is isotopic to ˇ on S if and only if there exists an annulus on S whose boundary is equal to ˛ [ ˇ. This is a classical result, see e.g. [27]. A collection of pairwise disjoint essential curves on S is called a system of curves if the curves in the collection are pairwise nonisotopic. Note that every subcollection of a system of curves on S is also a system of curves.


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In the rest of this chapter, unless otherwise indicated, a simple closed curves will be assumed to be essential, and in general we shall simply call it a curve. If C is a finite collection of curves on S then SC will denote the compact surface obtained from S by cutting S along C . A pants decomposition of S is a collection of curves C on S such that each component of SC is a pair of pants. Note that every pants decomposition of S is a system of curves on S. The surface S has a pants decomposition if and only if S is not a sphere with at most two holes nor a closed torus. Moreover, on such a surface S , if C is a system of curves, then there exists a pants decomposition on S containing C . A nonempty system of curves C on S is a maximal system of curves on S if and only if one of the following two situations occurs: (1) S is a closed torus and C consists of exactly one nonseparating curve on S. (2) S is not a closed torus and C is a pants decomposition of S . In this case, the cardinality of C is equal to 3g 3 C b. Suppose that C is a pants decomposition of S. Let R be a regular neighborhood of jC j. The closure of the complement of R on S is a disjoint union of pairs of pants on S . These pairs of pants are called pairs of pants of C . Note that the pairs of pants of a pants decomposition are defined only up to isotopy relative to C . Suppose that P is a pair of pants of a pants decomposition C of S . Then P is contained in a unique component Q of SC . We say that P is an embedded pair of pants of C if the natural quotient map W SC ! S embeds the pair of pants Q in S . Let C be a pants decomposition of S . We say that C is an embedded pants decomposition of S if every pair of pants of C is embedded. For example, if S is a closed surface of genus two and C is a disjoint union of three nonisotopic nonseparating curves on S , then C is an embedded pants decomposition of S.

2.2 Arcs An arc ˛ on S is a subspace of S which is homeomorphic to the interval Œ0; 1. The endpoints of ˛ are the images of 0 and 1 under a homeomorphism from Œ0; 1 to ˛. We say that an arc ˛ is properly embedded if ˛ intersects the boundary of S precisely at its endpoints. Unless otherwise indicated, all arcs on S will be assumed to be properly embedded. Let ˛ be an arc on S. Suppose that one endpoint of ˛ lies on @i and the other endpoint of ˛ lies on @j . Then we say that ˛ joins @i to @j , and that ˛ is an arc of type fi; j g. Let ˛ be an arc on S of type fi; j g and ˇ be an arc on S of type fk; lg. Note that if ˛ is isotopic to ˇ, then fi; j g D fk; lg. An arc ˛ on S is essential if there does not exist an embedded closed disk on S whose boundary is equal to the union of ˛ with an arc contained in the boundary of S .

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Any arc on S joining two distinct boundary components of S is essential. Likewise, if g > 0, b D 1, then any arc on S that intersects a simple closed curve on S transversely and at exactly one point is essential. In the rest of this chapter, unless otherwise indicated, all curves will be assumed to be essential, and we shall simply use the terminology “curve” instead of “essential curve”.

2.3 Boundary graphs Let ˛ be an arc on S joining @i to @j . The boundary graph of ˛, G˛ , is the union @i [ ˛ [ @j (see Figure 1). We refer to @i and @j as the boundary components of G˛ . Note that G˛ has one boundary component if ˛ joins a boundary component of S to itself (i.e. if i D j ) and two boundary components if ˛ joins distinct boundary

(1)

(2)

(3)

Figure 1. The three types of boundary graphs, drawn in bold lines: (1) a boundary graph of an arc joining two distinct boundary components of S; (2) a boundary graph of an arc joining a boundary component of S to itself with at least one of the two boundary components of a regular neighborhood of the boundary graph being a boundary component of S; (3) a boundary graph of an arc joining a boundary component of S to itself with a regular neighborhood of the boundary graph having two essential curves on S on its boundary.

components of S (i.e. if i ¤ j ). Since arcs are, by definition, properly embedded, @i [ @j D G˛ \ @S . A boundary graph is a boundary graph of an arc and an essential boundary graph is a boundary graph of an essential arc.

2.4 Geometric intersection number Definition 2.1. Let ˛ and ˇ be curves on S . The geometric intersection number of ˛ and ˇ on S is the minimum number i.˛; ˇ/ D iS .˛; ˇ/ of points in ˛ 0 \ ˇ 0 where ˛ 0 and ˇ 0 are curves on S that are isotopic to ˛ and ˇ respectively. Definition 2.2. Let f˛1 ; : : : ; ˛n g be a collection of curves on S . We say f˛1 ; : : : ; ˛n g is in minimal position on S if for all 1 i < j n, the geometric intersection number of ˛i and ˛j on S is equal to the number of points in ˛i \ ˛j .


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The following fact is often useful. If f˛1 ; : : : ; ˛n g is a finite collection of curves on S , then there exists a collection fˇ1 ; : : : ; ˇn g of curves on S such that ˛i is isotopic to ˇi on S , 1 i n and fˇ1 ; : : : ; ˇn g is in minimal intersection position on S . For a proof, we can assume without loss of generality that the curves f˛1 ; : : : ; ˛n g intersect transversely, and then apply an innermost disk elimination argument, to eliminate non-essential intersection points. For the existence of such disks, see e.g. [27]. Alternatively, we can prove this fact by equipping S with a hyperbolic structure and replacing each curve ˛i by the unique closed geodesic ˇi which is homotopic to it. The following proposition will also be useful. Proposition 2.3. Let C be a collection of pairwise disjoint and non-homotopic essential curves on S and let ˛ 2 C . Then there exists a curve on S such that i.˛; / ¤ 0 and i.ˇ; / D 0 for all ˇ 6D ˛, ˇ 2 C . Proof. Since S contains an essential curve, S is not a sphere with at most three holes. Suppose, on the one hand, that S is a closed torus. Then any two disjoint essential curves on S are isotopic, ˇ is isotopic to ˛ for each ˇ in C . Since any essential curve on S is nonseparating, ˛ is nonseparating. Hence, S˛ is an annulus and there exists a curve on S that intersects ˛ transversely and has exactly one point of intersection with ˛. It follows that i.; ˇ/ D i.; ˛/ D 1 for each ˇ in C . If S is not a sphere with at most three holes or a closed torus, we can find a pants decomposition C1 of S such that each curve in C is homotopic to a curve in C1 . If ˛1 is a curve of the pants decomposition C1 of S , there exists a curve on S such that i.; ˛1 / ¤ 0 and i.; ı/ D 0 for each curve ı in C1 n f˛1 g, see Figure 2.

Figure 2. Choosing the curve associated to ˛1 in the proof of Proposition2.3. The figure represents the cases where ˛1 is on the boundary of one and two pairs of pants respectively.

It follows that for each ˇ in C , i.; ˇ/ D i.; ˇ1 / ¤ 0 if and only if ˇ is isotopic to ˛ on S (i.e. if and only if ˇ1 D ˛1 ). Geometric intersection number is an effective obstruction for isotoping curves to one another. We shall use this obstruction repeatedly throughout this chapter. More precisely, we shall use the following two propositions.

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Proposition 2.4. Let ˛ and ˇ be essential curves on S. Then ˛ is not isotopic to ˇ if and only if there exists a curve on S such that i.˛; / D 0 and i.ˇ; / ¤ 0. Proof. If ˛ is isotopic to ˇ on S , then i.˛; / D i.ˇ; / for every curve on S. Assume now that ˛ is not isotopic to ˇ on S. First, suppose that i.ˇ; ˛/ ¤ 0. Then, ˛ is a curve on S such that i.˛; ˛/ D 0 and i.ˇ; ˛/ ¤ 0. Now, suppose that i.ˇ; ˛/ D 0. Then there exists a curve ˇ1 on S which is isotopic to ˇ and disjoint from ˛. Therefore, ˛ and ˇ1 are disjoint nonisotopic essential curves on S. Let C D f˛; ˇ1 g. By Proposition 2.3, there exists a curve on S such that i.˛; / D 0 and i.ˇ; / ¤ 0. Therefore, in any case, there exists a curve on S such that i.˛; / D 0 and i.ˇ; / ¤ 0. Proposition 2.5. Suppose that S is not a sphere with at most four holes or a torus with at most one hole. Let ˛ and ˇ be nonisotopic essential curves on S. Then there exists a curve on S such that is not isotopic to ˛, i.; ˛/ D 0 and i.; ˇ/ ¤ 0. Proof. Since S contains an essential curve, S is not a sphere with at most three holes. Since S is also not a closed torus, we can extend ˛ to a pants decomposition F of S . For each curve ı in F n f˛g choose a curve ı 0 on S such that ı 0 is disjoint from every curve in F n fıg and has nonzero geometric intersection with ı; see Figure 2 again. Let F 0 be the union of F n f˛g with fı 0 j ı 2 F n f˛gg. Suppose that i.; ˇ/ D 0 for each curve of F 0 . Without loss of generality, we may assume that F 0 [ fˇg is in minimal position. Since S is not a sphere with four holes or a torus with one hole, then each component of the surface obtained from S by cutting S along F 0 is either a disk or an annulus on S containing a boundary component of S, or an annulus on S containing ˛, or a pair of pants on S containing ˛ such that two of its boundary components are boundary components of S. Since i.; ˇ/ D 0 for each curve of F 0 and F 0 [ fˇg is in minimal position, ˇ is contained in the interior of one of the components M of the cut surface. Since ˇ is an essential curve on S contained in M , M cannot be a disk on S or an annulus on S containing a boundary component of S. Hence, M is either an annulus on S containing ˛, or a pair of pants on S containing ˛ such that two of its boundary components are boundary components of S . Since ˛ and ˇ are essential curves on S contained in M , it follows that ˇ is isotopic to ˛. This is a contradiction. Hence, i.; ˇ/ ¤ 0 for some curve of F 0 . Note that each curve of F 0 is disjoint from ˛ and not isotopic to ˛ on S . Hence, is not isotopic to ˛, i.; ˛/ D 0, and i.; ˇ/ ¤ 0.


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2.5 Orientation Proposition 2.6. Suppose that S is not a sphere with at most three holes and let H W S ! S be a homeomorphism of S. If H preserves the isotopy class of every essential curve on S , then H is orientation-preserving. Proof. First, consider the case where the genus of S is zero. Since S is not a sphere with at most three holes, there exists a sphere with four holes X embedded in S such that three of the four boundary components of X are boundary components C1 , C2 , and C3 of S and the remaining boundary component C0 of X is either a boundary component of S or an essential curve on S . As in Section 4.2 of [60], we recall the lantern relation discovered by M. Dehn [25] and rediscovered and popularized by D. Johnson [63]. To do this, we choose an orientation on S and we let ˛ij D ˛j i , 1 i < j 3 be an arc on S joining Ci to Cj . We can suppose that ˛12 , ˛23 , and ˛31 are disjoint. The surface obtained from X by cutting along ˛12 [ ˛23 [ ˛31 contains a unique component D which is a disc. Suppose that D is on the left of ˛12 as we travel along ˛12 from C1 to C2 , as in Figure 3.

C0 C1

C12

C31

˛12 C3

˛31 ˛23

C2

C23

Figure 3. The lantern relation: t0 B t1 B t2 B t3 D t12 B t23 B t31 , where ti (respectively tj k ) is the isotopy class of a twist map about Ci (respectively Cj k ).

Let Cij D Cj i be the unique essential boundary component of a regular neighborhood Pij D Pj i in X of Ci [ ˛ij [ Cj . Let Ti W S ! S and Tj k D Tkj W S ! S denote right Dehn twist maps supported on regular neighborhoods on S of Ci and Cj k . Let ti and tj k D tkj be the isotopy classes of Ti and Tj k D Tkj respectively. Then, we have the following relation, usually called the lantern relation: t0 B t1 B t2 B t3 D t12 B t23 B t31 D t23 B t31 B t12 D t31 B t12 B t23 :

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Since C1 , C2 , and C3 are boundary components of S , it follows that for i D 1; 2; 3, ti is equal to the identity element of .S/. Hence t0 D t12 B t23 B t31 D t23 B t31 B t12 D t31 B t12 B t23 : Let h 2 .S/ be the isotopy class of H W S ! S . Suppose for contradiction that H W S ! S is orientation-reversing. The mapping class h B tij B h1 is equal to sij where sij is a left Dehn twist about H.Cij /. On the other hand, since H preserves the isotopy class of every essential curve on S, H.Cij / is isotopic on S to Cij . It follows that sij D tij1 . Likewise, if C0 is essential on S , then h B t0 B h1 D t01 . On the other hand, if C0 is a boundary component of S , then t0 is equal to the identity element of .S / and, hence, h B t0 B h1 D t01 . In any case, h B t0 B h1 D t01 . We conclude that 1 1 1 t01 D h B t0 B h1 D h B .t12 B t23 B t31 / B h1 D t12 B t23 B t31 : This implies t0 D t31 B t23 B t12 : It follows from the above equations that t31 B t12 B t23 D t31 B t23 B t12 which implies t12 B t23 D t23 B t12 : Hence, the right Dehn twists t12 and t23 about the essential curves C12 and C23 on S commute. It follows from Lemma 4.3 of [96] that i.C12 ; C23 / D 0. Since this geometric intersection number is equal to two, this is a contradiction. Hence, H is orientation-preserving. Now the case where the genus of S is positive is handled similarly, by using a torus with one hole on S instead of a sphere with four holes on S. Suppose that S has positive genus. Then there exist transverse essential curves ˛ and ˇ on S such that ˛ and ˇ have exactly one point of intersection. Let T˛ W S ! S be a homeomorphism representing t˛ and be the image of ˇ under T˛ . Then, since T˛ is orientation-preserving, t˛ B tˇ B t˛1 D t :

˛ ˇ

Figure 4. A conjugation relation: t˛ B tˇ B t˛1 D t .


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Since ˇ is essential on S and T˛ W S ! S is a homeomorphism, is essential on S . Assume for contradiction that H is orientation-reversing. Then, as before, by conjugating by h, we conclude that t˛1 B tˇ1 B t˛ D t1 : This implies

t˛1 B tˇ B t˛ D t :

It follows from the above equations that t˛ B tˇ B t˛1 D t˛1 B tˇ B t˛ which implies

t˛2 B tˇ D tˇ B t˛2 :

As before, it follows from Lemma 4.3 of [96] that i.˛; ˇ/ D 0. Since this geometric intersection number is equal to one, this is a contradiction. Hence, H is orientationpreserving. In any case, H is orientation-preserving.

2.6 Domains A subsurface X of S is a surface with boundary contained in S such that every boundary component of X is either a boundary component of S or disjoint from the boundary of S . A domain on S is a connected compact subsurface X of S which is not equal to S and each of whose boundary components is either contained in @S or is essential on S . The peripheral boundary components of X are those which are contained in @S. The following properties of domains follow easily from the definitions: Proposition 2.7. Let X be a domain on S . Then: • X is not a disk; • no boundary component of X bounds a disk on S ; • there does not exist an annulus on S whose boundary is equal to the union of a boundary component of X with a boundary component of S; • X has at least one essential boundary component on S. Let C be a curve on S. A regular neighborhood of C on S is an annulus R in the interior of S such that C is a curve on R which is homotopic to each of the two boundary components of R. A regular neighborhood of a curve C on S is a domain on S if C is an essential curve on S . We say that a domain on S is peripheral if it has at least one peripheral boundary component. We say that a domain on S is monoperipheral if it has exactly one

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peripheral boundary component, and biperipheral if it has exactly two peripheral boundary components. More generally, for k 0, we say that a domain on S is k-peripheral if it has exactly k peripheral boundary components. Let X be a domain on S . The inside of X, denoted by X , is the complement in X of the union of the essential boundary components of X. Let f@i j 1 i ng be the collection of all essential boundary components of X on S . Let A D fAi j 1 i ng be a collection of disjoint annuli on X such that Ai \ @X D @i , 1 i n. Let Y be the closure in X of the complement of the annuli in A. We say that Y is obtained from X by shrinking X on S. Note that Y is a domain on S which is contained in the inside X of X on S and that Y is isotopic to X on S . Proposition 2.8. Let X be a domain on S and Y be a subsurface of X . If Y is a domain on X, then Y is a domain on S . Proof. By Proposition 2.7, S is not a disk and no boundary component of X bounds a disk on S . Clearly Y is a compact, connected, orientable subsurface of S which is not equal to S . Let @ be a boundary component of Y . Since Y is a domain on X , @ is either a boundary component of X or an essential curve on X . Suppose, on the one hand, that @ is a boundary component of X. Since X is a domain on S , it follows that @ is either a boundary component of S or an essential curve on S . Suppose, on the other hand, that @ is an essential curve on X . In particular, @ is in the interior of X. Hence, @ is in the interior of S . Suppose that @ is not an essential curve on S . Then @ either bounds a disk D on S or cobounds an annulus A on S with a boundary component of S . Suppose that @ bounds a disk D on S. Since no boundary component of X bounds a disk on S , no boundary component of X can be contained in D. Hence, D is disjoint from the boundary of X but intersects the interior of X, since it contains @. Since D is connected, this implies that D is contained in X. Since @ is an essential curve on X, this is a contradiction. Hence, @ does not bound a disk on S. Hence, @ cobounds an annulus A on S with a boundary component of S. Since no boundary component of X can bound a disk on S or cobound an annulus with the boundary component of S, no boundary component of X can be contained in the interior of A. Hence, An is disjoint from the boundary of X but intersects the interior of X, since it contains @. Since A n is connected and X is compact, this implies that A is contained in X. As before, this is a contradiction, and, hence, @ is an essential curve on S . This shows that each boundary component of Y is either a boundary component of S or an essential curve on S, completing the proof that Y is a domain on S. Proposition 2.9. Let X be a domain on S and let ˛ and ˇ be curves on X . Then the geometric intersection number of ˛ and ˇ in X is equal to the geometric intersection number of ˛ and ˇ in S.


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Proof. Let m be equal to the geometric intersection number of ˛ and ˇ in X and n be equal to the geometric intersection number of ˛ and ˇ in S. Without loss of generality, we may assume that ˛ and ˇ are transverse with exactly m points of intersection. Then there does not exist a disk on X whose boundary is the union of an arc of ˛ and an arc of ˇ. Since ˛ and ˇ meet transversely at m points, we have m n. Suppose that m > n. Then there exists a disk D on S whose boundary @D is the union of an arc of ˛ and an arc of ˇ. Let C be a component of @X. Since C is connected and disjoint from @D, C is either contained in the interior of D or the complement of D in S . Since C does not bound a disk on S , it follows that C is in the complement of D. It follows that D is disjoint from @X. Since D is connected and disjoint from @X , D is either contained in the interior of X or the complement of X on S. Since @D is contained in the interior of X, it follows that D is contained in the interior of X . Hence, D is a disk on X whose boundary @D is the union of an arc of ˛ and an arc of ˇ. This is a contradiction. Thus, m n and, hence, m D n. Note that one can also prove Proposition 2.9 using hyperbolic geometry. Proposition 2.10. Let X be a domain on S and ˛ be an essential curve on X. Then there exists an essential curve ˇ on X such that the geometric intersection number of ˛ and ˇ on S is not equal to zero. Proof. Since ˛ is an essential curve on X , the domain X is not a sphere with at most three holes. Then, there exists a domain X 0 on X which is a sphere with four holes or a torus with one hole, such that ˛ is an essential curve on X 0 . We can find on X 0 a curve with the required properties, and use Proposition 2.9. Proposition 2.11. Let X and Y be domains on S . Suppose that X is isotopic on S to a domain on Y . Then Y is not isotopic on S to a domain on X . Proof. Let X1 be a domain on Y such that X is isotopic to X1 on S. Suppose that Y is isotopic on S to a domain Y1 on X. It follows that X is isotopic to a domain X2 on S such that Y is a domain on X2 . Since X1 is a domain on Y and Y is a domain on X2 , it follows from Proposition 2.8 that X1 is a domain on X2 . Thus, X1 has an essential boundary component ˛ on X2 . Since ˛ is an essential curve on X2 , it follows from Proposition 2.10 that there exists an essential curve ˇ on X2 such that the geometric intersection number of ˛ and ˇ on S is not equal to zero. Since X1 is isotopic to X2 on S, the essential boundary component ˛ of X1 on S is isotopic to an essential boundary component of X2 on S . Hence, ˛ is isotopic to a curve ˛1 on S which is disjoint from X2 . It follows that i.˛; ˇ/ D i.˛1 ; ˇ/. Since ˛1 is disjoint from X2 and ˇ is contained in X2 , ˛1 is disjoint from ˇ and, hence, i.˛1 ; ˇ/ D 0. Hence, i.˛; ˇ/ D 0, which is a contradiction. Hence, Y is not isotopic on S to a domain on X.

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Proposition 2.12. Let X be a domain on S and let Y be the complement of the interior of X in S . Then: (1) if ˛ is an essential curve on X , then ˛ is an essential curve on S; (2) if ˛ is an essential curve on X, then ˛ is not isotopic to any curve on S contained in Y ; (3) if U and V are distinct components of Y and ˛ is an essential curve on U , then ˛ is not isotopic to any curve on V ; (4) if U is a component of Y and U is isotopic to X, then U and X are annuli meeting along their common boundary, Y D U , and S is a closed torus; The proof is easy, and we leave it to the reader. The following is a weak converse for Proposition 2.8. Proposition 2.13. Let X be a domain on S and Y be a subsurface of X . If Y is a domain on S, then Y is isotopic on S to either X , or a domain on X , or a regular neighborhood of an essential boundary component of X . Proof. Assume that Y is not isotopic on S to either X or a regular neighborhood of an essential boundary component of X. Let @ be a boundary component of Y . We may assume that @ is not a boundary component of X. Hence, @ is not a boundary component of S . Since Y is a domain on S, it follows that @ is an essential curve on S . Suppose that @ is not an essential curve on X. Since @ is an essential curve on S it cannot bound a disk on S . Hence, it cannot bound a disk on X. Hence, @ must cobound an annulus A on X with a boundary component of X. Since @ is essential on S and A is an annulus on S, is not a boundary component of S . Hence, since X is a domain on S , is an essential boundary component of X on S . Since Y is contained in X and the interior of Y is disjoint from the complement of @ [ in A, the interior of Y is contained in either the interior of A or the complement of A in X. Hence, Y is contained in either A or the closure of the complement of A in X. Suppose that Y is contained in A. Since Y is a domain on S and A is contained in the interior of S, it follows that Y is isotopic on S to A. Since Y is not isotopic to a regular neighborhood of an essential boundary component of X on S , this is a contradiction. Hence, Y is contained in the closure of the complement of A in X . Let Y 0 D Y [ A. Note that Y 0 is a domain on S which is isotopic to Y on S, is contained in X, and has one less boundary component in the interior of X than Y . It follows, by induction, that Y is isotopic to a domain on S which is contained in X and which has all of its boundary components in the boundary of X . In other words, Y is isotopic on S to X .


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The converse of Proposition 2.13 follows immediately from Propositions 2.8 and 2.13. Hence, we have the following equivalence. Proposition 2.14. Let X be a domain on S and Y be a subsurface of X. Then, Y is a domain on S if and only if Y is isotopic on S to either X, or a domain on X , or a regular neighborhood of an essential boundary component of X . Definition 2.15. Let X, J , and Y be disjoint domains on S . We say that X is tied to Y by J if the following three properties are satisfied (see Figure 5): (1) X and J have exactly one common boundary component; (2) J and Y have exactly one boundary component; (3) the interior of J is disjoint from X [ Y .

X

J

Y

Figure 5. X is tied to Y by J .

As an example, suppose that S is a closed torus. Then every domain on S is an annulus and any two disjoint domains on S, X and Y , are tied to one another by exactly two annuli on S, J and K. Moreover, S D X [ J [ Y [ K. Proposition 2.16. Let X and Y be disjoint domains on S . Then X is isotopic to Y on S if and only if X and Y are annuli and X is tied to Y by an annulus on S . We leave the proof to the reader. Definition 2.17. Let X be a domain on S. We say that X is elementary if it is either an annulus or a pair of pants on S . Otherwise, we say that X is nonelementary. Proposition 2.18. Let X be a domain on S . Then X is a nonelementary domain on S if and only if there exist curves ˛ and ˇ on S such that i.˛; ˇ/ ¤ 0 and ˛ and ˇ are contained in the interior of X. Proof. Suppose, on the one hand, that X is elementary and ˛ and ˇ are curves on S contained in the interior of X. Then ˛ is isotopic on S to a boundary components @ of X. Since ˇ is in the interior of X, @ and ˇ are disjoint. Hence, i.˛; ˇ/ D i.@; ˇ/ D 0.

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Suppose, on the other hand, that X is nonelementary. Since X is a domain on S , X has at least one essential boundary component on S . Suppose that the genus of X is positive. Then there exists an embedded torus with one hole Y in X which is either equal to X or is a domain on X. Let ˛ and ˇ be curves on Y which intersect transversely and have exactly one point of intersection. Then, by Proposition 2.9, iS .˛; ˇ/ D iX .˛; ˇ/ D iY .˛; ˇ/ D 1. Suppose now that the genus of X is zero. Since X is a domain on S , X has at least one essential boundary component on S . In particular, X is not a disc. Since X is nonelementary, X is not an annulus or a pair of pants. Hence, X has at least four boundary components. Thus there exists an embedded sphere with four holes Y in X which is either equal to X or is a domain on X . Let ˛ and ˇ be curves on Y which intersect transversely, have exactly two points of intersection, and are such that the complement of ˛ [ ˇ in Y has exactly four components, each of which contains exactly one boundary component of Y . Then, by Proposition 2.9, iS .˛; ˇ/ D iX .˛; ˇ/ D iY .˛; ˇ/ D 2. Definition 2.19. Let F be a collection of pairwise disjoint domains on S. Let Y be the closure of the complement of jF j in S. A codomain of F is a components of Y . Note that the codomains of a collection of pairwise disjoint domains on S are themselves domains on S . Proposition 2.20. Let X be a domain on S and Y be a codomain of X on S . If Y is an annulus, then Y is a nonseparating annulus on S and both boundary components of Y are essential boundary components of X . Proof. Since Y is a codomain of X , every essential boundary component of Y on S is a boundary component of X , and Y has at least one boundary component which is also a boundary component of X. Let @1 be such a boundary component of Y and let @2 be the other boundary component of Y . Since @1 is a boundary component of X, it is essential on S, which implies that @2 is not a boundary component of S . Thus, @2 is also a boundary component of X, and therefore both @1 and @2 are essential boundary components of X. Since X is a connected subset of S and Y is a codomain of X on S , there is a path in S connecting the two boundary components of Y and whose image is in the complement of the interior of Y in S . This shows that Y is nonseparating on S . Corollary 2.21. Suppose that X is a domain on S and X is an annulus. Then X has a codomain which is an annulus if and only if S is a closed torus. Proof. Suppose that X is a domain on S , that X is an annulus, and that X has a codomain Z which is an annulus. By Proposition 2.20, X and Z have their two boundary components in common. Since S is orientable, the union of X and Z is a closed torus. Since S is connected, S is equal to that torus. The converse is clear.


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A nonempty collection of pairwise disjoint domains on S is called a system of domains on S if the domains in the collection are pairwise nonisotopic. The following is an immediate corollary of Proposition 2.16. Corollary 2.22. Let F be a collection of pairwise disjoint domains on S . Then the following are equivalent: (1) F is a system of domains on S; (2) no two distinct annuli in F are isotopic on S ; (3) no two distinct annuli in F are tied to one another by an annulus on S . The collection of codomains of a system of domains on S is a collection of pairwise disjoint domains. However, the collection of codomains of a system of domains on S is not necessarily a system of domains, since two distinct such codomains may be isotopic.

Figure 6. An example of a system of domains (the three non-shaded pieces) and their codomains (the two shaded pieces). Two distinct codomains of a system of domains might be isotopic, as in this figure.

Proposition 2.23. Suppose that F is a collection of disjoint domains on S. Then F is a system of domains on S if and only if there does not exist two distinct annular domains of jF j which are joined by an annular codomain of jF j.

3 Simplicial complexes 3.1 Abstract simplicial complexes In this section, we introduce some standard terminology from the theory of abstract simplicial complexes, and we complement it for later use in this chapter. A reference for some of the classical material is the book [105] by Munkres.

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Definition 3.1. Let V be a set. An abstract simplicial complex K with vertex set V is a collection of nonempty finite subsets of V such that: (1) if v 2 V , then fvg 2 K; (2) if is an element of K and is a subset of , then is an element of K. In this chapter, the term simplicial complex shall refer to an abstract simplicial complex, unless otherwise specified. Let K be a simplicial complex with vertex set V . If x is an element of V , then we say that x is a vertex of K. If is an element of K and x is an element of , then we say that is a simplex of K and x is a vertex of . Note that each vertex of each simplex of K is a vertex of K. If has k C 1 vertices, then we say that is a k-simplex of K. If x is an element of V , then we also say that the corresponding 0-simplex fxg of K is a vertex of K. If e is a 1-simplex of K, then we say that e is an edge of K. If

is a 2-simplex of K, then we say that is a triangle of K. Definition 3.2. Let K be a simplicial complex and let F be a subcollection of K. We say that F is a subcomplex of K if each subset of an element of F is an element of F . If F is a subcomplex of a simplicial complex K, then F is itself a simplicial complex, and the vertex set of F is a subset of the vertex set of K. Proposition 3.3. Let K be a simplicial complex with vertex set V and W be a subset of V . Let KW be the set of all simplices of K that have all of their vertices in W . Then KW is a subcomplex of K with vertex set W . Definition 3.4. Let K, W , and KW be as in Proposition 3.3. We say that KW is the subcomplex of K induced by the subset W of the set of vertices V of K. Note that the subcomplex of a simplicial complex induced by a subset of its vertices is itself a simplicial complex. Moreover, it is completely determined by the simplicial complex K and its vertex set W . Let K be a simplicial complex. For each nonnegative integer n, the n-skeleton Kn of K is the subcomplex of K consisting of all k-simplices of K with k n. Its vertex set is equal to the support jKn j of Kn (i.e. the union of all the k-simplices of K with k n). The 1-skeleton of a simplicial complex K is a graph, sometimes called the underlying graph of K. Note that if F is a subcomplex of an abstract simplicial complex K and n is any nonnegative integer, then the n-skeleton Fn of F is a subcomplex of the n-skeleton Kn of K. If 2 K, then we say that is a face of . A maximal simplex of a simplicial complex K is a simplex which is not a proper face of any simplex of K.


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The simplicial complex K is finite-dimensional if there exists an integer N such that every simplex of K is a k-simplex for some k N . If K is finite-dimensional, then the dimension of K is the minimum such integer N . If the dimension of K is N , then a top-dimensional simplex of K is an N -simplex of K. A simplicial complex of dimension one is a simplicial graph, or, more briefly, a graph. Definition 3.5. A simplicial complex K is a flag complex if the following holds: If fx0 ; : : : ; xn g is a subset of K0 such that fxi ; xj g is an edge of K for all 0 i < j n, then fx0 ; : : : ; xn g is a simplex of K. Definition 3.6. Let ˛, ˇ, and ı be simplices of a simplicial complex K. We say that ˛ is joined to ˇ by ı if ı D ˛ [ ˇ. Note that if ˛ is joined to ˇ by simplices ı and of K, then ı D . Note also that two simplices of a simplicial complex are joined by a vertex of that simplicial complex if and only if they are both equal to that vertex. Definition 3.7 (The join of two simplices). Let ˛ and ˇ be simplices of a simplicial complex K which are joined in K by a simplex ı of K. Then we say that ı is the join of ˛ and ˇ in K, and we write ı D ˛ ˇ. Definition 3.8 (The star of a simplex). Let ˛ be a simplex of a simplicial complex K. The star of ˛ in K is the subcomplex St.˛/ D St.˛; K/ of K whose simplices are the simplices of K which contain the simplex ˛ together with all the faces of such simplices of K. In particular, if x is a vertex of a simplicial complex K, then the star of x in K is the subcomplex St.x; K/ of K whose simplices are the simplices of K which contain the vertex x together with all the faces of such simplices of K. Let K be a simplicial complex and ˛ be a simplex of K. Note that the 0-skeleton St 0 .˛; K/ of St.˛; K/ is the set of all vertices w of K such that ˛ [ fwg is a simplex of K. Proposition 3.9. Let K be a flag complex. Let ˛ and ˇ be simplices of K. Then the following are equivalent: (1) St.˛; K/ D St.ˇ; K/. (2) St0 .˛; K/ D St 0 .ˇ; K/. Proof. Clearly (1) implies (2). We shall now show that (2) implies (1). To this end, suppose that (2) holds and let be a simplex of St.˛; K/. We need to show that is a simplex of St.ˇ; K/. In other words, we need to show that ˇ [ is a simplex of K. Since K is a flag complex, it suffices to show that any two distinct vertices of ˇ [ are joined by an edge of K. To this end, let x and y be distinct vertices of ˇ [ . If x

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and y are both vertices of the simplex ˇ of K, then fx; yg is an edge of the simplex ˇ of K and, hence, an edge of K. Likewise, if x and y are both vertices of the simplex of K, then fx; yg is an edge of the simplex of K and, hence, an edge of K. Suppose that x is a vertex of ˇ and y is a vertex of . Since is a simplex of St.˛; K/, y is a vertex of St.˛; K/ and, hence, of St.ˇ; K/. Since y is a vertex of St.ˇ; K/, ˇ [ fyg is a simplex of K. Since x is a vertex of ˇ, it follows that fx; yg ˇ [ fyg. Since fx; yg is a face of the simplex ˇ [ fyg of K, it follows that fx; yg is a simplex of K. Hence, every two distinct vertices of ˇ [ are joined by an edge of K. Since K is a flag complex, this implies that ˇ [ is a simplex of K. In other words, is a simplex of St.ˇ; K/. This proves that St.˛; K/ is a subcomplex of St.ˇ; K/. By a symmetric argument, it follows that St.ˇ; K/ is a subcomplex of St.˛; K/. This proves that St.˛; K/ D St.ˇ; K/. Definition 3.10 (The link of a simplex). Let be a simplex of a simplicial complex K. The link of in K is the subcomplex Lk. / D Lk. ; K/ of K whose simplices are the simplices of St. ; K/ which have empty intersection with . The proof of the following proposition is similar to that of Proposition 3.9. Proposition 3.11. Let K be a flag complex. Let ˛ and ˇ be simplices of K. Then the following are equivalent: (1) Lk.˛; K/ D Lk.ˇ; K/. (2) For each vertex x of K, x is a vertex of Lk.˛; K/ if and only if x is a vertex of Lk.ˇ; K/. Note that if D ;, then Lk. ; K/ D St. ; K/ D K. Suppose that is a simplex of a simplicial complex K. Note that is joined to each of the simplices of Lk. /; and the simplices of St. / are precisely the faces of the joins of with the simplices of Lk. /. In particular, if x is a vertex of a simplicial complex K, then the link of x in K is the subcomplex Lk.x; K/ of K whose simplices are the simplices of St.x; K/ that do not have x as a vertex. Suppose that x is a vertex of a simplicial complex K. Note that fxg is joined to each of the simplices of Lk.x; K/, and the simplices of St.x; K/ are precisely the faces of the joins of fxg with the simplices of Lk.x; K/. Definition 3.12 (The link of a subcomplex). Let F be a subcomplex of a simplicial complex K. The link of F in K, denoted by Lk.F; K/, is defined to be T fLk. ; K/ j 2 F g:


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Remark 3.13. If is the empty simplex of K, then F . / is the empty subcomplex of K and Lk. ; K/ D K D Lk.F . /; K/. Hence, there is no ambiguity regarding the “link of the empty set”. Remark 3.14. Let F be a subcomplex of K and be a simplex of F . Then Lk.F; K/ Lk. ; K/. Remark 3.15. Let F and G be subcomplexes of a simplicial complex K. If F G, then Lk.G; K/ Lk.F; K/. Proposition 3.16. Let x be a vertex of a simplicial complex K. If there exists a simplex

in K such that Lk. / D fxg, then Lk.Lk.x// D fxg. Proof. Suppose that there exists a simplex in K such that Lk. / D fxg. Then, in particular, x 2 Lk. /. Hence, is a simplex of Lk.x/. This implies that Lk.Lk.x// Lk. /. Hence, Lk.Lk.x// fxg. On the other hand, x 2 Lk.Lk.x//. Thus, Lk.Lk.x// D fxg. Definition 3.17. Let K be a simplicial complex with vertex set V and L be a simplicial complex with vertex set W . A simplicial map from K to L is a map ' W V ! W such that, for each simplex of K, '. / is a simplex of L. Let ' W V ! W be a simplicial map from a simplicial complex K with vertex set V to a simplicial complex L with vertex set W . The rule 7! '. / determines a map from K to L. We denote this map by ' W K ! L and we say that ' W K ! L is a simplicial map from K to L. If we need to distinguish between ' W V ! W and ' W K ! L, then we shall say that ' W V ! W is the vertex correspondence associated to ' W K ! L. Note that the map ' W K ! L is both determined by and determines the map ' W V ! W . The map ' W K ! L is injective if and only if ' W V ! W is injective. If ' W K ! L is surjective, then ' W V ! W is surjective. The converse, however, is not necessarily true. For instance, if L has at least one edge e and K is equal to the zero skeleton L0 of L, then the vertex set V of K is equal to the vertex set W of L, the identity map ' W V ! W is surjective, but the corresponding map ' W K ! L is not surjective, since the edge e of L is not in the image of ' W K ! L. If ' W K ! L is a simplicial map, then '.K/ is a subcomplex of L. Definition 3.18. Let K and L be abstract simplicial complexes. A simplicial isomorphism ' W K ! L is a simplicial map ' W K ! L for which there exists a simplicial map W L ! K such that ' W K ! L and W L ! K are inverse functions. In the case where K D L, we call a simplicial isomorphism ' W K ! L a simplicial automorphism of K. Note that a simplicial map ' W K ! L is a simplicial isomorphism if and only if it is bijective. By the previous observations, if ' W K ! L is bijective, then ' W V ! W is also bijective. The converse need not be true.

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The following naturality property follows easily from the definitions: Proposition 3.19. Let K be a simplicial complex, ' 2 Aut.K/, and x be a vertex of K. Then '.St.x; K// D St.'.x/; K/ and '.Lk.x; K// D Lk.'.x/; K/.

3.2 Exchange automorphisms We shall use the following notion of exchange automorphisms of abstract simplicial complexes, and the related results. Definition 3.20 (Simple exchange automorphism). Let K be a simplicial complex, fx; yg be a pair of vertices of K, and ' W K ! K be an automorphism of K. We say that ' is a simple exchange of K exchanging the vertices x and y of K if '.x/ D y, '.y/ D x, and '.z/ D z for every vertex z of K which is neither equal to x nor equal to y. Let ' W K ! K be a simple exchange of a simplicial complex K exchanging the vertices x and y of K. Note that ' W K ! K is equal to the identity map idK W K ! K of K if and only if x D y. In this case, we say that ' is a trivial simple exchange. Otherwise, ' is said to be a nontrivial simple exchange. Of course, not all simplicial complexes admit nontrivial simple exchanges. Let K be a simplicial complex, ' W K ! K be a simple exchange of K exchanging the vertices x and y of K, and W K ! K be a simple exchange of K exchanging the vertices u and v of K. Then ' D if and only if either x D y and u D v or fx; yg D fu; vg. In particular, a nontrivial simple exchange of K exchanges a unique pair of distinct vertices of K. Example 3.21. Let K.n/ denote the simplicial complex of all subsets of the set f1; : : : ; ng. Then, for every pair of distinct vertices i and j of K.n/, the standard transposition .i; j / in the group of permutations of f1; : : : ; ng extends to a simple exchange of K.n/ which exchanges i and j . These simple exchanges generate the group of simplicial automorphisms of K.n/, which is naturally isomorphic to the symmetric group †n , the group of permutations of the vertex set f1; : : : ; ng of K.n/. Definition 3.22 (Exchangeable vertices). Let K be a simplicial complex and let x and y be two vertices of K. We say that x and y are exchangeable in K if there exists a simple exchange of K exchanging x and y. Example 3.23. Let V D Z f1; 0; 1g and K be the one-dimensional simplicial complex on V , illustrated in Figure 8, whose edges are the pairs f.m; 0/; .m C 1; 0/g and f.m; 0/; .m; /g with m 2 Z and 2 f1; 1g. Then two distinct vertices x and y of K are exchangeable if and only if fx; yg D f.m; 1/; .m; 1/g for some m 2 Z. Note that for any subset W of Z there is a unique automorphism 'W W K ! K such

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Figure 7. Three complexes: The one to the left has all its vertices exchangeable. The one in the middle has no distinct vertices exchangeable. The one to the right has exactly one pair of distinct vertices that are exchangeable.

that 'W .m; t/ is equal to .m; t / if m 2 W and .m; t / otherwise. In particular, '; D idK W K ! K. If U and V are subsets of Z, then 'U B 'V D 'U 4V (where 4 denotes symmetric difference). In particular, 'W B 'W D idK W K ! K. It follows that the collection f'W jW Zg of automorphisms of K is a subgroup BK of the automorphism group, Aut.K/, of K, which is naturally isomorphic to the Boolean algebra B.Z/ of all subsets of Z.

Figure 8. The complex used in Example 3.23: a line of edges.

Let DK be the group of automorphisms of K generated by the translation .m; n/ 7! .m C 1; n/ and the involution .m; n/ 7! .1 m; n/. Note that this involution has no fixed vertices in K. The subgroup DK of Aut.K/ is naturally isomorphic to the infinite dihedral group D1 of isometries of Z equipped with its standard metric. The group of automorphism of K, Aut.K/, is a split extension of its subgroup DK by its normal subgroup BK , and we have the following commutative diagram: 1

/ B.Z/ '

1

/ BK

/ B.Z/ Ì D1 '

/ Aut.K/

/ D1 D Isom.Z/

/1

'

/ DK

/ 1.

On the set of vertices of any simplicial complex, we define a relation, , called the exchange relation, in which x y if and only if x and y are exchangeable. Proposition 3.24. Let K be a simplicial complex. Then the exchange relation on K is an equivalence relation on K0 .

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Proof. Let x 2 K0 . Then the identity map idK W K ! K is a simplicial automorphism of K exchanging x and x. Hence, x x. Suppose that x; y 2 K0 and x y. Let ' W K ! K be a simplicial automorphism of K which exchanges x and y. Then the same automorphism ' W K ! K is a simplicial automorphism of K which exchanges y and x. Hence, y x. Suppose that x; y; z 2 K0 , x y, and y z. Let ' W K ! K be a simplicial automorphism of K which exchanges x and y and W K ! K be a simplicial automorphism of K which exchanges y and z. Then the conjugate ' B B ' W K ! K of W K ! K by ' W K ! K is a simplicial automorphism of K which exchanges x and z. Hence, x z. The following result gives a basic necessary and sufficient condition for two vertices of a simplicial complex to be exchangeable. Proposition 3.25. Let K be a simplicial complex and x and y be vertices of K. Let F be the subcomplex of K consisting of all simplices of K that have neither x nor y as a vertex. Then the following are equivalent: (1) x and y are exchangeable in K. (2) St.x; K/ \ F D St.y; K/ \ F . Proof. First, we prove that (1) implies (2). To this end, suppose that x and y are exchangeable in K. Then, there is a unique automorphism ' W K ! K such that '.x/ D y, '.y/ D x, and '.z/ D z for every vertex z of F . Suppose that is a simplex of St.x; K/\F . In other words, suppose that fxg[ is a simplex of K and is a simplex of F . Then '.fxg [ / D f'.x/g [ '. / D fyg [ is a simplex of K. Since fyg [ is a simplex of K and is a simplex of F , it follows that is a simplex of St.y; K/ \ F . This proves that St.x; K/ \ F St.y; K/ \ F . Likewise, St.y; K/ \ F St.x; K/ \ F and, hence, St.x; K/ \ F D St.y; K/ \ F . This proves that (1) implies (2). Now we prove that (2) implies (1). To this end, suppose that St.x; K/ \ F D St.y; K/ \ F . Consider the bijection ' W K0 ! K0 defined by the rule '.x/ D y, '.y/ D x, and '.z/ D z for every vertex z of F . Since ' W K0 ! K0 is an involution of K0 , it suffices to prove that ' extends to a simplicial map ' W K ! K. In other words, it suffices to prove that for every simplex of K, '. / is a simplex of K. To this end, suppose that is a simplex of K. If x and y are both vertices of , then '. / is equal to the simplex of K. Likewise, if neither x nor y is a vertex of , then '. / is equal to the simplex of K. Suppose that x is a vertex of and y is not a vertex of . Let D n fxg. Since x is a vertex of and D n fxg, it follows that D fxg [ . Since is a simplex of K, this implies that is a simplex of St.x; K/. Since y is not a vertex of and D n fxg, is a simplex of F . This implies that is a simplex of St.x; K/ \ F and, hence, of St.y; K/ \ F . Since is a simplex of St.y; K/, fyg [ is a simplex of K.


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Since is a simplex of F and D fxg[ , it follows that '. / D f'.x/g['. / D fyg [ . Hence, './ is a simplex of K. This shows that if x is a vertex of and y is not a vertex of , then '. / is a simplex of K. Likewise, if y is a vertex of and x is not a vertex of , then '. / is a simplex of K. In any case, './ is a simplex of K. This proves that ' W K0 ! K0 extends to a simplicial map ' W K ! K. Since ' W K0 ! K0 is an involution, its extension ' W K ! K is an involution. Hence, this extension ' W K ! K is a simplicial automorphism exchanging x and y. This proves that (2) implies (1). Remark 3.26. One can visualize the above exchangeability condition, Condition (2), as a sort of reflection across the hyperplane F defined in the statement of Proposition 3.25. The two vertices x and y of K can be reflected across F since they have been symmetrically joined to F along a subcomplex G of F (i.e. along F \ Lk.x; K/ D F \ Lk.y; K/) and, in the case where fx; yg is an edge of K, to one another. The following propositions are refinements of Proposition 3.25 corresponding to the situations where fx; yg is or is not an edge of K. Proposition 3.27. Let K be a simplicial complex and x and y be distinct vertices of K that are not connected by an edge of K. Then the following are equivalent: (1) x and y are exchangeable in K. (2) Lk.x; K/ D Lk.y; K/. Proof. Let F be the subcomplex of K consisting of all simplices of K which have neither x nor y as a vertex. Since x and y are not joined by an edge of K, it follows that Lk.x; K/ D St.x; K/ \ F and Lk.y; K/ D St.y; K/ \ F . Proposition 3.28. Let K be a flag simplicial complex and x and y be vertices of K which are connected by an edge of K. Then the following are equivalent: (1) x and y are exchangeable in K. (2) St.x; K/ D St.y; K/. Proof. Let F be, as before, the subcomplex of K consisting of all simplices of K which have neither x nor y as a vertex. Suppose that St.x; K/ D St.y; K/. Then St.x; K/ \ F D St.y; K/ \ F . It follows from Proposition 3.25 that x and y are exchangeable. This proves that (2) implies (1). We shall now show that (1) implies (2). Suppose that x and y are exchangeable in K. It follows from Proposition 3.25 that St.x; K/ \ F D St.y; K/ \ F . We must show that St.x; K/ D St.y; K/. To this end, suppose that is a simplex of St.x; K/. Let D [ fxg. Since is a simplex of St.x; K/, is also a simplex of K.

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Let D [ fyg. We shall show that is a simplex of K. Since K is a flag complex, it suffices to show that any two distinct vertices of are joined by an edge of K. To this end, let w and z be vertices of . If neither w nor z is equal to y, then w and z are vertices of the simplex of K and, hence, are joined by an edge of K. Hence, we may assume that z D y. This implies that w is not equal to y. It follows that x and w are both vertices of the simplex of K. Hence, w is a vertex of St.x; K/. If w D x, then w and z are vertices of the simplex fx; yg of K. Hence, we may assume that w is not equal to x. Since w is a vertex of St.x; K/ and w is not equal to x or y, it follows that w is a vertex of St.x; K/ \ F and, hence, of St.y; K/ \ F . It follows that w is a vertex of St.y; K/. Since w is not equal to y, this implies that w and y are joined by an edge of K. In other words, w and z are joined by an edge of K. In any case, w and z are joined by an edge of K. This shows that is a simplex of K. Since is a face of the simplex of K and y is a vertex of , it follows that is a simplex of St.y; K/. This shows that St.x; K/ St.y; K/. Likewise, St.y; K/ St.x; K/. Hence, St.x; K/ D St.y; K/. This proves that (1) implies (2). Proposition 3.29. Let K be a simplicial complex. Let E be a collection of exchangeable pairs of distinct vertices of K with the property that no two distinct pairs in E have a common vertex. Then there exists a unique automorphism 'E W K ! K such that (1) for each pair fx; yg in E, 'E .x/ D y and 'E .y/ D x; (2) 'E .z/ D z for every vertex z of K which is not an element of some pair in E. Proof. Let ' W K0 ! K0 be the unique involution which exchanges the two vertices in each pair in E and fixes every other vertex of K. Let be a simplex of K. We shall now show that './ is a simplex of K. Let 0 be the set of all vertices x of such that there does not exist a vertex y of K such that fx; yg 2 E, let 1 be the set of all vertices x of such that there exists a vertex y of K such that fx; yg 2 E and fx; yg \ D fxg and let 2 be the set of all vertices x of such that there exists a vertex y of K such that fx; yg 2 E and fx; yg \ D fx; yg. Note that D 0 [ 1 [ 2 . From the definition of ', '.i / D i , i D 0; 2. Let n be the number of elements of 1 . Suppose, on the one hand, that n D 0. Then, 1 D ; and, therefore, '. / D '.0 [ 2 / D '.0 / [ '.2 / D 0 [ 2 D . Hence, '. / is equal to the simplex of K. Suppose, on the other hand, that n > 0. Let 1 D fxj j 1 j ng. For each integer j with 1 j n, let yj be the unique vertex of K such that fxj ; yj g 2 E. From the definition of ', '.1 / D fyj j 1 j ng. It follows that '. / D 0 [ 2 [ fyj j 1 j ng.


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Let j be an integer with 1 j n. Since fxj ; yj g 2 E, fxj ; yj g is an exchangeable pair of vertices of K. Hence, there exists a simple exchange 'j W K ! K of K exchanging xj and yj . Since the distinct pairs fxj ; yj g, 1 j n, are in E, they are disjoint. It follows that the composition '1 B : : : B 'n W K ! K is an automorphism of K such that .0 / D 0 , .2 / D 2 and .xj / D yj , 1 j n. This implies that '. / D ./. Since W K ! K is an automorphism of K and is a simplex of K, it follows that ./ is a simplex of K; that is to say, '. / is a simplex of K. This shows that the involution ' W K0 ! K0 extends to a simplicial map ' W K ! K. Since ' W K0 ! K0 is an involution, its simplicial extension ' W K ! K is also an involution and, hence, a simplicial automorphism of K. Hence, ' W K ! K is a simplicial automorphism of K that satisfies Properties (1) and (2) of Proposition 3.29. Since these conditions on ' W K ! K determine the restriction ' W K0 ! K0 , and since any two simplicial maps which agree on the vertices of their common domain are equal, it follows that ' W K ! K is the unique such automorphism of K. Definition 3.30 (Generalized exchange). Let K, E, and 'E W K ! K be as in Proposition 3.29. We call the automorphism 'E W K ! K of K the generalized exchange of K associated to E. If F and G are subsets of E, then 'F B 'G D 'F 4G . Hence, by Proposition 3.29, we have the following result. Proposition 3.31. Let K be a simplicial complex. Let E be a collection of exchangeable pairs of distinct vertices of K with the property that no two distinct pairs in E have a common vertex. Then there exists a monomorphism ˆ from the Boolean algebra B.E/ of all subsets of E to Aut.K/ such that ˆ.F / D 'F for every subset F of E. Definition 3.32 (Boolean subgroup). Let K be a simplicial complex. Let E be a collection of exchangeable pairs of distinct vertices of K with the property that no two distinct pairs in E have a common vertex. The Boolean subgroup of Aut.K/ corresponding to E, denoted by BE , is the image ˆ.B.E// of the Boolean algebra B.E/ under the monomorphism ˆ of Proposition 3.31. In particular, the Boolean subgroup BE is naturally isomorphic to the Boolean algebra B.E/. Proposition 3.33. Let K be a simplicial complex. Let E be a collection of exchangeable pairs of distinct vertices of K with the property that no two distinct pairs in E have a common vertex. Let ' 2 Aut.K/, F E and G D '.F /. Then G is a collection of exchangeable pairs of distinct vertices of K with the property that no two distinct pairs in G have a common vertex. Moreover, ' B 'F B ' 1 D 'G .

3.3 Quotient complexes In this section, we develop a notion of quotient complex which will be used in our study of the complex of domains.

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We start with the following result on building a simplicial map out of a map defined on the vertex set of a simplicial complex. Proposition 3.34. Let K be a simplicial complex on the vertex set V , W be a set, and W V ! W be a map of V onto W . Let L be the collection of all subsets of W for which there exists a simplex of K such that D . /. Then L is a simplicial complex with vertex set W and W V ! W is a simplicial map from K to L Proof. First, we show that L is a simplicial complex. For this, we must show that each singleton subset of W is an element of the collection L and every subset of an element of L is an element of L. Since W V ! W is surjective, each element w of W is the image under of a vertex v of K. Thus, fwg D .fvg/. Since K is a simplicial complex and v is a vertex of K, fvg is a simplex of K. Hence, by the definition of L, fwg is an element of L. Suppose that is an element of L and that is a subset of . By the definition of L, there exists a simplex of K such that D . /. Let ı D 1 ./ \ . Since ı is contained in the simplex of the simplicial complex K, ı is a simplex of K. Since D . / and is contained in , it follows that D .ı/. Hence, by the definition of L, is an element of L. This shows that L is a simplicial complex. Next, we show that W V ! W is a simplicial map from K to L. To this end, let be a simplex of K and D . /. By the definition of L, is a simplex of L. This proves that W V ! W is a simplicial map from K to L. Definition 3.35 (Quotient complex). Let K, V , W , W V ! W , and L be as in Proposition 3.34. We say that L is the quotient complex of K and W K ! L is the natural projection associated to the vertex correspondence W V ! W . Definition 3.36 (Simplicial quotient map). Let K be a simplicial complex with vertex set V , L be a simplicial complex with vertex set W , and W K ! L be a simplicial map. We say that W V ! W is a simplicial quotient map if for every subset of W , is a simplex of L if and only if there exists a simplex of K for which . / D . Proposition 3.37. Let W V ! W be a simplicial quotient map from a simplicial complex K to a simplicial complex L. Then W V ! W maps V onto W . Proof. Let w be an element of W . Then fwg is a simplex of L. Since W K ! L is a simplicial quotient map, it follows that there exists a simplex of K such that fwg D . /. Thus, there exists a vertex x of such that w D .x/. Since x 2 V , x 2 V . Hence, there exists an element x of V such that w D .x/. This proves that W V ! W maps V onto W . Proposition 3.38. Let W V ! W be a simplicial quotient map from a simplicial complex K to a simplicial complex L and ˛ W V ! Z be a simplicial map from


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K to a simplicial complex M . Suppose that ˛ W V ! Z is constant on each fiber 1 .w/, w 2 W , of W V ! W (i.e. that ˛.x/ D ˛.y/ whenever .x/ D .y/, x; y 2 V ). Then there exists a unique simplicial map ˇ W W ! Z from L to M such that ˛ D ˇ B W V ! Z. Proof. By Proposition 3.37, W V ! W maps V onto W . Since W V ! W is surjective and ˛ W V ! Z is constant on the fibers of W V ! W , there exists a unique map ˇ W W ! Z such that ˛ D ˇ B W V ! Z. It remains only to show that ˇ W W ! Z is a simplicial map from L to M . To this end, suppose that is a simplex of L. Since W K ! L is a simplicial quotient map, there exists a simplex of K such that D . /. Thus ˇ./ D ˇ.. // D .ˇ B /. / D ˛. /. Since ˛ W K ! M is a simplicial map and is a simplex of K, it follows that ˛. / is a simplex of M ; that is to say, ˇ./ is a simplex of M . This proves that ˇ W W ! Z is a simplicial map from L to M , which completes the proof. Proposition 3.39. Let W V ! W and ˛ W V ! Z be simplicial quotient maps from a simplicial complex K to a simplicial complex L and a simplicial complex M respectively. Suppose that W V ! W and ˛ W V ! Z have the same fibers (i.e. for each pair of elements, x and y, of V , .x/ D .y/ if and only if ˛.x/ D ˛.y/). Then there exists a unique simplicial isomorphism ˇ W W ! Z from L to M such that ˛ D ˇ B W W ! Z. Proof. By Proposition 3.37, W V ! W and ˛ W V ! Z are surjective. By Proposition 3.38, there exist unique simplicial maps ˇ W W ! Z from L to M and W Z ! W from M to L such that ˛ D ˇ B and D B ˛. It remains only to show that ˇ W W ! Z and W Z ! W are inverse maps. To this end, let ı D B ˇ W W ! W and D ˇ B W Z ! Z. Note that ı B D . B ˇ/ B D B .ˇ B / D B ˛ D W V ! W . Since W V ! W is surjective and ı B D W V ! W , it follows that ı W W ! W is equal to the identity map of W . Likewise, W Z ! Z is equal to the identity map of Z. This proves that ˇ W W ! Z and W Z ! W are inverse maps, completing the proof. Proposition 3.39 shows that any two quotient complexes of a given simplicial complex corresponding to simplicial quotient maps with the same fibers are canonically isomorphic. We now construct a canonical model for the isomorphism class of any such quotient complex. Definition 3.40. Let be an equivalence relation on the vertex set V of an abstract simplicial complex K. Let Vz be the set of equivalence classes of on V and W V ! Vz be the associated natural projection which maps each vertex x of K to its equivalence class Œx D fy 2 V j y xg. Let Kz be the quotient complex of K and W K ! Kz be the natural projection associated to the vertex correspondence W V ! Vz . We say that Kz is the quotient of K by and W K ! Kz is the natural projection from K z to K.

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We have the following immediate corollary of Proposition 3.39. Proposition 3.41. Let ˛ W V ! Z be a simplicial quotient map from a simplicial complex K to a simplicial complex M . Let be the equivalence relation on V defined by the rule x y if and only if ˛.x/ D ˛.y/. Let Kz be the quotient of K by z Then there exists a unique and W K ! Kz be the natural projection from K to K. simplicial isomorphism ˇ W Vz ! Z from Kz to M such that ˛ D ˇ B W V ! Z. Definition 3.42 (Derived complex). Let be the exchange relation on the vertex set V of an abstract simplicial complex K. We denote the quotient complex Kz by K 0 and say that K 0 is the derived complex of K. Proposition 3.43. Let K be a simplicial complex and W K ! K 0 be the natural projection from K to the derived complex K 0 of K. Let ' W K ! K be an automorphism of K. Then there exists a unique automorphism ' 0 W K 0 ! K 0 such that B ' D '0 B W K ! K 0. Definition 3.44 (Derived automorphism). Let K, W K ! K 0 , ' W K ! K, and ' 0 W K 0 ! K 0 be as in Proposition 3.43. We say that the automorphism ' 0 W K 0 ! K 0 of K 0 is the automorphism of K 0 derived from ' W K ! K. Proposition 3.45. Let K be a simplicial complex and K 0 be its derived complex. Then there is a homomorphism W Aut.K/ ! Aut.K 0 / defined by the following rule: for each automorphism ' W K ! K of K, .'/ is equal to the automorphism of K 0 derived from ' W K ! K. Definition 3.46 (Derivation homomorphism). Let W Aut.K/ ! Aut.K 0 / be as in Proposition 3.45. We say that W Aut.K/ ! Aut.K 0 / is the derivation homomorphism from Aut.K/ to Aut.K 0 /. Definition 3.47 (Exchange automorphisms group). Let K be a simplicial complex, W Aut.K/ ! Aut.K 0 / be the derivation homomorphism from Aut.K/ to Aut.K 0 /. We call the kernel of W Aut.K/ ! Aut.K 0 / the group of exchange automorphisms of K and any element in this kernel an exchange automorphism of K. Example 3.48. If x and y are exchangeable vertices of E, then the simple exchange of K exchanging the vertices x and y of K is an exchange automorphism. More generally, let E be a collection of exchangeable pairs of distinct vertices of K with the property that no two distinct pairs in E have a common vertex. Then the generalized exchange 'E W K ! K of K associated to E is an exchange automorphism, and the Boolean subgroup of Aut.K/ corresponding to E, BE , is a subgroup of the group of exchange automorphisms Aut.K/ of K. Definition 3.49 (Generalized exchange automorphism). Let K, E, and 'E W K ! K be as in Proposition 3.29. We call the automorphism 'E W K ! K of K the generalized exchange of K associated to E.


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Proposition 3.50. Let K be a simplicial complex. Let E be a collection of exchangeable pairs of distinct vertices of K with the property that each pair of distinct vertices in E is an edge of K and no two distinct edges in E have a common vertex. Let be the equivalence relation on the vertex set V of K defined by the rule x y if and only if either x D y or fx; yg 2 E. Let Kz be the quotient of K by the equivalence relation on V and W K ! Kz be the associated natural projection. Let be a subset of the z Then is a simplex of Kz if and only if 1 . / is a simplex of K. vertex set Vz of K. Proof. Suppose, on the one hand, that 1 . / is a simplex of K. Since W V ! Vz is surjective, .1 .// D . Since 1 ./ is a simplex of K and .1 . // D , it z follows from the definition of Kz that is a simplex of K. z Suppose, on the other hand, that is a simplex of K. It follows from the definition of K= that there exists a simplex of K such that . / D . Since is a simplex of K, there exists a nonnegative integer k and k C 1 distinct vertices, xi , 0 i k, of K such that D fxi j 0 i kg. Let i be an integer with 0 i k. Let yi D xi if xi is not a vertex of some pair of distinct vertices of K in the collection E. Otherwise, let yi be the unique vertex of K such that fxi ; yi g is one of the edges of K in the collection E. If xi D yi , then St.xi ; K/ D St.yi ; K/. Otherwise, fxi ; yi g is an edge of K with an exchangeable pair of vertices and, hence, by Proposition 3.28, St.xi ; K/ D St.yi ; K/. Thus, in any case, St.xi ; K/ D St.yi ; K/. By our choices of yi , 0 i k, it follows from the definition of that 1 . / D fxi ; yi j 0 i kg. Let i be an integer with 0 i k. Let i D [ fyj j 0 j i g. We shall prove, by induction on i , that i is a simplex of K. First, consider the case where i D 0. Since x0 2 , is a simplex of St.x0 ; K/ and, hence, of St.y0 ; K/. Since is a simplex of St.y0 ; K/, [ fy0 g is a simplex of K. Hence, we may let 0 D [ fy0 g. Now suppose that 0 i < k. Assume, by induction, that i is a simplex of K. Since 0 < i C 1 k, xiC1 is a vertex of and, hence, of i . Thus, i is a simplex of St.xi C1 ; K/ and, hence, of St.yiC1 ; K/. Since i is a simplex of St.yiC1 ; K/, i [ fyi C1 g is a simplex of K. Since iC1 D i [ fyiC1 g, it follows that iC1 is a simplex of K. This proves, by induction, that k is a simplex of K. Since k D [ fyj j 0 j kg and D fxi j 0 i kg, it follows that k D fxi ; yi j 0 i kg. In other words, k D 1 ./. It follows that 1 . / is a simplex of K, which completes the proof. The following propositions will be useful below for our study of the automorphism groups of the complex of domains. z be as in Proposition 3.50 and let AutE .K/ be Proposition 3.51. Let K, E, , K, the stabilizer of E in Aut.K/. If ' 2 AutE .K/, then there exists a unique simplicial z automorphism ' W Kz ! Kz such that ' B D B ' W K ! K.

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Proof. Let ' 2 AutE .K/; that is to say, suppose that '.E/ D f'.e/ j e 2 Eg D E. It follows that the fibers of both simplicial maps W V ! Vz and B ' W V ! Vz from K to Kz are the equivalence classes of the equivalence relation on the vertex set V of K. Since the simplicial maps W V ! Vz and B' W V ! Vz from K to Kz have the same fibers and W K ! Kz is a simplicial quotient map, it follows from Proposition 3.38 that z there exists a unique simplicial map ' W Kz ! Kz such that ' B D B ' W K ! K. z It remains only to show that ' W Kz ! Kz is a simplicial automorphism of K. To this end, let D ' 1 W K ! K. Since '.E/ D E, it follows that .E/ D E. Hence, by the argument above, there exists a unique simplicial map W Kz ! Kz such z that B D B W K ! K. Note that . B ' / B D B .' B / D B . B '/ D . B / B ' D . B / B ' D B . B '/ D B idV D W V ! Vz . Since W V ! Vz is surjective and . B ' / B D W V ! Vz , it follows that B ' W Vz ! Vz is the identity map of Vz . Likewise, ' B W Vz ! Vz is the identity map of Vz . This proves that ' W Vz ! Vz and W Vz ! Vz are inverse simplicial maps from z completing z z It follows that ' W Vz ! Vz is a simplicial automorphism of K, K to K. the proof. z be as in Proposition 3.50 and Proposition 3.51. Proposition 3.52. Let K, E, , K, z be the stabilizer of .E/ Let AutE .K/ be the stabilizer of E in Aut.K/ and Aut .E/ .K/ z z in Aut.K/. Then there exists a unique homomorphism W AutE .K/ ! Aut .E/ .K/ such that for each automorphism ' 2 Aut E .K/, .'/ is the unique simplicial autoz Moreover, there exists a morphism ' W Kz ! Kz such that ' B D B ' W K ! K. natural short exact sequence z ! 1; 1 ! BE ! AutE .K/ ! Aut.E/ .K/ corresponding to inducing automorphisms of Kz from automorphisms of K which preserve E. Proof. The existence and uniqueness of such a homomorphism W Aut E .K/ ! z follows from Proposition 3.51. Since BE is by definition a subgroup Aut .E/ .K/ of AutE .K/, the homomorphism BE ! AutE .K/ is injective. That the kernel of z is equal to the image of the the natural homomorphism W Aut E .K/ ! Aut .E/ .K/ natural homomorphism BE ! AutE .K/ follows from the definition of the natural z projection W K ! Kz and the definition of W Aut E .K/ ! Aut.E/ .K/.

3.4 Topology One can associate to any abstract simplicial complex K a topological space called the geometric realization of K, and, usually, the topological properties of K refer to those of K. However, there are certain topological properties of K that have a simple


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definition in terms of K, and we shall use this approach. Thus, we use the following terminology: We shall say that K is connected if for any two vertices v and w in K, there exists a finite sequence of vertices v0 D v; v1 ; : : : ; vn D w such that for all k D 0; 1; n 1, fvk ; vkC1 g is an edge of k. We shall say that the sequence v0 D v; v1 ; vn D w is an edge-path of length n joining v and w. Note that a simplicial complex is connected if and only if its 1-skeleton is connected. We shall say that K is bounded if there exists an integer n such that any two vertices of K are connected by an edge-path of length n. We shall say that K is unbounded if it is not bounded. We shall say that K is locally finite if for any vertex v of K, there are only finitely many edges containing it.

4 Some complexes associated to surfaces In this section, we shall discuss some simplicial complexes associated to the surface S . In most cases, these complexes are finite-dimensional flag complexes whose simplices are finite collections of isotopy classes of subspaces of S of a certain type, which can be represented by disjoint subspaces. The extended mapping class group .S / of S acts naturally on each of these complexes via the natural action of the group of homeomorphisms of S on the relevant subspaces. For each of these complexes we say that an automorphism of the complex is geometric if it is induced by a homeomorphism of S. A question that has been thoroughly studied is whether every automorphism of such a complex is geometric. The affirmative answer to this question is known to hold for a number of complexes associated to S, and we shall review several of these results in this section. The complex of domains plays a special role in this theory, because the answer for that complex is negative, as we shall see (Theorem 8.8). Let us note that there are simplicial complexes associated to a surface that we do not touch upon in this chapter; for instance, the disk complex, on which there has been recent work, see the work of Korkmaz and Schleimer [77] and of Masur and Schleimer [92]. In the rest of this section, we define the complexes, we state the corresponding rigidity results, and we survey some results on the geometry of these complexes and on their relation to Teichmüller space.

4.1 The curve complex C.S / As already said, we shall use the term “curve” to denote an essential simple closed curve, except in some special cases where this is explicitly mentioned.

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Definition 4.1. The curve complex of S, C.S /, is the simplicial complex whose nsimplices, for every n 0, are the collections of n C 1 distinct isotopy classes of disjoint curves on S. Note that a finite collection of vertices of C.S / forms a simplex of C.S / if and only if each pair of vertices in this collection can be represented by disjoint curves on S . In other words, C.S/ is a flag complex. The curve complex was introduced by Harvey in 1978, with the idea that this complex encodes some boundary structure of Teichmüller space, in analogy with Tits buildings which encode boundary structure of symmetric spaces. A recent and valuable survey by Lizhen Ji, included in Volume IV of this Handbook, explores the relation between curves complexes and Tits buildings, see [62]. In addition to its relation with the boundary structure of Teichmüller space, the curve complex turned out to be an extremely interesting object, and several of its properties were revealed by Ivanov, Masur, Minsky, Hammenstädt, Bowditch and others. From the point of view of the boundary structure of Teichmüller space, we can mention right away the following two facts: (1) The curve complex encodes the boundary strata of augmented Teichmüller space, a union of Teichmüller space with the set of stable marked Riemannian surfaces, that is, Riemann surfaces in which a certain set of disjoint simple closed curves have been pinched to points. The strata of augmented Teichmüller space are Teichmüller spaces of surfaces of lower genera. (2) The curve complex can be embedded in a canonical way in Thurston’s boundary. We shall elaborate on this below. Let us first consider some special cases. If S is a sphere with at most three holes, then C.S / is empty. If S is a sphere with four holes or a torus with at most one hole, then there are infinitely many isotopy classes of curves on S , but no two such curves are disjoint and nonisotopic. Hence, in these cases, C.S/ is an infinite set of vertices. In all the other cases, C.S/ is connected. This connectedness result was stated by Harvey in [38], and proofs in print were given by Harer in [39] and by Masur and Minsky in [88], Lemma 2.1. The proof by Masur and Minsky uses induction on the geometric intersection number between curves, and it provides an upper bound for the distance between two vertices in terms of the intersection number of the curves that represent them. The essence of this induction process is already contained in Lickorish’s proof of the fact that the mapping class group is generated by Dehn twists [79]. Ivanov gave in [59] another proof of the connectedness that uses Cerf theory. We also refer to the paper by Putman [115] for a unified proof of the connectedness of the curve complex and of several other simplicial complexes. The curve graph is the 1-skeleton of the curve complex. A valuable set of notes on the curve complex is Schleimer’s [124].


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A maximal simplex in the curve complex C.S / is represented by a family C of disjoint essential curves on S such that the surface SC obtained by cutting S along C is a disjoint union of pairs of pants. (Note that in a pants decomposition of S, some of the curves, namely, the curves that are boundary components of the surface, are not essential.) Thus, a maximal simplex in C.S / is naturally a pants decomposition of the surface. In the following proposition, we summarize the information on the dimension of C.S /. Proposition 4.2 (The dimension of the curve complex C.S /). For any surface S , the complex C.S/ is finite-dimensional. If S is a sphere with at most three holes, then C.S / is empty. If S is a closed torus, then C.S / is an infinite set of vertices. Otherwise, all maximal simplices of C.S/ have the same number of vertices, which is 3g 3 C b, and dim.C.S// D 3g 4 C b. This proposition follows from the fact that there is an upper bound for the number of pairwise disjoint and pairwise non-isotopic essential curves on S , and that this upper-bound can be computed. For g 2 or b 3, by an Euler characteristic count, the maximal number of non-isotopic essential curves on S is 3g 3 C b (which is also the number of essential curves in a pants decomposition of S). Therefore, the dimension of C.S/ is equal to 3g 4 C b. Note that this value is 1 provided S is not a sphere with at most four holes or a torus with at most one hole. The extended mapping class group acts simplicially on C.S / in the following natural manner: if 2 .S/ is the isotopy class of a homeomorphism f of S and if is a simplex of C.S/ which is represented by a collection of curves C1 ; : : : ; Ck , then . / is the simplex represented by the collection of curves f .C1 /; : : : ; f .Ck /. The complex C.S/ is not locally finite, provided it is connected. The reason is that as soon as a surface contains an essential curve, it contains infinitely many such curves. Thus, if ˛ is an essential curve on S and if C.S / is connected (which means that there is at least one essential curve in a component of the surface S˛ obtained from S by cutting it along ˛), then there are infinitely many distinct essential curves on S˛ , and therefore the vertex representing ˛ in C.S / belongs to infinitely many edges. From the natural action of the extended mapping class group .S / on C.S /, we obtain a natural homomorphism from .S / into the group Aut.C.S // of simplicial automorphisms of C.S/. The basic result on the automorphism group of C.S / is due to N. Ivanov, who first proved that for any g 2, the natural homomorphism .S / ! Aut.C.S // is an isomorphism provided S is not the closed surface of genus 2. In the case of genus 2, Ivanov proved that the homomorphism is surjective and its kernel is Z2 , generated by the hyperelliptic involution (see [53]). Korkmaz continued the analysis made by Ivanov by studying the case of surfaces of genus 0 and 1. He proved in [74] that for such surfaces, any automorphism of C.S / is induced by an element of .S/ if S is not a sphere with 4 holes or a torus with 2 holes.

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In the cases where S is a torus with one hole or a sphere with four holes, there are automorphisms of C.S/ that are not geometric since in each of these cases the curve complex is an infinite countable set of vertices, and therefore its automorphism group is uncountable. Luo in [82] analyzed a delicate remaining case, namely, the case where the surface S is a torus with two holes. He proved that in that case the map .S / ! Aut.C.S // is not surjective. Luo also gave a new proof of the theorem that included all cases. Let us say a few words about the last case. Luo noticed that there is an isomorphism C.S1;2 / ! C.S0;5 / induced by the projection map W S1;2 ! S1;2 = , where is a hyperelliptic involution of S1;2 , and where S0;5 is identified with the complement of the singular locus of in S1;2 = . It follows that the automorphism group of C.S1;2 / is isomorphic to the automorphism group of C.S0;5 /. Now it is known that the extended mapping class groups .S1;2 / and .S0;5 / are not isomorphic. More precisely, .S1;2 / is an order-two extension of a subgroup of index 5 in .S0;5 /. Thus, we have .S1;2 / 6' Aut.C.S1;2 //. One can further understand the situation as follows. Consider a hyperelliptic involution of the torus with two holes which exchanges the two holes. The quotient surface is a sphere with one hole, and the quotient map is ramified over four points. In this way, the mapping classes of the torus with two holes correspond to the mapping classes of the sphere with five holes that preserve one of the holes and that permute the four others. This gives rise to a subgroup of index five. The extended mapping class group of the torus with two holes is an extension of that group by the hyperelliptic involution. The homomorphism .S1;2 / ! Aut.C.S1;2 // is also not injective, since the hyperelliptic involution acts trivially on C.S1;2 /. (This was already known from works of Birman and of Viro, cf. [12] and [129].) The following theorem summarizes the results on the automorphism group of the complex C.S/. Theorem 4.3 (Ivanov–Korkmaz–Luo). Consider a surface Sg;b whose curve complex C.S / has positive dimension. (Equivalently, the curve complex of C.S / is connected; equivalently, S is not a sphere with at most four holes or a torus with at most one hole.) Then, we have the following: (1) For .g; b/ 62 f.1; 2/; .2; 0/g, the natural homomorphism .Sg;b / ! Aut.C.Sg;b // is an isomorphism. (2) The homomorphism .S2;0 / ! Aut.C.S2;0 // is surjective and its kernel is of order two, generated by the hyperelliptic involution. (3) The homomorphism .S1;2 / ! Aut.C.S1;2 // is neither surjective nor injective. The kernel of this homomorphism is of order two, generated by the hyperelliptic involution, and its image is a subgroup of index 5 in Aut.C.S1;2 //. The image consists in the simplicial automorphisms of C.S1;2 / that preserve the set of vertices represented by nonseparating curves.


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As already mentioned, Luo, in his paper [82], gave a proof of Theorem 4.3 that includes all the cases and which is different from the proofs by Ivanov and by Korkmaz. Luo’s proof uses induction, and it is in the spirit of Grothendieck’s reconstruction principle. For a survey of the relation with Grothendieck’s ideas, see Chapter 17 of Volume II of this Handbook [84], written by Luo. We note that for any domain X on S, we have a natural simplicial embedding C.X/ ,! C.S /: The mapping class group action on the curve complex and the above rigidity result (Theorem 4.3) have been used successfully to obtain rigidity results for mapping class actions on other simplicial complexes, some of which are described in later subsections. Let us mention that there are rigidity results for non-simplicial actions of the mapping class group that have been obtained by inducing from these actions an action on the curve complex, and applying Theorem 4.3. We mention as examples the mapping class group action by homeomorphisms on the space of unmeasured foliations of the surface, see Papadopoulos [108], the action by PL homeomorphisms on the space of measured foliations of the surface, see Papadopoulos [109], the action on the space of measured foliations of the surface that preserves the geometric intersection function, see Luo [83], the action by isometries of Thurston’s asymmetric metric on Teichmüller space, see Walsh [130] and the action on the reduced Bers boundary of Teichmüller space, see Ohshika [106]. Let us now present some further results on the curve complex. In her paper [47], Irmak showed that a sufficient condition for a simplicial map of the curve complex to be geometric is that this map is superinjective that is, if it preserves disjointness of curves. More precisely, Irmak introduced the following notion: a simplicial map C.S/ ! C.S/ is said to be superinjective if for any two vertices ˛ and ˇ of C.S/ we have the equivalence i.˛; ˇ/ 6D 0 () i.˛; ˇ/ D 0. It is easy to see that a simplicial superinjective map is injective. In the paper [47], Irmak proved that if S is closed of genus 3, then a simplicial map C.S / ! C.S/ is superinjective if and only if it is induced by a homeomorphism of S . As a corollary, Irmak obtained the following rigidity result: if K is a finite index subgroup of .S/ and if f W K ! .S/ is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of .S /. In her paper [48], Irmak extended this result to all surfaces Sg;b with g 2 except the cases where .g D 2; b D 0/ and .g D 2; b D 1/. In her paper [49] she extended this result to the remaining cases, .g D 2; b D 0/ and .g D 2; b D 1/. Shackleton [120] studied simplicial embeddings between curve complexes. He showed that except for a few surfaces of low genus and small number of boundary components, any simplicial embedding from C.Sg;b / ! C.Sg 0 ;b 0 / is geometric (that is, it is induced by maps between surfaces) provided 3g 3 C b 3g 0 3 C b 0 . Shackleton deduced from this fact a so-called strong local co-Hopfian property for

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mapping class groups, extending the results by Irmak [47], [49], [48] on superinjective maps. Let us give a quick review of some other properties and applications of the curve complex. Harer [41] showed that the curve complex is homotopy equivalent to a bouquet of spheres, and Ivanov and Ji [61] showed that this bouquet of spheres contains infinitely many spheres. Ivanov [58] then used the rigidity of the automorphism group of the curve complex (Theorem 4.3) to give a new and geometric proof of the famous theorem by Royden saying that the isometry group of the Teichmüller metric (except for a few surfaces of low genus and small number of components, as in the hypothesis of Theorem 4.3) coincides with the natural image of the extended mapping class group in that isometry group. Ivanov’s proof of Royden’s result made a relation between the curve complex and some boundary structure of Teichmüller space, a relation that was first suspected by Harvey. Ivanov’s proof is in the spirit of the classical proof of Mostow strong rigidity for irreducible compact locally symmetric spaces of rank 2 using Tits buildings. Another important result on the geometry of the curve complex is its hyperbolicity. This result was obtained by Masur and Minsky in [88]. In their proof of this result, Masur and Minsky introduced the notion of “tightness” of geodesics, and they showed that between any two points in the curve complex there are only finitely many tight geodesics. In the same paper, Masur and Minsky considered the electric Teichmüller space, defined by attaching a cone to every thin part of that space, and they showed that this space is quasi-isometric to the curve complex, and therefore it is Gromovhyperbolic. Bowditch gave another proof of the hyperbolicity of the curve complex in his paper [16]. This proof by Bowditch is more quantitative than the one by Masur and Minsky; it gives an upper bound for the hyperbolicity constant by a logarithmic function of the complexity of the surface (genus + number of boundary components). In the same paper, Bowditch gave a criterion for recognizing when two geodesic segments are close (in terms of the Hausdorff distance). In the paper [17], Bowditch made the Masur–Minsky result on tightness more quantitative, and he used it to prove that the action of the mapping class group .S / on the curve complex C.S/ is acylindrical, in the sense that for any r > 0, there exist two nonnegative constants R and N such that for all vertices a and b in C.S / whose distance is R, there are at most N distinct elements g in such that d.a; ga/ r and d.b; gb/ r. In the same paper, Bowditch obtained other results on the action of the mapping class group on the curve complex. For instance, he proved that there exists an integer m that depends only on the genus and number of boundary components of S such that for any pseudo-Anosov mapping class g, its power g m preserves some bi-infinite geodesic in the curve graph. As a corollary, Bowditch proved that the stable lengths of the action of pseudo-Anosov elements on the curve graph are rational with bounded denominator.


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A third proof of the hyperbolicity of the curve complex was given by Hamenstädt in Volume I of this Handbook [33]. Bell and Fujiwara, in [6], used Bowditch’s results on the curve complex to show that the asymptotic dimension of that complex is finite. In their paper [89], Masur and Minsky used the hyperbolicity of the curve complex and the fact that some algorithmic questions have good solutions in hyperbolic spaces to study quasi-geodesics in mapping class groups via their actions on curve complexes. Klarreich identified the Gromov boundary @C.S / of the curve complex with the subspace of geodesic laminations that are maximal (in the sense that every complementary component is an ideal triangle) and minimal (in the sense that every leaf is dense in the support of the lamination), in the quotient space of measured lamination space obtained by forgetting the transverse measure; see [72]; see also Hamenstädt [35] and [36] for another proof of Klarreich’s result; see also the survey in Chapter 10 by Hamenstädt in Volume I of this Handbook [33]. The Gromov boundary @C.S / is also identified with the space of ending laminations, see Leininger and Schleimer [78]. Gabai showed in [30] that @C.S/ is connected and path-connected, for any surface which is not a sphere with at most four holes or a torus with at least one hole. The same result was obtained by Leininger and Schleimer [78], in the cases of surfaces of genus at least four and of closed surfaces of genus at least two. Kim [71] obtained a result that made a relation between convergence to Thurston’s boundary in the Teichmüller space T .S/ and convergence to points in the Gromov boundary of the curve complex. For some fixed constant L ( a “Bers constant”) she defined a map ˆ on T .S/ which associates to each element in T .S / a pants decomposition of a hyperbolic surface representing that point whose total length is bounded by L. There is a quotient map u W ML.S/ ! UML.S / from the measured lamination space ML.S/ to the space of unmeasured laminations UML.S /, that is, the quotient space of the space of ML.S/ by the operation of forgetting the transverse measure. The map ˆ has the property that for each 2 T .S / converging to a projective class of measured laminations Œ in Thurston’s boundary, if ˆ. i / converges to a geodesic lamination in the Hausdorff metric topology, then u.Œ / is contained in . Furthermore, if Œ is a filling lamination, then each pants curve in ˆ. i / converges to u.Œ / in C.S / [ @C.S/, where @C.S/ is the Gromov boundary of C.S /. The curve complex, its hyperbolicity and the description of its Gromov boundary were used by Brock, Canary and Minsky as an essential ingredient in the proof of Thurston’s ending lamination conjecture. The conjecture states that a hyperbolic 3manifold with finitely generated fundamental group is determined by its topological type and its end invariants. At the 2004 European Congress of Mathematics, Bowditch gave a survey on the ending lamination conjecture with its relations to the curve complex, see [15]. Masur and Wolf [87] proved that (again, except for some finite set of surfaces of low genus and small number of boundary components), the isometry group of the Weil–Petersson metric on Teichmüller space coincides with the natural image in that

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group of the extended mapping class group. Their proof involves the fact that a Weil– Petersson isometry gives rise to a simplicial map of the curve complex and the result by Ivanov, Korkmaz and Luo on the rigidity of automorphisms of curve complexes (Theorem 4.3 above). Using the curve complex and the action of the mapping class group on it, Behrstock and Margalit obtained in [5] existence and nonexistence results of homomorphisms between mapping class groups. Combined with earlier work by Korkmaz, Irmak, Bell and Margalit, they proved that for S D S0;b with b D 2; 3; 4 or S1;b where b D 0; 1; 2, there exists an isomorphism of a finite-index subgroup of .S / into .S / which is not the restriction of an inner automorphism. In all the other cases, every injection of a finite index subgroup into .S/ is the restriction of an inner automorphism. Rafi and Schleimer proved in [117] that any quasi-isometry of the curve complex of an orientable, connected, compact surface of genus g and b boundary components satisfying 3g 3 C b 2 is at bounded distance from a simplicial automorphism. They noted that a consequence of this results is that one can reconstruct the topology of the surface from the quasi-isometry type of its curve complex. In their paper [116], the same authors gave nontrivial examples of quasi-isometric embeddings between curve complexes, obtained by taking orbifold covers. More precisely, they proved that if S 0 ! S is a covering map,1 then there is a multi-valued covering map W C.S / ! C.S 0 / which is a quasi-isometric embedding, with a quasi-isometric constant depending only on the genus and number of boundary components of S and the degree of the covering map. A result of the same flavor (quasi-isometric rigidity) which applies to the mapping class group is proved in the paper [4] by Behrstock, Kleiner, Minsky and Mosher, which also contains other applications of the hyperbolicity of the curve complex to the study of the large-scale geometry of the mapping class group. The authors show that any self quasi-isometry of the mapping class group of S (with the exception of some surfaces of low genus and small number of boundary components) is at bounded distance from a homomorphism defined by left-multiplication in the group. As a consequence, they obtained a new proof of a theorem of Hamenstädt saying that if a group is quasi-isometric to a mapping class group, then the two groups are isomorphic up to taking finite-index subgroups and quotients with finite kernel. To state other results, let d1 denote the simplicial metric on the curve graph C1 .S / and let dT be the Teichmüller metric on T .S /. Hamenstädt in [34] defined a map ‡ W T .S/ ! C1 .S / which has the property that there exists a number L > 0 such that for every x and y in T .S /, we have d1 .‡.x/; ‡.y// LdT .x; y/ C L: 1As

a matter of fact, in their paper S and S 0 might be 2-dimensional orbifolds and not only surfaces.


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The map is coarsly equivariant with respect to the mapping class group actions on the domain and the range. Maps with a similar flavor are also contained in Masur–Minsky [88], Hamenstädt [36] and Hamenstädt [35]. Maher [85] used the curve complex to study random walks in the mapping class group. He used a result of Masur and Minsky, obtained in [88], stating that there is a “relative metric” on the mapping class group which makes that group quasi-isometric to the curve complex. A relative metric on a group is a word metric with respect to an infinite generating set consisting of the union of a finite generating set with a finite collection of subgroups. In the case considered, the groups consist of stabilizers of simple closed curves. Maher’s result is that a random walk in the mapping class group makes linear progress with respect to this relative metric. Let us also mention an application of curve complexes to 3-manifolds. In 1997, Hempel introduced a quantity called now the “Hempel distance” associated to a Heegaard splitting of 3-manifold defined as follows. A Heegaard splitting is determined by the choice of two maximal systems C1 and C2 of essential pairwise non-homotopic curves on a surface S that correspond to curves that bound discs in a handlebody bounded by S. The set of curves C1 and C2 , considered as sets of vertices in the curve complexes C.S/ respectively, span two subcomplexes K1 and K2 of C.S /. In the paper [45], Hempel studied the relation between the geometric and combinatorial properties of the subcomplexes K1 and K2 with topological properties of the 3-manifold equipped with the given Heegaard splitting. The Hempel distance of the given Heegaard splitting is the distance between K1 and K2 in the 1-skeleton C1 .S / of C.S /. Hempel showed that there exist Heegaard splittings of closed oriented 3manifolds that are at distance n for arbitrarily large n. He also showed that for any splitting of a 3-manifold which is Seifert fibered or which contains an essential torus, the subcomplexes K1 and K2 are at a distance at most two apart. There are several variants of this Hempel distance. For instance, given a Heegaard splitting of a 3manifold M defined by an embedded surface S bounding two handlebodies V and W in M , one can measure the minimal distance in C.S / between two curves on S that bound disks in V and in W respectively. Hartshorn [37] proved that any Heegaard splitting of a closed Haken 3-manifold M that contains an incompressible surface g has Hempel distance at most 2g. This result can be considered as a generalization of a classical result of Haken saying that if a 3-manifold is reducible (that is, if M contains an essential surface of genus 0) then any Heegaard splitting of M is reducible (that is, has zero Hempel distance). Scharlemann gave another version of this result that is valid in the case where M has boundary, cf. [123]. There is also recent work by Masur and Schleimer on the Hempel distance, see [92]. As other instance of the use of the curve complex in 3-manifold theory, we mention the result by Masur and Minsky who considered in their paper [90] the subset of compression disks of C.S/, when the (closed) surface S is considered as the boundary of a 3-manifold. They showed that this subset is quasi-convex, with a quasi-convexity constant that only depends on the genus of S. The result is obtained as a special case of a general result on quasi-convex subsets of C.S /.

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Curve complexes can also be used to obtain presentations of mapping class groups, see Hirose [46] and Benvenuti [7]. Chapter 10 by Hamenstädt in volume I of this Handbook surveys some applications of the curve complex in the study of the large-scale geometry of Teichmüller space, cf. [33]. A survey on several other applications of the curve complex are considered in detail in the chapter by Lizhen Ji in Volume IV of this Handbook [62]. In his paper [100], Minsky used the combinatorial structure of curve complexes to establish new relations between the geometry of a hyperbolic 3-manifold and the asymptotic geometry of its ends. The results in that paper is used in the proof of the ending lamination conjecture. We note finally that the curve complex of a non-orientable surface has also been studied, see e.g. the paper by Scharlemann [122]. Szepietowski used the curve complex of a non-orientable surface to find a presentation of the mapping class group of such a surface [126].

4.2 The arc complex A.S / As already said, we shall use the term “arc” to denote an essential arc on the surface. Definition 4.4. The arc complex of S, A.S /, is the simplicial flag complex whose n-simplices are collections of n C 1 pairwise distinct isotopy classes of disjoint arcs on S . The arc complex was introduced by Harer in [41]. Since homeomorphisms and isotopies of S take homotopic arcs to homotopic arcs and disjoint arcs to disjoint arcs, the extended mapping class group of S acts simplicially on A.S/. Let us first consider a few cases of surfaces of low genus and small number of boundary components. The complex A.S/ is empty if either b D 0 or g D 0, b D 1, and it is reduced to a single vertex if g D 0, b D 2. For g D 0; b D 3, A.S/ is a finite 2-dimensional simplicial complex having six vertices, nine edges and four 2-cells, see Figure 9. In all the other cases, A.S/ is a locally infinite connected complex with infinitely many vertices. A maximal simplex in the arc complex A.S / is represented by a family A of disjoint essential arcs on S such that the surface SA obtained by cutting S along A is a disjoint union of hexagons. A maximal simplex in A.S / is naturally an ideal triangulation in the following sense: Definition 4.5 (Ideal triangulation). An ideal triangulation of S is a system of disjoint and pairwise non-isotopic arcs in S that is maximal with respect to inclusion.


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Figure 9. The finite simplicial complex on the left-hand side represents the arc complex of the sphere with three holes. The six vertices of this complex are the isotopy classes of the arcs represented on the right-hand side.

Each connected component of the surface S cut open along an ideal triangulation is a hexagon, that is, a disk with six distinct points on its boundary, dividing this boundary into six arcs which we call the (distinguished) edges, where three non-consecutive edges arise from three arcs on the surface, and where the other edges are segments in the boundary of S . We shall call such a hexagon an ideal hexagon. The names “ideal triangulation” and “ideal hexagon” stem from the fact that if we pinch each boundary component of S to a point, obtaining, as a quotient surface, a closed surface with distinguished points arising from the boundary components of S , then each ideal hexagon becomes, in the quotient surface, a triangle whose vertices are at the set of distinguished points (hence called an ideal triangle), and the ideal triangulation of S becomes a decomposition into triangles having all of their vertices at the distinguished points (that is, an ideal triangulation in the usual sense). We shall study a graph called the ideal triangulation graph in Section 4.5 below. Proposition 4.6 (The dimension of the arc complex A.S /). If S is a closed surface or a sphere with one hole, then A.S/ is empty. If S is a sphere with two holes, then A.S / is a singleton. In all the other cases, all maximal simplices of A.S / have the same number of vertices, which is 6g 6 C 3b. Consequently, we have dim.A.S // D 6g 7 C 3b. Proof. This follows by a standard Euler characteristic count. Irmak and McCarthy gave a complete description of the automorphism group of the arc complex. They proved the following: Theorem 4.7 (Irmak–McCarthy [50]). Let Sg;b be a surface with nonempty boundary and with negative Euler characteristic. Then, the natural homomorphism W .S/ ! Aut.A.S //

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is surjective. This map is an isomorphism provided .g; b/ 62 f.1; 1/; .0; 3/g. In the excluded cases, the kernel of is the centre of .S /; that is, we have the following: (1) if S is a pair of pants, the kernel of is Z2 , generated by the isotopy class of any orientation-reversing involution of S that preserves each boundary component of this surface; (2) if S is a torus with one hole, the kernel of is Z2 , generated by the hyperelliptic involution of S . We note that the proof of this result, given in [50], does not make use of the corresponding result for the curve complex (Theorem 4.3 above). This contrasts with the proofs of other rigidity results, e.g. Theorem 4.13 below. We also note that in the same paper, Irmak and McCarthy obtained a stronger result, namely, that any injective simplicial self-map of A.S / is induced by a homeomorphism of S . Irmak obtained an analogous theorem for non-orientable surfaces [51]. Let us mention a few natural maps between arc complexes and curve complexes. There is an operation of doubling a surface S D Sg;b with nonempty boundary along one or several of its boundary components. It is defined as follows. We choose a subset @0 of the boundary @S of S , and we assume @0 has cardinality k. The double of S along @0 is then a surface S@d0 of genus 2g C k 1 having 2.b k/ boundary components, equipped with a system @00 of k distinguished curves in its interior. The surface S@d0 cut along @00 consists of two copies of S . The image of the boundary curves @0 by the two natural inclusions of S into S@d0 is the union of curves @00 , and there is an orientation-reversing involution of S@d0 that fixes pointwise the set j@00 j and exchanges the two copies of S in S@d0 . Given a surface S with nonempty boundary @ D @S, and given a collection @0 @ of k components of @, we denote by A.S; @0 / the subcomplex of A.S / induced by the vertices represented by arcs on S that have their two endpoints on @0 . The union of any arc in S having its two endpoints on @0 with the image of this arc in the double S@d0 by the natural involution is a curve in S@d0 . This gives a natural simplicial embedding A.Sg;b ; @0 / ,! C.S2gCk1;2.bk/ /: In particular, if we take @0 to be the union of all the boundary components of S D Sg;b , then the resulting surface S@d D S2gCb1;0 is the double of S D Sg;b . The union of any arc on Sg;b with its image by the natural involution of S d is a curve on SgCb;0 , and this association defines a natural simplicial embedding A.Sg;b / ,! C.S2gCb1;0 /: If X is a domain on S such that a nonempty set of boundary components of X consists of boundary components of S , then we have a natural simplicial embedding A.X; @X \ @S / ,! A.S /:

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Let S1 be a surface with boundary which has infinite topological type, and consider an exhaustion of S1 by subsurfaces of finite type with boundary, S0 S1 S2 ; such that for every i 0, Si is a domain on SiC1 and S1 D

S1

iD1

Si and such that

@S0 @S1 @S2 : Then, we have a sequence of natural embeddings A.S0 ; @S0 \ @S1 / ,! A.S1 ; @S1 \ @S2 / ,! A.S2 ; @S2 \ @S3 / ,! : These maps are used in the study of arc complexes of surfaces of infinite type. There are several relations between the arc complex and Teichmüller and moduli spaces. The arc complex was used in the determination of the virtual Euler characteristic of moduli space, cf. Penner [110] and Harer & Zagier [42]. There is an identification between the (geometric realization of the) arc complex and a product space of Teichmüller space with an open simplex. Such an identification was used by Kontsevich in his proof of the Witten conjecture [73]. Looijenga worked out the details of this identification and he gave an interpretation of the arc complex in terms of the Deligne–Mumford compactification of moduli space [80]. More precisely, Looijenga interpreted the arc complex in terms of the product of a quotient of the Deligne–Mumford compactification of Teichmüller space with a closed simplex. Penner included the arc complex in the setting of his decorated theory of Teichmüller and moduli spaces. In [111], he showed that the arc complex can be used as a combinatorial compactification of moduli space. Penner also showed that there is a homeomorphism between the quotient by the mapping class group of an open dense subspace of the (topological realization of the) arc complex of a punctured surface with moduli space; see the paper [112] for precise statements.2 In a recent paper [113] Penner studied the topology of the arc complex with endpoints at distinguished points, namely, he gave a characterization of those arc complexes that are PL spheres. In the paper [104], Mondello used hyperbolic length of maximal arc systems as parameters for Teichmüller space and he expressed the Weil–Petersson Poisson structure of that space in terms of these parameters. Thus, maximal cells of the arc complex appear in this work as parameter spaces for Teichmüller space, cf. also Mondello [102]. In [101], Mondello used these arcs in the description of the so-called tautological classes in moduli space cf. also Mondello [103]. Relations between the arc complex and Teichmüller space, based on the geometry of hyperbolic right-angles hexagons, are also made in the recent paper [31] by Guo and Luo; see also the paper [127] by Ushijima. 2 One has to be careful about the definitions used in these papers, where a distinction is made between arc complexes of punctured surfaces and those of bordered surfaces. There are also variants, for instance, the endpoints of arc complexes on bordered surfaces are required to be in a certain set of distinguished points, and so on. In fact, in the literature cited here, there are several definitions of an “arc complex” that are not all equivalent.

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Like for curve complexes, there are relations between arc complexes and 3-manifolds. In the paper [119], Saito andYamamoto consider a function they call “translation distance” on open book decompositions, defined in a way similar to the Hempel distance defined on Heegaard splittings using the curve complex, that we recalled in §4.1. The authors prove some properties of the translation distance in relation with the topology of the ambient 3-manifold. They obtain topological properties of open book decompositions of translation distance 0 and 1. Arc complexes have been used to construct operads, see the paper by Kaufmann, Livernet and Penner [66], and the survey by Kaufmann in Volume IV of this Handbook [65].

4.3 The arc and curve complex AC.S / We now introduce an abstract simplicial complex in which the curve complex and the arc complex naturally embed. Definition 4.8 (The arc and curve complex). The arc and curve complex of S , denoted by AC.S /, is the flag simplicial complex whose k-simplices, for each k 0, are the collections of k C 1 distinct isotopy classes of one-dimensional submanifolds which are either curves or arcs in S , and such that this collection can be represented by disjoint curves or arcs on the surface. This complex was studied by Hatcher in [43], who proved that it is contractible. Note that if b D 0, then there are no arcs on S D Sg;b , and in that case AC.Sg;0 / D C.Sg;0 /. When talking about the arc and curve complex, we shall assume that b 1. Unlike the cases of the curve complex and the arc complex, maximal simplices in the arc and curve complex do not all have the same dimension, as we see in the following; Proposition 4.9 (Maximal simplices in the arc and curve complex [75]). A maximal simplex of AC.S/ that has maximal dimension consists of arcs, i.e. it is an ideal triangulation of S. The dimension of such a simplex is 6g7 C 3p. The dimension of a maximal simplex in AC.S/ that has minimal dimension is 3g 4 C 2p. There exist maximal simplices in AC.S/ of all dimensions between 3g 4 C 2p and 6g 7 C 3p. This proposition, which is easy to prove, immediately gives the following rigidity result: Theorem 4.10. If the arc and curve complexes AC.S / and AC.S 0 / of two surfaces S and S 0 are simplicially isomorphic, then S is homeomorphic to S 0 . Proof. Assume S D Sg;n and S 0 D Sh;p are two surfaces of types .g; n/ and .h; p/ respectively. From Proposition 4.9, if AC.S / and AC.S 0 / are simplicially isomorphic,


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or if their geometric realizations are homeomorphic, we have 6gC3n7 D 6hC3p7 and 3g C 2n 4 D 3g C 2n 4. The two equations imply g D h and n D p; that is, the surfaces are homeomorphic. Note that in the statement of Theorem 4.10, the hypothesis that AC.S / and AC.S 0 / are simplicially isomorphic can be replaced by the weaker hypothesis that their geometric realizations are homeomorphic. There are natural simplicial maps from the curve complex C.S / and from the arc complex A.S/ into the arc and curve complex, which extend the natural inclusions at the level of the vertices. These maps are injective, and the complexes C.S / and A.S / can be naturally considered as subcomplexes of the arc and curve complex. The extended mapping class group .S / acts naturally in a simplicial way on the complex AC.S/, and its is clear that the resulting map from .S / into the simplicial automorphism group Aut.AC.S // of AC.S / is a homomorphism. We have the following: Theorem 4.11 (Korkmaz–Papadopoulos [75]). If the surface Sg;b is not a sphere with one, two or three holes nor a torus with one hole, then the natural homomorphism .Sg;b / ! Aut.AC.Sg;b // is an isomorphism. This result says in particular that there are no automorphisms of the arc and curve complex that send a vertex represented by an arc (respectively a curve) to a vertex represented by a curve (respectively an arc). In fact, this can be seen directly by counting the dimensions of maximal simplices containing a vertex which is an arc or a curve respectively, and noting that these dimensions are different. This is used in the proof of Theorem 4.11 given [75]. Two different proofs of Theorem 4.11 are given in [75]; one proof uses the induced action of an automorphism of AC.S / on the curve complex and the rigidity result on its automorphism group (Theorem 4.3), and another one uses the induced action on the arc complex, and the corresponding rigidity result (Theorem 4.7). Let us say a few words on the surfaces that are excluded by the hypothesis of Theorem 4.11. If S is a sphere with one hole, then AC.S / is empty. If S is a sphere with two holes, then AC.S / D A.S / is a single point. If S is a sphere with three holes, then AC.S / D A.S / is a finite complex (see Figure 9 above). In the last two cases, by Theorem 4.7, the natural homomorphism from .Sg;b / to Aut.AC.Sg;b // is surjective and its kernel is Z2 D Z=2Z, which is the center of .Sg;b /. Finally, in the case where S is a torus with one hole, the natural homomorphism Aut.AC.Sg;b // ! Aut.A.Sg;b // is an isomorphism, which implies that we have an isomorphism .Sg;b /=Z2 ' Aut.AC.S//. In analogy with the case of the simplicial map A.Sg;b / ! C.S2gCb1;0 / defined in §4.2, there is a natural injective simplicial map AC.Sg;b / ! C.S2gCb1;0 / ob-

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tained by doubling the surface. In fact, taking doubles along subsets of the set of boundary components (see the definition in §4.2), gives a set of simplicial injections AC.Sg;b / ! AC.S2gCk1;2.bk/ /, defined for every k and b satisfying 1 k b.

4.4 The pants decomposition graph P1 .S / An elementary move between two pants decompositions on S is a transformation of a pants decomposition in which a single curve C is modified (that is to say, the initial pants decomposition and the one obtained from it by the elementary move contain the same set of curves except for that curve C ), such that C and the curve C 0 obtained from C by the move have the smallest possible intersection number. This number i.C; C 0 / is equal to 1 or 2, depending on whether C is on the boundary of one or of two pairs of pants of the decomposition. (Note that C and C 0 have the same number of adjacent pairs of pants, 1 or 2.) The two types of elementary moves are represented in Figure 10. We regard an elementary move as an operation which is defined up

Figure 10. The two types of elementary moves between pants decompositions.

to isotopy, so we can talk about two isotopy classes of pants decompositions that are obtained from each other by a elementary move. We shall say that the elementary move is performed on the curve C that is transformed in the pair of pants decomposition. Note that the number of possible elementary moves performed on a given pants decomposition is infinite, provided we can perform one such move. Definition 4.12 (The pants decomposition graph). The pants decomposition graph, or, more briefly, the pants graph, P1 .S/ of S is the simplicial graph whose vertices are isotopy classes of pants decompositions and where two vertices are joined by an


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edge if the two pants decompositions that represent them (up to homotopy) differ by an elementary move. If S is a sphere with at most two holes or a closed torus, then P1 .S / is empty. If S is a pair of pants, then P1 .S/ consists of one vertex. In all the other cases, P1 .S / is locally infinite: each vertex is contained in infinitely many edges. The pants decomposition graph P1 .S/ was introduced by Hatcher and Thurston in the appendix to their paper [44]. Hatcher and Thurston proved that for any closed surface S of genus g 1, the pants decomposition graph is connected, that is, any two isotopy classes of pants decompositions on a given surface can be obtained from each other by a finite sequence of elementary moves. Another proof of this connectedness result is contained in Putman [115]. The index 1 in the notation P1 .S/ reflects the fact that the pants decomposition graph is the 1-skeleton of a thicker CW complex that has been defined by Hatcher and Thurston by attaching to P1 .S/ three kinds of two-cells (“triangles”, “squares” and “pentagons”), called the pants decomposition complex and denoted by P .S/. Hatcher and Thurston proved in [44] that for any closed surface S of genus 1, the complex P .S/ is simply connected. The square two-cells of the pants decomposition complex represent the commutation relation of elementary moves of disjoint supports. Diagrams representing the triangles and the pentagons are given in the chapter by Funar, Kapoudjian and Sergiescu in this volume ([32], p. 595–664), where a pants decomposition complex is defined for surfaces of infinite topological type and used in the theory of Thompson’s groups. Margalit proved the following rigidity result: Theorem 4.13 (cf. [86]). Let Sg;b be a surface of negative Euler characteristic. If .g; b/ 62 f.0; 3/; .1; 1/; .1; 2/; .2; 0/; .0; 4/g, then the homomorphism .Sg;b / ! Aut.P1 .Sg;b // is an isomorphism. Furthermore, in the excluded cases, we have the following: (1) The homomorphism .S0;3 / ! Aut.P1 .S0;3 // is not injective. (This is because P1 .S0;3 / is reduced to a point, and therefore Aut.P1 .S // is trivial, whereas .S0;3 / is not trivial: it is an order-two extension of the permutation group on three elements.) (2) The homomorphism .S0;4 / ! Aut.P1 .S0;4 // is surjective, and its kernel is Z2 ˚ Z2 , generated by two hyperelliptic involutions. (3) In the case where .g; b/ D .1; 1/; .1; 2/ or .2; 0/, the homomorphism .Sg;b / ! Aut.P1 .Sg;b // is surjective, and its kernel is Z2 , generated by a hyperelliptic involution. Note that a pants decomposition of Sg;b can be regarded as a maximal simplex in the curve complex C.Sg;b / of Sg;b . The graph P1 .Sg;b / can be regarded as a

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subcomplex of the dual complex to the curve complex C.Sg;b /. The proof by Margalit of Theorem 4.13 does not use this fact but nevertheless it uses the result of Ivanov, Korkmaz and Luo on the automorphism rigidity of the curve complex (Theorem 4.3 above). The pants graph has interesting properties that are related to Teichmüller theory, and we list a few of them. In the paper [21], Brock proved that the pants graph, endowed with its natural simplicial metric, is quasi-isometric to the Teichmüller space of S endowed with its Weil–Petersson metric. In the paper [22], Brock and Margalit gave a new point of view on the result due to Masur and Wolf [87] saying that the isometry group of the Weil–Petersson metric on Teichmüller space coincides (except for a few special types of surfaces) with the natural image of the extended mapping class group in that space, by showing that a Weil–Petersson isometry induces a simplicial map of the pants complex and using Margalit’s rigidity result (Theorem 4.13) on the automorphisms of pants complexes. In the paper [18], Brock and Farb used the metric properties of the pants graph to prove that the Weil–Petersson metric on the Teichmüller space T .Sg;b / is Gromovhyperbolic if and only if 3g 3 C b 2. Thus, unlike the case of the curve complex, Gromov hyperbolicity for the pants graph holds in a very limited number of cases. One instance is when the surface is a five-holed sphere, and in that case Shackleton made a detailed study of the geometry of geodesics in that space. In particular, he showed that any two distinct points of the Gromov boundary can be joined by a geodesic [121]. In the paper [91], Masur and Schleimer showed that in the case where S is closed of of genus g 3, the pants graph of S D Sg , equipped with its natural simplicial metric, has only one end. Here, a path metric space X is said to have one end if for any point O in X and any R > 0, the complement of the ball of centre O and radius R has only one unbounded component. Having one end is a quasi-isometry invariant of a metric space. From this and from Brock’s result on the quasi-isometry between the pants graph and the Weil–Petersson metric of the Teichmüller space of Sg , the authors deduce that Teichmüller space endowed with its Weil–Petersson metric has only one end. In the paper [1], Aramayona proved that (again, with a small number of exceptional surfaces) any locally injective simplicial map between pants graphs is induced by a 1 injective map between the underlying surfaces. In the papers [2] and [3], Aramayona, Parlier and Shackleton proved a rigidity result for some graph embeddings in the pants graphs, namely, the Farey graphs and the products of two Farey graphs. They showed that any simplicial embedding of such a graph in a pants graph is totally geodesic. In the paper [93], Mj describes a sequence of complexes that interpolate between the curve graph and the pants graph. Concerning the mapping class group action on the pants graph, a good question is to know whether pseudo-Anosov elements have invariant axes. The question is natural and can be asked for mapping class group actions on a variety of spaces. The question is especially natural in the present setting since we know that the pants graph


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is quasi-isometric to Teichmüller space equipped with its Weil–Petersson metric, and since the analogous question for the Weil–Petersson metric has a positive answer, due to work of Yamada [131] and Daskalopoulos and Wentworth [24]. Shackleton studied recently this question, and he obtained a positive answer in the special case where the surface is a five-holed sphere [121]. (Recall that this is one of the very rare instances where the pants graph is Gromov-hyperbolic.)

4.5 The ideal triangulation graph T.S / In this section, S is a surface with nonempty boundary. We already defined the notion of an ideal triangulations of S (Definition 4.5). We shall consider elementary moves between ideal triangulations of S . These moves are described in Figure 11. Thus, in an elementary move, we replace some

Figure 11. An elementary move on an ideal triangulation: A pair of adjacent hexagons is replaced by a new pair of adjacent hexagons by replacing one arc of the ideal triangulation by another one. The segments in bold lines that are on the boundary of the hexagon represent the arcs that are edges of the triangulation, an the other segments on the boundary of the hexagon are contained in the boundary of the surface.

(homotopy class of) edge of a triangulation by a different one and we keep the other (homotopy classes of) edges unchanged. The (homotopy class of) edge that is transformed by the move is said to be exchanged by the move, and the move is said to be performed on that edge. Definition 4.14 (The ideal triangulation graph). The ideal triangulation graph of S , T .S /, is the simplicial graph whose vertices are the isotopy classes of ideal triangulations of S in which an edge connects two vertices whenever these vertices differ by an elementary move. The ideal triangulation graph has been studied by several authors, including Harer [39] and Hatcher [43]. The rigidity result for the automorphism group of this graph was obtained by Korkmaz and Papadopoulos in [76] (see Theorem 4.16 below).

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Let be an ideal triangulation on S . An arc on S which is an element of the system of arcs defining will be called an edge of . There is an important distinction between exchangeable and non-exchangeable edges of . This notion is defined as follows. An edge of is said to be exchangeable if an elementary move can be performed on it, giving rise to a new ideal triangulation. The edge is said to be non-exchangeable if no elementary move can be performed on it. Non-exchangeable edges on S are those that are on the boundary of a unique hexagon. The configuration is represented in Figure 12. e

Figure 12. The edge e is non-exchangeable in the ideal triangulation of which two arcs are represented. The figure is planar, the two circles represented are boundary components of the surface, there are two arcs joining them, which are part of an ideal triangulation, and the edge e is on the boundary of a unique ideal hexagon.

The following rigidity result follows easily from the distinction made between exchangeable and non-exchangeable edges. Theorem 4.15 ([76]). Let T .Sg;n / and T .Sh;m / be the two ideal triangulation graphs associated of two surfaces Sg;n and Sh;m respectively. Then, T .Sg;n / and T .Sh;m / are homeomorphic if and only if the surfaces Sg;n and Sh;m are homeomorphic. Proof. The non-trivial direction is the “only if” direction, and it follows easily from the following valency considerations in the ideal triangulation graph. We define the valency of a vertex in T .Sg;n / as the number of edges abutting (locally) at that vertex. An ideal triangulation that represents a vertex of maximal valency in the ideal triangulation graph is an ideal triangulation that does not contain any nonexchangeable edge. It is easy to see that there exist such triangulations on any surface. It is also easy to see that an ideal triangulation representing a vertex of minimal valency contains a configuration of the form represented in Figure 13. The configuration is maximal in the sense that all the boundary curves of S are involved. Such a triangulation exists on any surface, provided the surface has at least two boundary components. From the preceding two facts, it easily follows by an Euler characteristic count that for any g and n, the maximum valency at a vertex of T .Sg;n / is 6g C 3n 6 and the minimal valency is 6g C 3n 6 .n 1/ D 6g C 2n 5. Thus, if the two graphs T .Sg;n / and T .Sh;m / are homeomorphic, we have 6g C 3n 6 D 6h C 3m 6 and 6g C 2n 5 D 6h C 2m 5. The two equations imply that g D h and n D m; that is, the surfaces Sg;n and Sh;m are homeomorphic. (A special and easy argument is needed in case one of the surfaces has only one boundary component.)


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Figure 13. On a surface Sg;n with n 2, an ideal triangulation containing such a configuration involving all the boundary components of Sg;n represents a vertex of T .Sg;n / that has minimal valency.

The extended mapping class group .S / acts naturally on the graph T .S / by simplicial automorphisms, and we have the following. Theorem 4.16 (Korkmaz–Papadopoulos [76]). Let S be a connected orientable surface with at least one hole and which is not a sphere with three holes or a torus with one hole. Then the natural homomorphism .S / ! Aut.T .S // is an isomorphism. The proof of Theorem 4.16 given in [76] uses the rigidity theorem for the arc complex obtained by Irmak and McCarthy (Theorem 4.7). It is shown that any automorphism of the ideal triangulation graph induces an automorphism of the arc complex, and that this correspondence between the two automorphism groups is an isomorphism. Although this fact is not used in the proof of Theorem 4.16 , we note that the graph T .S / is the 1-skeleton of the complex dual to the arc complex A.S /, and therefore any automorphism of A.S/ induces an automorphism of the graph T .S /. We also note that T .S / is a strict subcomplex of the dual complex of A.S /, and a priori its automorphism group could be larger than the automorphism group of A.S /. Theorem 4.16 shows that this is not the case. The surfaces that admit ideal triangulations and that are excluded by the hypothesis of Theorem 4.16 are the sphere with three holes and the torus with one hole. These cases are also analyzed in the paper [76], and the situation is as follows: If S D S0;3 is a sphere with three holes, the graph T .S0;3 / is finite, and it is homeomorphic to a tripod, whose central vertex is represented by the unique (up to isotopy) ideal triangulation of S0;3 in which every edge is exchangeable. The automorphism group Aut.T .S0;3 // is isomorphic to the permutation group on three elements, the mapping class group of .S0;3 / is also isomorphic to the permutation group on three elements (the holes of S0;3 ), and the natural homomorphism .S0;3 / ! Aut.T .S0;3 // is an isomorphism. Hence, the natural homomorphism .S0;3 / ! Aut.T .S0;3 // is surjective and its kernel is the center of .S0;3 /, a cyclic group of order two.

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If S D S1;1 is a torus with one hole, then its ideal triangulation graph is the regular infinite tree in which every vertex has valency three. The automorphism group of this tree is uncountable. Thus, the natural homomorphism .S1;1 / ! Aut.T .S1;1 // is highly non-surjective. As we remarked in the preceding sections, we can double the surface S D Sg;b along its boundary components and obtain a closed surface SgCb;0 . There is a natural simplicial embedding T .Sg;b / ,! P1 .SgCb;0 / obtained by doubling the ideal triangulations on Sg;b , an operation that gives pairs of pants decompositions on SgCb;0 , and noting that the double of an elementary move between ideal triangulations on Sg;b is an elementary move between the two corresponding pairs of pants decompositions of SgCb;0 . An ideal triangulation is also a maximal simplex in the arc complex and as such, it can be used to parametrize the Teichmüller space of a surface with boundary. We already mentioned in §4.2 the works of Penner, Mondello and others on that subject.

4.6 The Schmutz graph of nonseparating curves G.S / In [125], Paul Schmutz Schaller introduced and studied a new 1-dimensional simplicial complex G.S/ associated to S which we call the Schmutz graph. There are two different definitions, depending on whether the genus of S is 0 or 1. Definition 4.17 (The Schmutz graph). Let S D Sg;b be a surface of negative Euler characteristic which is not a pair of pants. Then: (1) If g 1, the vertex set of G.S/ is the set of isotopy classes of nonseparating simple closed curves on S , and two vertices are related by an edge whenever their geometric intersection number is 1. (2) If g D 0, the vertex set of G.S/ is the set of isotopy classes of simple closed curves on S which separate S into two components one of which is a pair of pants. (Note that two of the boundary components of this pair of pants are boundary components of S, and therefore such a vertex does not exist if b 1.) In this case, two vertices are related by an edge whenever their geometric intersection is equal to two. Schmutz Schaller proved that G.S/ is connected and that its simplicial automorphism group is equal to the extended mapping class group modulo its centre. More precisely, he proved the following: Theorem 4.18 (Schmutz Schaller [125]). Let S D Sg;b be a surface of negative Euler characteristic which is not a pair of pants. Then, if .g; b/ 62 f.0; 4/; .1; 1/; .1; 2/; .2; 0/g, the natural homomorphism .S/ ! G..S // is an isomorphism.


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Furthermore, in the exceptional cases, the situation is as follows: (1) for .g; b/ 2 f.1; 1/; .1; 2/; .2; 0/g, the homomorphism is surjective, and its kernel is Z2 , generated by the hyperelliptic involution of S ; (2) for .g; b/ D .0; 4/, the homomorphism is surjective, and its kernel is Z2 ˚ Z2 , generated by two hyperelliptic involutions. In his work on surfaces, one of the main motivations of Schmutz Schaller was the study of systoles. In the paper where he defined and studied the graph G.S/, [125], he introduced a complex which he called the “systolic complex”, denoted by SC.S /. The definition is as follows. Definition 4.19 (The systolic complex). Let S D Sg;b be a surface of negative Euler characteristic which is not a pair of pants. Then: (1) If b D 0 or b D 1, then for every k 0, a k-simplex of SC.S / is a set of k C 1 nonseparating curves that mutually intersect at most once. (2) If b D 2, then simplices are as above, except that we allow here separating curves that separate a pair of pants from the rest of the surface. Furthermore, such a separating curve is allowed to intersect the other curves in the same simplex at most twice. Schmutz Schaller noted that the graph G.S/ is a subgraph of the systolic complex SC.S /. He also noted that although not every k-simplex of SC.S / corresponds to the intersection pattern of a set of systoles for some hyperbolic metric on Sg;b , but that “SC.S / is the natural combinatorial object which contains all interesting sets of systoles”. He proposed the following: Conjecture ([125], p. 244). The automorphism group of the systolic complex is isomorphic to the extended mapping class group.

4.7 The complex of nonseparating curves N.S / In this section, we assume that the genus g of S D Sg;p is 2. Definition 4.20 (The complex of nonseparating curves). The complex N.S / of nonseparating curves of S is the flag simplicial complex whose k-simplices, for every k 0, are the collections of k C 1 isotopy classes of nonseparating curves that can be represented by disjoint and pairwise non-isotopic curves. The complex N.S / admits a canonical simplicial injection as the subcomplex of the curve complex C.S/ induced by the set of vertices that are isotopy classes of nonseparating simple closed curves. Irmak proved the following:

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Theorem 4.21 (Irmak [49]). If S is not the closed surface of genus 2, then the natural homomorphism .S/ ! Aut.N.S // is an isomorphism. In the case where S is the closed surface of genus 2, the automorphism group of N.S / is .S/=Z2 , where Z2 is generated by the hyperelliptic involution of S . Note that although the vertex set of the complex N.S / of nonseparating curves is the same as the vertex set of the Schmutz graph G.S/, the 1-skeleton of N.S / is not simplicially equivalent to the Schmutz graph. However, the proof of Theorem 4.21 by Irmak uses the corresponding theorem by Schmutz Schaller (Theorem 4.18). Another proof of the connectedness of the complex of nonseparating curves for closed surfaces of genus 2 is contained in Putman [115].

4.8 The cut system graph HT1 .S / A cut system on the surface S is (the isotopy class of) a system of curves such that S cut along this system is a connected sphere with holes. Note that each of the curves defining a cut system is necessarily nonseparating, and that the cardinality of such a system is equal to the genus of the surface S (by one of the definitions of the genus). In particular, if the genus of S is 0, then there is no cut system on S. Thus, for the rest of this section, we suppose that the genus of S is 1. To simplify notation, we shall often identify a cut system and the set of homotopy classes of curves in that system. In their paper [44], Hatcher and Thurston introduced the following notion of elementary move between cut systems. A elementary move is the operation of replacing the (homotopy class of) a curve in a cut system by a new (homotopy class of a) curve such that the result is again a cut system, and such that the geometric intersection number between the old and the new (homotopy class of) curve is equal to one, see Figure 14. We already noted in Section 4.1 that Irmak extended the rigidity property of automorphisms of the curve complex to the setting of superinjective simplicial maps of that complex. Irmak also studied superinjective maps of the nonseparating curve complex. In her paper [49], she showed that for any surface S D Sg;b with g and b satisfying g 2 and b g 1, any simplicial superinjective map of N.S / is induced by a homeomorphism of S. Definition 4.22 (The cut system graph). The cut system graph, HT1 .S /, of S is the simplicial graph whose vertex set is the set of cut systems on S and whose edges are the pairs of cut systems that are related by an elementary move. The cut system graph is also called the Hatcher–Thurston graph, hence the notation HT1 .S /. As in the case of the pants decomposition graph, the index 1 in the notation


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Figure 14. Replacing one curve with another one, producing an elementary move in the cut system graph. (The old and the new curve are represented in this figure; they intersect in one point.)

HT1 .S / reflects the fact that the cut system graph is the 1-skeleton of a thicker CW complex that has been defined by Hatcher and Thurston by attaching to HT1 .S / three kinds of two-cells (triangles, squares and hexagons), called the cut system complex (or the Hatcher–Thurston complex) and denoted by HT .S /. Hatcher and Thurston proved in [44] that for any closed surface S of genus 1, the graph HT1 .S / is connected, and the complex HT .S/ is simply connected. Another proof of the connectedness result is contained in Putman [115]. In the case where the genus of S is 1, a cut system on S is reduced to a single nonseparating curve, and the cut system graph coincides with the Schmutz graph G.S/ defined in §4.6 above. The automorphism group of the cut system graph was studied by Irmak and Korkmaz, who proved in [52] that its group Aut.HT1 .S // of simplicial automorphisms is the extended mapping class group modulo its centre. More precisely, they obtained the following. Theorem 4.23 (Irmak and Korkmaz [52]). Let S D Sg;b be a compact surface of genus g 1 with b 0 boundary components. If S is not a torus with at most two holes or a closed surface of genus 2, then the natural map .S/ ! Aut.HT1 .S // is an isomorphism. In the excluded cases, this map is surjective and its kernel is Z=2, the centre of .S/. Irmak and Korkmaz proved Theorem 4.23 by passing through the Schmutz complex G.S /. A beautiful ingredient in their proof is the encoding of nonseparating simple closed curves in S by vertices and edges in the cut system graph. More precisely, a pair .v; e/, where v is a vertex of HT1 .S/ and e an edge containing v, determines in a natural way a homotopy class of a nonseparating curve, namely, the homotopy class

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of a curve in a cut system representing v that is transformed by the elementary move representing the edge e. (Recall that every curve in a cut system is nonseparating.) Irmak and Korkmaz used this fact to associate to each automorphism f of HT1 .S / an automorphism fQ of the Schmutz graph. They proceeded as follows. Start with an automorphism f of HT1 .S/. Take an (isotopy class of) nonseparating curve C1 on S . Complete it into a cut system C D fC1 ; C2 ; : : : ; Cg g. Perform an elementary move on C1 in the cut system C , replacing C1 by some curve D. Then, the collection C 0 D fD; C2 ; : : : ; Cg g (that is, the system obtained from C by replacing C with D) is also a cut system on S, and, as a vertex of HT1 .S /, this cut system is connected to the vertex C by an edge. Now since f W HT1 .S / ! HT1 .S / is simplicial, the vertices f .C / and f .C 0 / are also connected by an edge. Irmak and Korkmaz defined fQ.C1 / as the unique (homotopy class of) nonseparating curve that is in f .C / and that is not in f .C 0 /. They then proved that the resulting map fQ is independent of all the choices involved. The map fQ is then showed to be an automorphism of HT1 .S / that sends any pair of isotopy classes of nonseparating curves whose geometric intersection number is equal to one to a pair satisfying the same property. From this, Irmak and Korkmaz obtained a homomorphism from Aut.HT1 .S // to the Schmutz complex G.S/, and they finally proved that this map is an isomorphism. Irmak stated in her paper [49], p. 84, that the isomorphism result (Theorem 4.23) can also be deduced from her result on the automorphism group of the complex of nonseparating curves (Theorem 4.21 above), instead of the result on the Schmutz complex, using the same methods of proof. We note that the automorphism rigidity theorem stated in the paper [52] by Irmak and Korkmaz concerns the full Hatcher–Thurston CW complex and not its onedimensional skeleton, but the proof given in that paper works equally for the Hatcher– Thurston cut system graph. The cut system complex was used by Harer in his computation of the stable second homology of the mapping class group of a surface of genus g with b boundary coms I Z/ D ZsC1 for g 5. ponents and s punctures, namely, in his proof that H2 .g;b In fact, Harer attached 3-cells to the simply-connected Hatcher–Thurston cut system 2-complex, and he worked with the resulting simply-connected 3-complex. Harer’s proof involves the analysis of the cell stabilizers and the homology of the quotient of the 3-complex by the mapping class group. Note that the cell stabilizers are not finite, but they are mapping class groups of simpler surfaces; the details are in [40].

4.9 The complex of separating curves CS.S / Definition 4.24 (The complex of separating curves). The complex CS.S / of separating curves of S is the flag simplicial complex whose k-simplices, for every k 0, are the collections of k C 1 isotopy classes of curves that can be represented by disjoint and pairwise non-isotopic separating curves.


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The complex CS.S/ is canonically isomorphic to the full subcomplex of the curve complex C.S/ spanned by all vertices that are isotopy classes of separating curves. Ivanov and Farb announced in 2005 that for closed surfaces of genus 3, the complex CS.S/ is connected [29]. Other proofs of the connectedness were given by Masur and Schleimer in [91] and by McCarthy and Vautaw in [99]. Putman in [115] gave a new proof of the connectedness and he showed that this complex is simply connected for g 4. Looijenga proved that the complex CS.S / of a closed orientable surface of genus g is .g 3/-connected [81]. Looijenga and van der Kallen proved that the quotient of the complex CS.S / of a closed orientable surface S of genus g by Torelli group of S has the homotopy type of a bouquet of (g 2/-spheres [128]. Theorem 4.25 (Brendle & Margalit [19], Kida [67]). Suppose that the genus of S is 1, and that S is not a torus with at most two holes or a surface of genus two with at most one hole. Then, the natural homomorphism .S/ ! Aut.CS.S // is an isomorphism. Theorem 4.25 was obtained by Brendle and Margalit [19] for closed surfaces. The general form stated is due to Kida [67]. In his paper [68], Kida extended Irmak’s results on the rigidity of superinjective maps of the curve complex and of the nonseparating curve complex to the setting of superinjective maps of the separating curve complex. He proved that for any surface S D Sg;b satisfying one of the following three properties: .g 3; b 0/, .g D 2; b 2/, or .g D 1; b 3/, any simplicial superinjective map from CS.S / into itself is induced by an element of the extended mapping class group. As a corollary, he proved that for such a surface S, if K is any finite index subgroup of the Johnson kernel K.S / any injective homomorphism G ! K.S / is conjugation by some element in K. We recall that the Johnson kernel is the subgroup of the mapping class group that is generated by Dehn twists along separating curves.3 This implies that any finite index subgroup of K.S/ is co-Hopfian. (We recall that a group is said to be co-Hopfian if any injective homomorphism from that group to itself is an isomorphism.)

4.10 The Torelli complex T C.S / A bounding pair in S is a pair of nonseparating curves whose union separates S (see Figure 15). A Dehn twist along a bounding pair is defined as the product of a positive Dehn twist along one of the two curves in the bounding pair, and a negative Dehn twist along the other curve. 3 In fact, the Johnson kernel is usually defined as the kernel of the “Johnson homomorphism” defined on the Torelli group, and the equivalence with the above definition is a theorem due to Dennis Johnson.

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Figure 15. A bounding pair.

Dehn twists along bounding pairs play an important role in the study of the Torelli group, in particular because of a theorem of D. Johnson asserting that the Torelli group of any closed surface of genus 3 is generated by a finite collection of Dehn twists along bounding pairs, see [64]. Before Johnson obtained that result, Birman and Powell had proved that the Torelli group is generated by the infinite collection of all Dehn twists along separating curves and bounding pairs, cf. [11] and [114]. In some sense, the Torelli complex that we define now is an analogue for the Torelli group of the curve complex in its relation to the mapping class group. Definition 4.26 (The Torelli complex). The Torelli complex of S, denoted by TC.S /, is the flag simplicial complex whose vertices are of one of the following two types: (1) an isotopy class of a separating curve on S; (2) an isotopy class of a bounding pair on S . For k 1, a collection of k vertices is a .k 1/-simplex of TC.S / if and only if these vertices can be represented by curves or bounding pairs that are mutually non-isotopic and disjoint. In the case of surfaces of genus zero, the Torelli complex coincides with the curve complex, since any closed curve on such a surface is separating. Brendle and Margalit obtained in [19] the following theorem which was conjectured by Farb: Theorem 4.27 (Brendle–Margalit [19]). For any closed surface S of genus g 4 the natural homomorphism .S/ ! Aut.TC.S // is an isomorphism. In their paper [29], Farb and Ivanov had given an outline of the proof of this result, but with an additional structure on the vertices and on the two-simplices of that complex, and only for the case where the genus of S is 5. Kida obtained a more general version of that theorem, and we finally have the following: Theorem 4.28 (Brendle & Margalit [19], Kida [67]). Let S D Sg;b be a connected compact surface of genus g 1 with b 0 boundary components, such that S is not


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a torus with at most two holes or a surface of genus two with at most one hole. Then, the natural homomorphism .S/ ! Aut.TC.S // is an isomorphism. Note that the separating curve complex CS.S / is naturally a subcomplex of the Torelli complex TC.S/. In his paper [68], Kida proved an analogue of his results on superinjective maps of the separating curve complex that we described in Section 4.9, for superinjective maps of the Torelli complex. More precisely he proved that for any surface S D Sg;b satisfying .g 3; b 0/, .g D 2; b 2/, or .g D 1; b 3/, any simplicial superinjective map from T .S/ into itself is induced by an element of the extended mapping class group. As a corollary, he proved that for such a surface S , if I is any finite index subgroup of the Torelli group .S /, any injective homomorphism I ! .S / is conjugation by some element in I . This implies that any finite index subgroup of .S/ is co-Hopfian. In his paper [69], Kida obtained results on the rigidity of superinjective maps between the separating curve complex and the Torelli complex. He proved that for any surface S D Sg;b whose Euler characteristic .S / satisfies j.S /j 4, any simplicial superinjective map CS.S/ ! T .S/ is induced by an element of the extended mapping class group of S. As a corollary, Kida obtained that for such surfaces, any injective homomorphism from a finite index subgroup of the Johnson kernel K.S / into the Torelli group .S/ is induced by an element of the extended mapping class group. It follows from work done on the Torelli complex in the papers [19], [20] by Brendle and Margalit, [29] by Farb and Ivanov, and [67], [70] by Kida that for any surface S D Sg;b satisfying .g 3; b 1/ or .g D 2; b D 1/, any isomorphism between finite index subgroups of .S/ is a conjugation by an element of the extended mapping class group of S.

5 The complex of domains and its subcomplexes 5.1 The complex of domains D.S / Definition 5.1 (The complex of domains). The complex of domains of S , D.S/, is the flag simplicial complex whose k-simplices, for all k 0, are the collections of k C 1 distinct isotopy classes of disjoint domains on S. We note that the isotopy classes of any two disjoint domains are always distinct except in the case where the two domains are parallel annuli. Clearly, there is a natural simplicial embedding C.S / ! D.S/ obtained via the association to each curve on S of a regular neighborhood of that curve. We shall

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describe below injections of other simplicial complexes into D.S/. There is no natural injection from the arc complex into the complex of domains, but we shall describe a subcomplex of the arc complex, namely, the complex of boundary graphs, which is naturally injected in the complex of domains (see §5.3 below). From Proposition 2.8, for every domain X on S, we have a natural simplicial map D.X/ ,! D.S/: This map is an embedding. We shall study in detail the complex of domains. In particular, we shall describe its automorphism group in §8 below. Let us first briefly discuss the complex of domains associated to some surfaces of low genus and small number of boundary components. If S D S0;3 is a sphere with at most three holes, then S has no essential curves. Since a domain on a surface has at least an essential curve, D.S0;3 / is empty. Proposition 5.2. If S is a sphere with four holes, then D.S/ ' C.S / 2 where 2 is a triangle. Proof. Let S D S0;4 be a spheres with four holes. We recall that in that case C.S / is an infinite vertex set. Let X be a domain on S . Then X must have at least one essential boundary component on S . Suppose, on the one hand, that X has at least two essential boundary components C and D on S . Then, since any two non-homotopic essential curves on S have a nonempty intersection, there exists an annular domain A on S such that C and D are the two boundary components of A. Moreover, there are exactly two codomains, P and Q, of A on S, both of which are biperipheral pairs of pants on S. We may assume that P has C as its unique essential boundary component on S and Q has D as its unique essential boundary component on S. Since C and D are both boundary components of X it follows that X is equal to A. Suppose, on the other hand, that X has exactly one essential boundary component C on S . Then X must be one of the two biperipheral pairs of pants on S which have C as their unique essential boundary components on S . It follows that every domain on S is represented by either an annulus on S or a biperipheral pair of pants on S . This description of D.S/ exhibits this simplicial complex as a bundle over the infinite vertex set C.S/ with fiber a triangle 2 . There is a natural section of D.S/ corresponding to the biperipheral annuli on S . Proposition 5.3. If S is a closed torus, then D.S/ ' C.S / is an infinite set of vertices. Proof. Suppose that S D S1;0 is a closed torus. Then each domain on S is an annulus, and the natural map W C.S/ ! D.S/ is an isomorphism. Proposition 5.4. If S is a torus with one hole, then D.S/ ' C.S / 1 where 1 is an edge.


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Proof. Suppose that S D S1;1 is a torus with one hole. In this case, each domain on S is either an annulus or a monoperipheral pair of pants, and two domains on S are disjoint and nonisotopic if and only if one is a monoperipheral pair of pants and the other is an annulus in its complement. In the above description of the complex of domains of the torus with one hole, a component of D.S/ corresponding to the component f˛g 1 of C.S / 1 is the edge of D.S/ whose vertices correspond to a regular neighborhood of ˛ on S and to a monoperipheral pair of pants in its complement. This description of D.S/ exhibits D.S/ as a bundle over C.S/ with fiber an edge 1 . There are two natural sections of D.S/, one corresponding to the annuli on S , the other corresponding to the monoperipheral pairs of pants on S . This bundle is therefore trivializable with a natural trivialization. Proposition 5.5. If S is not a sphere with at most four holes or a torus with at most one hole , then D.S/ is connected. Proof. Let v be a vertex representing a domain C on S. Then, C has at least one essential boundary component. Let A be an annular domain representing this boundary component, and let w be the vertex in D.S/ represented by A. Since C and A are isotopic to disjoint surfaces, the vertices v and w are joined by an edge. Thus, any vertex in D.S/ can be joined by an edge to a vertex in the natural image of C.S / in D.S /. Since C.S/ is connected, this implies that D.S/ is connected. We now study maximal simplices in the complex of domains. One difference between the complex D.S/ and complexes such as A.S / or C.S / is that in A.S/ and C.S/, the maximal simplices are the top-dimensional simplices, whereas not all of the maximal simplices of D.S/ are top-dimensional simplices. In fact, in D.S/, there are maximal simplices of all dimensions between 1 and the dimension of the top-dimensional simplices. We next describe maximal simplices in D.S/. For these descriptions, it is helpful to distinguish between various types of vertices of D.S /, corresponding to some special domains. Domains of particular interest include annuli, nonannular domains, pairs of pants, peripheral pairs of pants, monoperipheral pairs of pants and biperipheral pairs of pants. Most of the propositions in the rest of this section are easy to prove, and the proofs are left to the reader. The following proposition gives a relation between pants decompositions and maximal simplices. Proposition 5.6. Suppose that S is not a sphere with at most three holes or a closed torus. Let C D fCi j 1 i ng be a maximal system of curves on S. Let X D fXi j 1 i ng be a collection of disjoint annuli on S such that Xi is a regular neighborhood of Ci on S, 1 i n. Let Y D fYj ; 1 j kg be the collection of components of the closure of the complement of X in S . Then:

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(1) n D 3g 3 C b; (2) k D 2g 2 C b; (3) the subsurfaces Xi , 1 i n are annular domains on S ; (4) the subsurfaces Yj , 1 j k, are pairs of pants on S; (5) the vertices xi , 1 i n and yj , 1 j k represented by, respectively, Xi , 1 i n and Yj , 1 j k are the vertices of a simplex of D.S/ which has exactly 5g 5 C 2b vertices and which is top-dimensional. It is easy to construct maximal simplices in D.S/ that are not top-dimensional. To describe the general maximal simplices in D.S/, we introduce the notion of tiling of a surface. A tiling F of S is a system of domains on S which is maximal with respect to inclusion. An element of such a tiling F is called a tile of F . The maximal simplices of D.S / are the unions of vertices representing a tiling of F . A tie of a tiling of S is a codomain of a tiling of S. Suppose first that S is a closed torus. If F is a tiling of S , then F has a unique tile and a unique tie, which are both annuli and which are glued along their two boundary components. If F is a collection of disjoint domains on a closed torus S , then the following are equivalent: (1) F is a tiling of S; (2) F is a system of domains on S; (3) jF j has exactly one codomain. Proposition 5.7. Suppose that S is not a closed torus, let F be a tiling of S and let T be a tie of F . Then T is an annulus on S with essential boundary components C and D such that there exists a unique pair of domains of F , N and A, such that C is an essential boundary component of N , D is an essential boundary component of A, N is not an annulus, A is an annulus, and A is isotopic to T . Proposition 5.8. Suppose that S is not a closed torus. Let F be a tiling of S , let X be a tile of F and C be an essential boundary component of X. Then there exists a unique tie T of F such that C is an essential boundary component of T . Proposition 5.9. Suppose that S is not a closed torus and let F be a collection of disjoint domains on S . Then F is a tiling of S if and only if every codomain of jF j is an annulus joining an annular domain of jF j to a nonannular codomain of F . Proposition 5.10 (Tilings and maximal simplices of D.S/). Suppose that S is not a sphere with at most three holes or a closed torus and let C D fCi j 1 i ng be a system of curves on S. Let X D fXi j 1 i ng be a collection of disjoint


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annuli on S such that for all 1 i n, Xi is a regular neighborhood of Ci on S . Let Y D fYj ; 1 j kg be the collection of components of the closure of the complement of X in S. Let F D fXi ; Yj j 1 i n; 1 j kg. Then: (1) the subsurfaces Xi , 1 i n are annular domains on S ; (2) the subsurfaces Yj , 1 j k, are thick domains on S; (3) the collection F is a tiling of S; (4) the components of the closure of Ri n Xi , 1 i n, are the ties of F ; (5) for 1 i n and 1 j k, the vertices xi and yj represented by, respectively, Xi and Yj , are the vertices of a simplex C of D.S/ which has exactly n C k vertices and which is a maximal simplex of D.S/. Definition 5.11 (The canonical maximal simplex of D.S/ associated to a system of curves). Suppose that S is not a sphere with at most three holes or a closed torus and let C be a system of curves on S. The simplex C provided by Proposition 5.10 is called the canonical maximal simplex of D.S/ associated to C . Proposition 5.12. Suppose that S is not a sphere with at most three holes or a closed torus and let be a maximal simplex of D.S/. Then there exists a system of curves C on S such that D C , where C is the canonical maximal simplex of D.S/ associated to C . Proof. Each maximal simplex contains a nonempty set of vertices which are represented by annular domains. We take C to be the system of curves that represent the homotopy classes of the union of these annular domains. Proposition 5.13. Suppose that S is not a sphere with at most three holes or a closed torus, let be a simplex of D.S/ and let C be a system of curves on S such that

D C is the canonical maximal simplex of D.S/ associated to C . Then the following are equivalent: (1) is a top-dimensional simplex of D.S /; (2) C is a pants decomposition of S. Proposition 5.14 (The dimension of the complex of domains D.S/). If S is a sphere with at most three holes, then D.S/ is empty. If S is a closed torus, then D.S/ is an infinite set of vertices. Otherwise, dim.D.S // D 5g C 2b 6. Proof. A sphere with at most three holes has no essential curves and, hence, no domains. Therefore, if S is a sphere with at most three holes, D.S/ is empty. If S is a closed torus, then the natural map W C.S / ! D.S/ is an isomorphism and, hence, D.S/ is an infinite set of vertices. Assume that S is not a sphere with at most three holes or a closed torus. Then, it follows from Proposition 5.6 that a top-dimensional simplex of D.S/ has 5g C 2b 5 vertices. Hence, dim.D.S// D 5g C 2b 6.

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We shall now construct a natural tiling of S associated to any system of domains on S . We assume that S is not a closed torus. Let F be a system of domains on S. Let A be the collection of annular domains in F . Let P be the collection of nonannular domains in F . Let D be the collection of annular codomains of F Let R be the collection of nonannular codomains of F Let D 2 D. Since D is an annular domain on S and annular domains do not have any peripheral boundary components, there exists a unique subset fF; Gg of F such that D joins F to G. Suppose, on the one hand, that F D G. In this case, we say that D is a coannulus of F attached to the domain F in F . Suppose that D is a coannulus of F attached to the domain F in F . Then F [ D is a domain on S with genus one greater than that of F , the same number of peripheral boundary components as F , and two less essential boundary components than F . In particular, if F is an annulus, then F [ D is a closed torus and, hence, S is a closed torus. Since S is not a closed torus, it follows that each such coannulus of F joins a nonannular domain F in F to itself (i.e. a domain F in P to itself). Suppose, on the other hand, that F ¤ G. In this case, we say that D is a coannulus of F attached to the distinct domains F and G in F . Note that in this case, it is possible that either F or G is an annular domain on S. Suppose that D is a coannulus of F attached to the distinct domains F and G in F . Then F [ D [ G is a domain on S with genus equal to the sum of the genera of F and G, with the same peripheral boundary components as F [ G, and two less essential boundary components than F [ G. Let Q be a collection of domains on S which is obtained from R by replacing each domain R in R by a domain Q which is obtained from R by shrinking R on S (see §2.6 for the notion of shrinking a domain). In particular, Q is contained in the interior of R and Q is isotopic to R on S . Hence, Q and R represent the same vertex of D.S /. Moreover, since R is not an annulus on S, Q is not an annulus on S. Note that Q has the same number of elements as R, that F and Q are disjoint collections of domains on S, and that F [ Q is a collection of disjoint domains on S . Let E be the collection of codomains of F [ Q. Suppose that E 2 E. Note that there exists a unique pair of distinct domains on S , fF; Qg, such that F 2 F and Q 2 Q such that E joins F to Q on S. Hence, E joins a domain F in F [ Q to a nonannular domain Q in F [ Q. Note that it is possible that F is also a nonannular domain on S. It follows from Proposition 2.23 that F [ Q is a system of domains on S . Note that D and E are disjoint sets, D [ E is a collection of disjoint annuli on S, and D [ E is the collection of codomains of the system of domains, F [ Q. Let Dnonann be the subcollection of D consisting of all annuli D in D which join a nonannular domain F in F to a nonannular domain G in F .


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Let B be a collection of domains on S which is obtained from Dnonann by replacing each domain D in Dnonann by a domain B which is obtained from D by shrinking D on S . In particular, B is contained in the interior of D and B is isotopic to D on S . Hence, B and D represent the same vertex of D.S/. Moreover, since D is an annulus on S , B is an annulus on S. Let Enonann be the subcollection of E consisting of all annuli E in E which join a nonannular domain F in F to a nonannular domain Q in Q. Let C be a collection of domains on S which is obtained from Enonann by replacing each domain E in Enonann by a domain C which is obtained from E by shrinking E on S . In particular, C is contained in the interior of E and C is isotopic to E on S. Hence, C and E represent the same vertex of D.S/. Moreover, since E is an annulus on S , C is an annulus on S. Note that F , Q, B, and C are disjoint collections of domains on S , and F [ B [ C is a collection of disjoint domains on S. Let G D F [ B [ C . Note that G contains F and G D P [ Q [ A [ B [ C . Let T be the collection of codomains of G on S . Suppose that T 2 T . Then, T is an annulus joining a nonannular domain X in G (i.e. a domain X in P [ Q) to an annulus Y in G (i.e. a domain Y in A [ B [ C ). Hence, from the above construction, we have the following result. Proposition 5.15. Let F be a system of domains on S . Let A be the collection of annular domains in F , let P be the collection of nonannular domains in F , let D be the collection of annular codomains of F on S and let R be the collection of nonannular codomains of F on S . Finally, let Q, B, and C be the collections of domains that are constructed as above and let G D P [ Q [ A [ B [ C . Then: (1) G is a tiling of S containing F ; (2) G is well-defined up to isotopies on S which fix the support jF j of F pointwise; (3) for each domain G 2 A [ B [ C, there exists a domain F 2 F and an essential boundary component @ of F on S such that G is isotopic to a regular neighborhood of @ on S; (4) for each domain F 2 Q and each essential boundary component @ of F on S, there exists a unique domain G 2 A [ B [ C such that G is isotopic to a regular neighborhood of @ on S ; (5) the number of elements of A [ B [ C is equal to the number of isotopy classes of essential boundary components of domains of S in F ; (6) for each domain Q 2 Q, there exists a unique domain R 2 R such that Q is contained in the interior of R and Q is isotopic to R on S ; (7) for each domain R 2 R, there exists a unique domain Q 2 Q such that Q is contained in the interior of R and Q is isotopic to R on S; (8) the number of elements of G is equal to the sum of the number of nonannular domains in F , the number of nonannular codomains of F on S , and the number of isotopy classes of essential boundary components of domains of S in F .

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Proposition 5.16. Let be a simplex of D.S/, let F be a system of domains on S whose elements represent the vertices of and let G be the tiling of S that is associated to F provided by Proposition 5.15. Then, the simplex of D.S/ whose vertices are represented by the domains in G is the unique maximal simplex of D.S/ which contains the simplex and which has the least number of vertices among all maximal simplices of D.S / containing . Definition 5.17 (The canonical maximal simplex). Let be a simplex of D.S/. The simplex of D.S/ that is provided by Proposition 5.16 is called the canonical maximal simplex of D.S/ containing . Proposition 5.18. Suppose that S is not a closed torus. Let X be a domain on S . Let Y1 ; : : : ; Yk be the k codomains of X on S. Let x; y1 ; : : : ; yk be the vertices of D.S/ represented by X; Y1 ; : : : ; Yk . Then fx; y1 ; : : : ; yk g is a k-simplex of D.S/. Proof. Let f@j j 1 j ng be the collection of all essential boundary components of X on S . Let fAj j 1 j ng be a collection of disjoint annuli on X such that Aj \ @X D @j , 1 j n. Let Z be the closure of the complement of jAj in X . Note that Z is a domain on S which is isotopic to X on S . In particular, Z represents the vertex x of D.S/. Note that fZ; Yi j 1 i kg is a collection of disjoint domains on S and that fAi j 1 i ng is the collection of codomains of fZ; Yi j 1 i kg. Suppose that fZ; Yj j 1 j kg is not a system of domains on S. It follows from Proposition 2.23 that there exists an annular codomain Ai of fZ; Yi j 1 i kg which joins the annulus Z to an annular codomain Yj of X on S, where 1 i n and 1 j k. Since X is a domain on S which is isotopic to the annular domain Z on S , it follows that X is an annular domain on S. Hence, Yj is an annular codomain of the annular domain X on S . It follows that S is a closed torus. This is a contradiction. Suppose that fZ; Yi j 1 i kg is a system of domains on S. It follows that fŒZ; ŒYi j 1 i kg is a k-simplex of D.S/; that is to say, fx; y1 ; : : : ; yk g is a k-simplex of D.S/. In the next subsections, we describe several subcomplexes of the complex of domains. For all these complexes, a finite collection of vertices forms a simplex if and only if each pair of vertices in this collection can be represented by disjoint domains on S . In other words, all these complexes are flag complexes.

5.2 The truncated complex of domains D 2 .S / Definition 5.19 (The truncated complex of domains). The truncated complex of domains of S , D 2 .S/, is the induced subcomplex of D.S/ corresponding to those vertices of D.S / that are not represented by biperipheral pairs of pants.


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Note that D 2 .S/ D D.S/ when b 1. In particular, D 2 .S / D D.S/ for any closed surface S. A biperipheral curve on S is a curve on S which is a boundary component of a biperipheral pair of pants. There is a unique projection W D.S/ ! D 2 .S / which sends each vertex of D 2 .S/ to itself and sends each remaining vertex of D.S/ to the vertex of D 2 .S/ represented by a regular neighborhood of the unique essential boundary component of any biperipheral pair of pants representing this vertex. For each vertex x of D 2 .S/ which is not represented by a regular neighborhood of a biperipheral curve on S, the fiber 1 .x/ of W D.S/ ! D 2 .S / above x is equal to fxg. Suppose that x is a vertex of D 2 .S/ which is represented by a regular neighborhood of a biperipheral curve on S. In the case where S is a sphere with four holes, the fiber 1 .x/ of W D.S/ ! 2 D .S / above x is the triangle of D.S/ induced by the vertices of D.S/ corresponding to a regular neighborhood of on S and the two biperipheral pairs of pants on S of which is a boundary component. Suppose that S is not a sphere with four holes. Then the fiber 1 .x/ of W D.S/ ! 2 D .S / above x is the edge of D.S/ induced by the vertices of D.S/ corresponding to a regular neighborhood of on S and the unique biperipheral pair of pants on S of which is a boundary component. For most of what concerns us here (in particular, for the rigidity result we prove in §8.2), these edge fibers “duplicate information”. Passing from D.S/ to D 2 .S/ or, what is essentially the same, applying the natural projection, amounts to removing this “duplication of information”. In the same way as for the complex of domains D.S/, there is a natural inclusion C.S/ ,! D 2 .S / which maps a vertex of C.S/ represented by a curve ˛ on S to the vertex of D 2 .S / represented by a regular neighborhood of ˛. We now briefly discuss the truncated complex of domains of a few surfaces of low genus and small number of boundary components. If S is a sphere with at most three holes, then D 2 .S / is empty. If S is a spheres with four holes, then, from Proposition 5.2 and the discussion that precedes it, D 2 .S/ ' C.S/ and, hence, D 2 .S / is an infinite set of vertices. If S is a closed torus, then, since S has no holes, the natural map W C.S / ! D 2 .S / is an isomorphism and D.S/ ' D 2 .S/ is an infinite set of vertices. If S is a torus with one hole, then D 2 .S / D D.S/ ' C.S / 1 where 1 is an edge. Proposition 5.20. If S is a surface of positive genus, then dim.D 2 .S// D dim.D.S // D 5g C 2b 6:

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Proof. If b 1, then D 2 .S/ D D.S/. Suppose b 2. If g D 1, then S is a torus with b holes, and we can find a pair of pants decomposition of S with no biperipheral curves by using the decomposition pictured in Figure 16. If g 2, then there exists a

Figure 16. The case where S has genus g 1 and b 2 boundary components. Either S is a torus with b holes, or there is a torus with b C 1 holes which is embedded in S. We can complete the system of curves represented in this picture to a pants decomposition of S in which no curve is biperipheral.

torus with b C1 holes embedded in S , and we can find a pair of pants decomposition of S with no biperipheral curves by using again the decomposition pictured in Figure 16. The tiling associated to such a pants decomposition defines at the same time a topdimensional simplex of S and a top-dimensional simplex of D 2 .S /. This proves the result. Proposition 5.21. If S is a sphere with at least four holes, then dim.D 2 .S // D dim.D.S // 2 D 5g C 2b 8. Proof. Suppose S is a sphere with at least four holes and let F be a tiling defining a topdimensional simplex of D 2 .S/. Let C be a boundary curve of a tile which is essential in S . Then, C is a separating curve on S . Let S1 and S2 be the two components of S n C . Each of these components is a sphere with at least two holes. Since the tiling F defines a top-dimensional simplex of D 2 .S /, each of the components S1 and S2 contains a tile which is a biperipheral annulus. The tiling F can be extended to a tiling F 0 of S defining a top-dimensional simplex of D.S/, by adding two biperipheral pants to the elements of F . This gives dim.D 2 .S // D dim.D.S // 2.

Figure 17. In any pants decomposition of a sphere with b 4 holes, there are necessarily two biperipheral pairs of pants.


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Proposition 5.22. The natural projection W D.S/ ! D 2 .S / is a simplicial quotient map. Proof. The map , from the vertex set of D.S/ to the vertex set of D 2 .S /, is surjective, and all what is needed is to show that a set of vertices of D 2 .S / is a simplex if and only if there exists a simplex of D.S/ such that D . /. If S is a sphere with four holes, then the simplicial complex D 2 .S / is reduced to its set of vertices, which are all annular vertices. Any simplex of D 2 .S / is a vertex of D 2 .S /, and its inverse image D 1 . / is a triangle of D.S/, whose elements are that annular vertex together with the two associated biperipheral pairs of pants. We have D . /. If S is not a sphere with four holes, then for any simplex of D 2 .S /, its inverse image D 1 ./ consists of the union of the vertices of considered as vertices in D.S / together with a unique biperipheral pair of pants for each biperipheral vertex of . Consider a system of domains F on S representing the simplex of D 2 .S /. We can complete F to a system of domains F 0 by adding to each biperipheral annulus in F a corresponding biperipheral pair of pants. The system of domains F 0 represents the vertex set , and therefore, is a simplex of D.S/. We have D . /. This completes the proof. We end this section by describing some maximal simplices in D 2 .S /. Proposition 5.23 (Pants decompositions). Suppose that S is not a sphere with at most three holes or a closed torus. Let C D fCi j 1 i ng be a maximal system of curves on S . Let X D fXi j 1 i ng be a collection of disjoint annuli on S such that Xi is a regular neighborhood of Ci on S , 1 i n. Let Y D fYj j 1 j pg be the collection of components of the closure of the complement of X in S which are not biperipheral pairs of pants on S. Then: (1) n D 3g 3 C b; (2) p D 2g 2 C b; (3) the subsurfaces Xi , 1 i n are annular domains on S ; (4) the subsurfaces Yj , 1 j k, are pairs of pants on S; (5) the vertices xi , 1 i n and yj , 1 j k represented by, respectively, Xi , 1 i n and Yj , 1 j k are the vertices of a simplex, 0C , of D.S/; (6) 0C has exactly n C p vertices; (7) if g D 0, then 0C has exactly 5g 7 C 2b vertices; (8) if g > 0, then 0C has exactly 5g 5 C 2b vertices; (9) 0C is a top-dimensional simplex.

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Definition 5.24 (The canonical maximal simplex of D 2 .S / associated to C ). Let C be a system of curves on S, let C be the canonical maximal simplex of D.S/ associated to C and let 0C be the simplex of D 2 .S / whose vertices are the vertices of C which are not represented by biperipheral pairs of pants on S. Then, 0C is called the canonical maximal simplex of D 2 .S / associated to C .

5.3 The complex of boundary graphs B.S / Definition 5.25 (The complex of boundary graphs). The complex of boundary graphs of S , B.S /, is the flag simplicial complex whose n-simplices are collections of n C 1 distinct isotopy classes of disjoint essential boundary graphs on S. Note that B.S/ is empty if b D 0. If S is a disk (i.e. a sphere with one hole), then B.S / is also empty. If S is an annulus, then S has a unique isotopy class of an essential arc, hence B.S/ has a unique vertex and no higher dimensional simplices. Note also that B.S/ is a nonempty finite set of vertices if g D 0, 2 b 3. If S is a torus with one hole, then B.S/ is, like A.S /, an infinite vertex set. If S is a surface of positive genus with two holes, then B.S / is disconnected and has higher dimensional simplices. Indeed, B.S / contains infinitely many components consisting of one vertex, corresponding to essential arcs on S which are contained in biperipheral pairs of pants on S. The boundary graph of any such arc necessarily intersects any other boundary graph. These are the only “isolated” vertices of B.S /. All other vertices correspond to arcs which are contained in domains on S which are monoperipheral pairs of pants on S . Since any such domain on S is disjoint from at least one other such domain on S, these vertices are contained in at least one edge of B.S /. When S is a pair of pants, then B.S/ has exactly six vertices, represented by the boundary graphs that are associated to the six isotopy classes of essential arcs in S (see Figure 9), and no higher dimensional simplices. Note that the boundary graph of an arc ˛ is not only determined by that arc but, in turn, it determines the arc. Indeed, ˛ is the closure in S of the complement of G˛ \ @S in G˛ . Let ˛ and ˇ be arcs on S . Suppose that there exists an isotopy ' W S Œ0; 1 ! S from G˛ to Gˇ . Note that the boundary components of the boundary graphs ' t .G˛ / remain constant throughout the isotopy. Hence, isotopic boundary graphs have the same boundary components. Since @S is invariant under any isotopy, it follows that ˛ is isotopic to ˇ if and only if G˛ is isotopic to Gˇ . Hence, there exists a natural bijection B0 .S/ ! A0 .S/. If G˛ is disjoint from Gˇ , then ˛ is disjoint from ˇ. Hence, this bijection extends to a natural simplicial inclusion i W B.S/ ! A.S /: This identifies B.S/ with a subcomplex of A.S / having the same vertex set as A.S /. Note, however, that the boundary graphs G˛ and Gˇ of disjoint arcs ˛ and ˇ are disjoint


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only when the boundary components of S joined by ˛ are distinct from those joined by ˇ. Hence, in general, the image subcomplex of A.S / has fewer simplices than A.S /. More precisely, the image subcomplex of A.S / is the subcomplex consisting of those simplices of A.S/ which have the property that each pair of distinct vertices of are represented by disjoint arcs ˛ and ˇ on S such that the boundary components of S joined by ˛ are distinct from those joined by ˇ. Assume now that b > 0 and that either g > 0 or b > 3. We already noted that A.S / is connected with infinitely many vertices. Since B0 .S / ' A0 .S /, B.S / has infinitely many vertices. Suppose that S has exactly one boundary component. It follows from our assumption that g > 0. Note that the boundary graphs of any two arcs on X must intersect, since they both contain the nonempty boundary of S. Hence, no two vertices of B.S / are joined by an edge of B.S/. Thus, B.S/, unlike A.S /, is an infinite set of vertices and in particular it is disconnected. Suppose that S has exactly two boundary components. Again, it follows that g > 0. Let ˛ be an arc on S joining the two boundary components @1 and @2 of S . Note that the boundary graph Gˇ of any arc ˇ on S must intersect G˛ . After all, at least one of the two boundary components of S is contained in both G˛ and Gˇ . It follows that the vertex of B.S/ represented by ˛ is not connected by an edge of B.S / to any other vertex of B.S/. Thus, B.S/, unlike A.S/ has at least two connected components. Actually, since g > 0, it can be shown that there are infinitely many distinct vertices of B.S / joining the two boundary components of S. It follows that B.S /, unlike A.S /, has infinitely many connected components. This shows, in particular, that the natural inclusion map B.S/ ! A.S/ need not be a homotopy equivalence. Proposition 5.26. If g 1 and b 3, or if g D 0 and b 4, then B.S / is connected. In Proposition 5.32 below, we describe in detail the complex B.S / in the case where S D S0;4 is a sphere with four holes. The dimension of the complex of boundary graphs is given below (Proposition 5.35). If S is a surface with nonempty boundary and if @0 @S is a union of components of @, then we denote by B.S; @0 / the subcomplex of B.S / induced by the vertices represented by boundary graphs S that are associated to arcs whose endpoints are on @0 . We have a natural simplicial embedding B.S; @0 / ,! B.S /: In particular, for any domain X on S having a nonempty collection of boundary components that are boundary components of S , we have a natural simplicial embedding B.X; @X \ @S / ,! B.S /:

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5.4 The complex of peripheral pairs of pants P@ .S / Definition 5.27. The complex of peripheral pairs of pants on S , P@ .S / is the subcomplex of D.S / induced by the set of vertices of D.S/ that are represented by peripheral pairs of pants on S. If S is a sphere with at most three holes or a torus with at most one hole, then P@ .S / is empty. If S is a sphere with four holes, then each peripheral pair of pants P on S is a biperipheral pair of pants, and we have a simplicial isomorphism P@ .S / ' C.S / 1 , where 1 is an edge.

Figure 18. A component of P@ .S0;4 / and domains representing its two vertices.

Now we describe a natural map W B.S/ ! P@ .S /: Suppose that S is not a sphere with at most three holes. Let ˛ be an arc on S. Let P˛ be a regular neighborhood on S of the boundary graph G˛ of ˛ such that P˛ is an essential surface on S. Since either g 1 or b 4, P˛ is an essential peripheral pair of pants on S. It follows that there is a natural map from the vertex set of B.S/ to the vertex set of P@ .S / which maps the vertex of B.S / represented by G˛ to the vertex of P@ .S/ represented by P˛ . The vertices of any simplex of B.S/ can be represented by arcs with disjoint boundary graphs. Furthermore, we may choose disjoint essential regular neighborhoods of these disjoint boundary graphs. It follows that the natural map B0 .S / ! P0 .S / extends to a natural simplicial map W B.S/ ! P@ .S /. Proposition 5.28. Suppose that S is not a sphere with at most three holes. Then the natural simplicial map W B.S/ ! P@ .S/ is surjective. We cannot always identify B.S/ with the complex of peripheral pairs of pants P@ .S / on S as the natural map W B.S/ ! P@ .S / need not be injective. Indeed, W B.S / ! P@ .S/ is injective precisely when b D 1; in which case, B.S / is either empty, when g D 0, or an infinite vertex set, when g 1.

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Figure 19 represents two boundary graphs, represented by arcs ˛ and ˇ, that have the same image in P@ .S/ by the natural inclusion .

P˛

Pˇ

˛

ˇ

Figure 19. The two boundary graphs ˛ and ˇ define the same boundary pair of pants.

The following proposition describes precisely the failure of W B.S / ! P@ .S / to be, in general, injective. Proposition 5.29. Suppose that S is not a sphere with at most three holes. Let ˛ be an essential arc on S, P˛ be an essential regular neighborhood of the boundary graph G˛ of ˛ on S, u be the vertex of B.S/ represented by G˛ , w be the vertex of P@ .S / represented by P˛ , and 1 .w/ be the fiber of over the vertex w of P@ .S /. Then: (1) if P˛ has exactly one boundary component on @S , then 1 .w/ is the single vertex, u, of B.S/; (2) if P˛ has exactly two boundary components on @S , then 1 .w/ is a set of diameter 2 in B.S/ consisting of three distinct vertices of B.S /, corresponding to arcs of S contained in P˛ . Proof. This follows easily from the classification of isotopy classes of arcs in pairs of pants. See Figure 9. Corollary 5.30. Suppose that S is not a sphere with at most three holes. Let W B.S / ! P@ .S/ be the natural simplicial map. If is a k-simplex of B.S /, then . / is a k-simplex of P@ .S/. Proof. Since W B.S/ ! P@ .S/ is simplicial, . / is an l-simplex of B.S / for some nonnegative integer l k. Suppose that l < k and, hence, that there exists a pair of distinct vertices u and v of such that .u/ D .v/. Let z D .u/. Since S is not a sphere with two or three holes and the distinct vertices u and v of B.S / are both in the fiber of over z, it follows from Proposition 5.29 that z is represented by a

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biperipheral pair of pants P on S and u and v are represented by the boundary graphs of arcs ˛ and ˇ on S contained in P . Note that each endpoint of ˛ and ˇ lies on one of the two peripheral boundary component of P . Since P is a domain on S and G˛ and Gˇ represent distinct vertices of the simplex of B.S /, we may assume that G˛ and Gˇ are disjoint. It follows that ˛ joins one of the two peripheral boundary components of P to itself, and ˇ joins the other of these two peripheral boundary components of S to itself. It follows from the classification of isotopy classes of arcs in pairs of pants that ˛ and ˇ intersect. This is a contradiction. Hence, l D k; that is to say, . / is a k-simplex of P@ .S /. Proposition 5.31. Let @i be a boundary component of S and be an essential curve on S . Then there exists an arc ˛ of type fi; i g on S such that is an essential boundary component of a regular neighborhood P˛ of G˛ on S. Proof. Since S is connected, there exists an embedded path J in S such that J \ @i is one endpoint of J and J \ is the other endpoint of J . Let G D @i [ J [ . Let N be an essential regular neighborhood of the graph G on S . Then N is an essential peripheral pair of pants on S with one of its essential boundary components isotopic to on S . By isotoping N on S, we may assume that is an essential boundary component of N . Let ˛ be an essential arc in N joining @i to itself. Then N is a regular neighborhood of G˛ . Suppose now that S D S0;4 is a sphere with four holes. The structure of B.S/ can be obtained from studying the natural map W B.S / ! P@ .S /. Let e be an edge of P@ .S/ and P and Q be disjoint biperipheral pairs of pants representing the vertices x and y of e. Since S is not a sphere with two or three holes and x is represented by the biperipheral pair of pants P on S, it follows from Propositions 5.28 and 5.29 that 1 .x/ is equal to the set of three vertices of B.S / corresponding to the boundary graphs on S contained in P . Note that no two of these three vertices of B.S/ are joined by an edge of B.S /. Likewise, 1 .y/ is equal to the set of three vertices of B.S/ corresponding to the boundary graphs on S contained in Q and no two of these three vertices of B.S / are joined by an edge of B.S /. Note, furthermore, that all the vertices of 1 .x/ are joined by edges to all the vertices of 1 .y/. Let G.X; Y / be a simplicial graph whose vertex set is the disjoint union of two nonempty sets, X and Y , and whose edges are the pairs fu; vg, u 2 X; v 2 Y . We say that G.X; Y / is the complete bipartite graph on X and Y . It follows that 1 .e/ is the complete bipartite graph G.1 .x/; 1 .y// on 1 .x/ and 1 .y/. Moreover, each edge of G.1 .x/; 1 .y// is mapped by W B.S / ! P@ .S / onto the edge e D fx; yg of P@ .S/. Let be an essential curve on S and A be a regular neighborhood of on S . Let P and Q be the two codomains of A on S , both of which are biperipheral pairs of pants on S . Let x and y be the vertices of P@ .S / represented by P and Q, and G D G.1 .x/; 1 .y//.


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The join K ? L of two simplicial complexes K and L is the simplicial complex whose simplices are disjoint unions ˛ t ˇ where ˛ and ˇ are simplices in K and L respectively. From these considerations, we deduce the following description of B.S /. Proposition 5.32. Let S D S0;4 be a sphere with four holes. Let G.3; 3/ be a complete bipartite graph on two sets of cardinality 3. C.S / is an infinite set of vertices and B.S/ ' G.3; 3/ ? C.S/, where the component of B.S / corresponding to the component G.3; 3/ fg of G.3; 3/ ? C.S / is equal to G .

Figure 20. A component of B.S0;4 / and boundary graphs representing its six vertices.

Let P be a peripheral pair of pants on S. If P is monoperipheral and @i is the unique boundary component of S which is a boundary component of P , then there exists an essential arc ˛ on P joining @i to itself. If P is biperipheral and @i and @j are the unique boundary components of S which are boundary components of P , then there exists an essential arc ˛ on P joining @i to @j . There is a natural inclusion i W P@ .S/ ! B.S / which maps the vertex of P@ .S/ corresponding to a peripheral pair of pants P on S to the vertex of B.S/ corresponding to the boundary graph G˛ of an essential arc ˛ on P joining the peripheral boundary components of P as above.

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Proposition 5.33. The composition B i W P@ .S / ! P@ .S / of the natural inclusion i W P@ .S / ! B.S/ with the natural map W B.S / ! P@ .S / is equal to the identity of P@ .S /. Proposition 5.34 (The dimension of the complex of peripheral pairs of pants). If S is a sphere with at most three holes or a closed surface, then P@ .S / is empty. If S is not a sphere with at most three holes or a closed surface and if g D 0, then dim.P@ .S // D b 3. If g 1, then dim.P@ .S // D b 1. Proof. We shall give the argument in the cases where S is not a sphere with at most three holes or a surface with at most one hole. In all of these cases, the calculation of dim.P@ .S// follows from the calculation of dim.D 2 .S //. In genus zero, this calculation shows, on the one hand, that each maximal collection of disjoint peripheral pairs of pants on S exhausts the boundary of S and has at least two biperipheral pairs of pants and, on the other hand, that there exists a maximal collection of disjoint peripheral pairs of pants on S with only two biperipheral pairs of pants. In positive genus, this calculation shows, on the one hand, that each maximal collection of disjoint peripheral pairs of pants on S exhausts the boundary of S and, on the other hand, that there exists a maximal collection of disjoint peripheral pairs of pants on S with no biperipheral pairs of pants. The calculation follows. Proposition 5.35 (The dimension of the complex of boundary graphs). If S is a closed surface or a sphere with one hole, then B.S / is empty. If S is a sphere with two holes, then B.S / is a singleton set. If S is a sphere with three holes, then B.S / is a set of six vertices. If S is not a closed surface or a sphere with at most three holes, then dim.B.S // D dim.P@ .S//. Hence, if g D 0, then dim.B.S // D b 3; and, if g 1, then dim.B.S// D b 1. Proof. We shall give the argument when S is not a closed surface or a sphere with at most three holes. Since S is not a sphere with two or three holes, it follows from Proposition 5.28 that dim.P@ .S// dim.B.S //. Likewise, it follows from Corollary 5.30 that dim.B.S// dim.P@ .S //. Hence, dim.B.S // D dim.P@ .S //. The result follows then from Proposition 5.34.

5.5 Other subcomplexes of D.S / Definition 5.36 (The complex of elementary domains). The complex of elementary domains E.S / of S is the subcomplex of D.S/ induced by the set of vertices of D.S/ which are represented by essential annuli and pairs of pants on S. Proposition 5.37 (The dimension of the complex of elementary domains). If S is a sphere with at most three holes, then E.S/ is empty. If S is a closed torus, then the natural map W C.S/ ! E.S/ is an isomorphism and, hence, E.S/ is an infinite set of vertices. Otherwise, dim.E.S// D dim.D.S // D 5g C 2b 6.


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Proof. Since E.S/ is a subcomplex of D.S /, we have dim.E.S // dim.D.S // D 5g C 2b 6. If S is neither a sphere with at most three holes nor a closed torus, we let C be a pants decomposition of S, and C the canonical maximal simplex of D.S / associated to C . The vertices of C are in E.S/, hence, C is also a simplex of E.S /. Its dimension is 5g C 2b 6. It follows that dim.E.S // 5g C 2b 6. This shows that dim.E.S// D dim.D.S// D 5g C 2b 6. Definition 5.38 (The complex of annular domains). The complex of annular domains R.S / of S is the subcomplex of D.S/ induced by the set of vertices of D.S/ which are represented by essential annuli on S . There is a natural simplicial isomorphism C.S/ ' R.S/ which maps the vertex of C.S/ represented by a curve ˛ on S to the vertex of R.S/ represented by a regular neighborhood of ˛ on S . We shall identify C.S/ with the complex of annuli R.S/ on S via this natural isomorphism. Definition 5.39 (The complex of pairs of pants). The complex of pairs of pants on S , R.S / is the subcomplex of D.S/ induced by the set of vertices of D.S/ which are represented by pairs of pants on S. Proposition 5.40 (The dimension of the complex of pairs of pants). If S is a sphere with at most three holes or a closed torus, then P .S/ is empty. Otherwise, one has dim.P .S // D 2g C b 3. Definition 5.41 (The complex of thick domains). The complex of thick domains TD.S / of S is the induced subcomplex of D.S/ corresponding to those vertices of D.S / which are represented by domains on S that are not annuli (i.e. which have negative Euler characteristic). Proposition 5.42 (The dimension of the complex of thick domains). If S is a sphere with at most three holes or a closed torus, then TD.S / is empty. Otherwise, one has dim.TD.S // D 2g C b 3.

6 Surface topology recognized by D.S / and D 2 .S / The aim of this section is to show on a series of specific cases how topological information on the surface S can be recognized by combinatorial information in the simplicial complexes D.S/ and D 2 .S/.

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The results of §6.4 and §6.5 below, entitled Recognizing annular vertices in D 2 .S / and Recognizing biperipheral edges in D.S/, will be used in the proofs of the rigidity results on the automorphisms od D 2 .S/ and D.S/ that we give in Sections 7 and 8. We start with the following:

6.1 Recognizing elementary vertices in D.S / We say that a vertex of D.S/ is elementary if it is represented by elementary domains on S . The significance of this notion appears in the following characterizations of elementary vertices of D.S/ and various subtypes of elementary vertices of D.S/. In particular, these characterizations imply that any automorphism of D.S/ preserves the subcomplex of D.S/ induced by the set of elementary vertices of D.S/. Proposition 6.1. Let x be a vertex of D.S/. If Lk.Lk.x// D fxg, then x is elementary. Proof. Let X be a domain on S representing x. Suppose that Lk.Lk.x// D fxg and x is not elementary. By Proposition 2.18, there exists a pair of curves ˛ and ˇ on S such that i.˛; ˇ/ ¤ 0 and ˛ and ˇ are contained in the interior of X. Let W be a regular neighborhood of ˛ in the interior of X and w be the vertex of D.S/ represented by W . Since i.˛; ˇ/ ¤ 0, W is not isotopic to any domain disjoint from X . Hence, w … Lk.x/. On the other hand, since W X, w 2 St.Lk.x//. Since w … Lk.x/ and w 2 St.Lk.x//, w 2 Lk.Lk.x//. Hence, w D x; that is to say, the annulus W is isotopic to X on S. Since X is not an annulus, this is a contradiction. Hence, x is elementary. This proves the proposition. Proposition 6.2 (Recognizing elementary vertices in D.S/). Suppose that S is not a closed torus and let x be a vertex of D.S/. Then the following are equivalent: (1) x is elementary. (2) There exists a simplex in D.S/ such that Lk. / D fxg. (3) Lk.Lk.x// D fxg. Proof. Proposition 6.1 shows that (3) implies (1) (with no need of the hypothesis that S is not a closed torus). Proposition 3.16 gives (2) ) (3). It remains only to prove that (1) implies (2). Suppose that x is elementary and let X be a domain on S representing x. Since x is elementary, X is either an annulus or a pair of pants. If X is an annulus, let be the simplex of D.S/ whose vertices are the vertices of D.S / which are represented by codomains of X on S. If X is a pair of pants, let be the simplex of D.S/ whose vertices are the vertices of D.S/ which are represented by codomains of X on S together with the


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vertices of D.S/ which are represented by regular neighborhoods of essential boundary components of X on S. Since X is isotopic to a domain on S which is disjoint from all the codomains of X on S and regular neighborhoods of all the essential boundary components of X on S, it follows that x 2 St. /. On the other hand, by Proposition 5.18, x … , hence, x 2 Lk. /. Suppose, conversely, that w 2 Lk. / and let W be a domain on S representing w. Since w 2 Lk. /, W is isotopic to a domain on S which is disjoint from all the codomains of X on S. Hence, without loss of generality, W is contained in X . Suppose that X is an annulus. Then W is isotopic to X on S . Suppose that X is a pair of pants. Suppose that W is not isotopic to X on S. Then W is isotopic to a regular neighborhood of an essential boundary component of X on S. This implies that w 2 . Since w 2 Lk. /, this is a contradiction. Hence, W is isotopic to X on S. In any case, therefore, W is isotopic to X on S; that is to say, w D x and, hence, Lk. / fxg. The following is an immediate corollary of Proposition 6.2. Corollary 6.3. Every simplicial automorphism of D.S/ preserves the subcomplex of D.S / induced by the set of elementary vertices of D.S/.

6.2 Recognizing nonseparating annuli in D.S / Proposition 6.4 (Recognizing nonseparating annuli in D.S/). Suppose that S is not a torus with at most one hole and let x be a vertex of D.S/. Then the following are equivalent: (1) There exists a nonseparating curve ˛ whose essential regular neighborhoods on S represent x. (2) There exists a vertex y of D.S/ such that Lk.y/ D fxg. Proposition 6.4 is vacuously true when S is a sphere with at most three holes and false when S is a torus with at most one hole. We thank Valentina Disarlo who simplified our initial proof of (2) implies (1). Proof. We first prove that (1) implies (2). Suppose that x is represented by an essential regular neighborhood X of a nonseparating curve ˛ on S . Since ˛ is nonseparating, the complement Y of the interior of X in S is a domain on S . Let y be the vertex of D.S/ represented by Y . Since ˛ is nonseparating and S is not a torus, Y is not isotopic to X on S ; that is to say, x ¤ y. Let M be a regular neighborhood of ˛ contained in the interior of X. Since M is isotopic to X on S, M also represents x. On the other hand, since M is disjoint from

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Y , fx; yg is a simplex of D.S/ (and x and y are distinct vertices). Since x is in this simplex but not equal to y, x 2 Lk.y/. Suppose that z 2 Lk.y/. Then we may choose an essential surface Z in S representing z and disjoint from Y . Since Z is disjoint from Y , Z is an essential surface contained in the interior of the annulus X. It follows that Z is isotopic to X on S ; that is to say, z D x. This proves that (1) implies (2). We now prove that (2) implies (1). Since Lk.y/ D fxg, we can find two disjoint domains X and Y on S representing the two vertices x and y of D.S/. By Proposition 6.4, X must be an elementary domain. Two cases are possible: (a) X is an essential annulus; (b) X is a pair of pants. In Case (a), let X be a regular neighborhood of some essential curve ˛. If ˛ were separating, then we would have two distinct connected components of S n X , and their closures would be two non-isotopic domains on S . These closures would define two distinct vertices on D.S/, each of which is joined to x. By connectedness of Y , one of these components would be disjoint from Y , therefore the vertex of D.S/ that it represents would belong to Lk.y/, which is a contradiction. Hence, ˛ is nonseparating. Assume now that we are in Case (b); that is, X is a pair of pants. Let 1 ; 2 ; 3 be the three boundary components of X, and let fN.i /giD1;2;3 be three disjoint regular neighborhoods of these boundary components. Three possible cases arise: (a) X is not a peripheral pair of pants; (b) X is a monoperipheral pair of pants; (c) X is a biperipheral pair of pants. In Case (a), the curves 1 ; 2 ; 3 are essential in S, and N.1 /; N.2 /; N.3 / are disjoint domains on S. If N.i / were homotopic to N.j / for some i 6D j , then the subsurface S n X would have two distinct connected components: the one homotopic to the annulus A joining i to j and the one containing a regular neighborhood of the third boundary curve of X. By connectedness, the domain Y must be contained in one and only one of these components. Hence, the other one gives a vertex in Lk.y/ which is different from x; this is a contradiction. If no N.i / were homotopic to another N.j /, then the fN.i /giD1;2;3 would give three different vertices in D.S/, and at least one of them would be contained in Lk.y/. This again gives a contradiction. In Case (b), assume that is contained in @S. If 2 were homotopic to 3 , then there would exist an annulus joining the two curves. Hence, S would be a torus with one boundary component, which is excluded by the hypothesis.


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Otherwise, N.2 / and N.3 / would give two different vertices on D.S/, and both of them are joined by an edge to x. Since Y is contained in S n X, then at least one of these two vertices can be realized as disjoint from Y , hence we get an element different from x in Lk.y/, which is a contradiction. In Case (c), X is a biperipheral pair of pants, and we may assume that 1 is its unique essential component. Hence, both of N.1 / and the closure of S n X are domains, and they represent two different vertices of D.S /. Each of these two vertices is joined to x by an edge, and exactly one of them belongs to Lk.y/. Again, we have a contradiction. This shows that (2) implies (1), and this ends the proof of Proposition 6.4. Corollary 6.5. Suppose that S is not a torus with at most one hole. Then every simplicial automorphism of D.S/ preserves the subcomplex of D.S/ induced by the set of vertices of D.S/ which are represented by regular neighborhoods of nonseparating curves on S .

6.3 Recognizing elementary vertices in D 2 .S / The proof of the following proposition is the same as that given for Proposition 6.1 and therefore we omit it. Proposition 6.6. Let x 2 D 2 .S/. If Lk.Lk.x// D fxg, then x is elementary. The next proposition is an analogue of Proposition 6.2 for D.S/. Proposition 6.7 (Recognizing elementary vertices in D 2 .S /). Let x 2 D 2 .S /. Suppose that S is neither a sphere with four holes nor a closed torus. Then the following are equivalent: (1) x is elementary. (2) There exists a simplex in D 2 .S/ such that Lk. / D fxg. (3) Lk.Lk.x// D fxg. Proof. Form Proposition 3.16, (2) implies (3). From Proposition 6.6, (3) implies (1). It remains only to prove that (1) implies (2). Suppose that x is elementary and let X be a domain on S representing x. Since x is elementary, X is either an annulus or a pair of pants. If X is an annulus, let be the simplex of D 2 .S / whose vertices are the vertices of D 2 .S / that are represented by codomains of X on S that are not biperipheral pairs of pants on S . If X is a pair of pants, let be the simplex of D 2 .S / whose vertices are the vertices of D 2 .S / that are represented by codomains of X on S that are not biperipheral pairs of pants on S together with the vertices of D 2 .S / that are represented by regular neighborhoods of essential boundary components of X on S.

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Since X is isotopic to a domain on S which is disjoint from all the codomains of X on S and regular neighborhoods of all the essential boundary components of X on S , it follows that x 2 St. /. By Proposition 5.18, x … . Hence, x 2 Lk. /. Suppose, conversely, that w 2 Lk. / and let W be a domain on S representing w. Suppose that X is an annulus. Since w 2 Lk. /, W is isotopic to a domain on S which is disjoint from all the codomains of X on S that are not biperipheral pairs of pants on S . Hence, without loss of generality, W is contained in the union U of X with all the codomains of X on S that are biperipheral pairs of pants. Suppose, on the one hand, that no codomain of X is a biperipheral pair of pants. It follows that U D X and, hence, W X. Since X is an annular domain and W is a domain contained in X, it follows that W is isotopic to X . Thus, w D x. Suppose, on the other hand, that X has a codomain Y which is a biperipheral pair of pants. It follows that X is a separating annulus and hence, has exactly one codomain Z other than Y . Since S is not a sphere with four holes, Z is not a biperipheral pair of pants. It follows that U D X [ Y . Since X is an annular domain on S and X \ Y is one of the essential boundary components of X, it follows that U is isotopic to Y . Hence, W is isotopic to a domain contained in Y . Since W represents a vertex of D 2 .S / and Y is a biperipheral pair of pants, it follows that W is not isotopic to Y . Since Y is a pair of pants and W is a domain contained in Y , it follows that W is isotopic to a regular neighborhood of the unique essential boundary component of Y . This implies that W is isotopic to X. Thus, w D x. This shows that w D x, if X is an annulus. Suppose that X is a nonbiperipheral pair of pants. Let B be the union of all the essential boundary components of X on S that are boundary components of codomains of X which are biperipheral pairs of pants. Since w 2 Lk. /, W is isotopic to a domain which is disjoint from B and all the codomains of X which are not biperipheral pairs of pants. It follows that W is contained in either X n B or Y n B for some biperipheral codomain Y of X . Suppose that W is contained in Y n B. Then W is a domain on S contained in Y . Since W represents a vertex of D 2 .S/ and Y is a biperipheral pair of pants, it follows that W is not isotopic to Y . Since Y is a pair of pants and W is a domain contained in Y , it follows that W is isotopic to a regular neighborhood of the unique essential boundary component of Y . Since w 2 Lk. / and hence, w is not in , this is a contradiction. It follows that W is contained in X n B. Then W is a domain on S contained in X. Suppose that W is not isotopic to X on S . Since X is a pair of pants and W is a domain on S contained in X, it follows that W is isotopic on S to a regular neighborhood of an essential boundary component of X . Since w is not in , this is a contradiction. Hence, W is isotopic to X on S. Thus, w D x.


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Proposition 6.8 (Distinguishing nonseparating annuli from regular neighborhoods of biperipheral curves in D 2 .S/). Suppose that the genus of S is positive and S is not a torus with at most one hole. Let ˛ be an essential curve on S which is either nonseparating or biperipheral and let x be the vertex of D.S/ represented by an essential regular neighborhood of ˛ on S. Then the following are equivalent: (1) ˛ is nonseparating. (2) There exists a top-dimensional simplex of D 2 .S / having x as one of its vertices. Proof. Suppose that ˛ is nonseparating. There exists a pants decomposition C of S with no biperipheral pairs of pants and such that ˛ is one of the curves in C . By Proposition 5.6, the corresponding simplex C of D.S/ is a top-dimensional simplex of D.S /. On the other hand, having no vertices corresponding to biperipheral pairs of pants, C is a simplex of D 2 .S/. Hence, it certainly is a top-dimensional simplex of D 2 .S /. This proves that (1) implies (2). Suppose that there exists a top-dimensional simplex of D 2 .S / having x as one of its vertices. By Proposition 5.20, dim.D 2 .S// D dim.D.S //. Hence, is a top-dimensional simplex of D.S/. It follows that is a maximal simplex of D.S/. By Proposition 5.12, there exists a system of curves C of S such that D C . By Proposition 5.13, C is a pants decomposition. Since x is a vertex of C , x corresponds to either a component of C or to a pair of pants of C . Since x corresponds to ˛, we may assume, by isotoping ˛, that ˛ is a component of C . Suppose that ˛ is biperipheral. Then the biperipheral pair of pants, Y , corresponding to ˛ must be a pair of pants of C . Hence, the corresponding vertex y of D.S/ is an element of C . Since C is a simplex of D 2 .C /, it follows that y is a vertex of D 2 .C /. Since no vertex of D 2 .C / corresponds to a biperipheral pair of pants, this is a contradiction. Hence, ˛ is nonseparating. This proves that (2) implies (1). Proposition 6.9. Suppose that S is neither a sphere with at most three holes nor a torus with at most one hole and let x be a vertex of D 2 .S / represented by a domain X on S . Suppose that X is not a regular neighborhood of a nonseparating curve on S nor a regular neighborhood of a biperipheral curve on S . Then the following are equivalent: (1) Either X is a regular neighborhood of a nonbiperipheral separating curve, or S is a torus with two holes and X is a nonperipheral pair of pants on S , or S is a sphere with five holes and X is a monoperipheral pair of pants on S . (2) There exists an edge e of D 2 .S/ such that Lk.e/ D fxg. Proof. In what follows, we use the notation ŒZ for a vertex represented by a domain Z. Suppose that X is a regular neighborhood of a nonbiperipheral separating curve. Let e be fŒY ; ŒZg where Y and Z are the distinct codomains of X . Since X is

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nonbiperipheral, neither Y nor Z are biperipheral pairs of pants. Hence, e is an edge of D 2 .S /. Since Y and Z are the codomains of X , it follows that Lk.e/ D fxg. Suppose that S is a torus with two holes and X is a nonperipheral pair of pants on S . Note that X has exactly two codomains, U and Z, where U is a biperipheral pair of pants and Z is a regular neighborhood of a nonseparating curve on S. Let Y be a regular neighborhood of the unique essential boundary component of U such that Y and Z are disjoint. Let e be fŒY ; ŒZg. Since Y and Z are not biperipheral pairs of pants on S, e D 2 .S /. Since Y is a regular neighborhood of a biperipheral curve on S and Z is a regular neighborhood of a nonseparating curve on S , Y and Z are nonisotopic on S . Hence, since Y and Z are disjoint domains on S, e is an edge of D 2 .S /. Since X is a pair of pants, X is isotopic to neither Y nor Z. Note that X is isotopic to a domain on S which is disjoint from Y [ Z. Hence, ŒX 2 Lk.e/. Let w 2 Lk.e/. Then there exists a domain W representing w which is disjoint from Y and Z and not isotopic to either Y or Z. Since W is disjoint from Y and Z, W is contained in either U or X. Suppose that W is contained in U . Since W is a domain on S and U is a biperipheral pair of pants on S, it follows that either W is isotopic to U or W is isotopic to a regular neighborhood of the unique essential boundary component of U . Since W represents a vertex of D 2 .S/, W is not isotopic to U . Hence, W is isotopic to a regular neighborhood of the unique essential boundary component of U . It follows that W is isotopic to either Y or Z. This is a contradiction. Hence, W is not contained in U . It follows that W is contained in X. Since W is a domain on S and X is a pair of pants on S , then W is isotopic to either X or a regular neighborhood of an essential boundary component of X. Suppose that W is isotopic to a regular neighborhood of an essential boundary component of X. Then W is isotopic to either Y or Z. This is a contradiction. Hence, W is isotopic to X on S; that is to say, w D ŒX . Hence, Lk.e/ D fŒX g. Suppose that S is a sphere with five holes and X is a monoperipheral pair of pants. Let e be fŒY ; ŒZg where Y and Z are disjoint regular neighborhoods of the two essential boundary components of X. Since S is not a torus with one hole, Y and Z are nonisotopic. Hence, e is an edge of D 2 .S /. Note that X [ Y [ Z is a domain on S isotopic to X with exactly two codomains, U and V , both of which are biperipheral pairs of pants on S. Moreover, the essential boundary components of U and V are isotopic to the essential boundary components of X. Since Y and Z are disjoint regular neighborhoods of essential boundary components of X, X is isotopic to a surface disjoint from Y and Z. Since X is a pair of pants and Y and Z are annuli, X is not isotopic to either Y or Z. Hence, ŒX 2 Lk.e/. Let w 2 Lk.e/. Then there exists a domain W representing w which is disjoint from Y and Z and not isotopic to either Y or Z. Since W is disjoint from Y and Z, W is contained in either X, U , or V . Since W represents a vertex of D 2 .S /, W is not a biperipheral pair of pants. Suppose that W is contained in U . Then, since W is


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not a biperipheral pair of pants, W must be isotopic to a regular neighborhood of the essential boundary component of U . Hence, W is isotopic to either Y or Z. This is a contradiction. Hence, W is not contained in U . Likewise, W is not contained in V . It follows that W is contained in X. Since X is a monoperipheral pair of pants on S , W is isotopic to either X or a regular neighborhood of an essential boundary component of X . The latter possibility would imply that W is isotopic to either Y or Z, leading to a contradiction. Hence, W is isotopic to X ; that is to say, w D ŒX . Hence, Lk.e/ D fxg. This establishes, in all cases, that (1) implies (2). Suppose that there exists an edge e of D 2 .S / such that Lk.e/ D fŒX g. Let e D fy; zg. Note that fx; y; zg is a triangle of D 2 .S /. Since Lk.e/ D fxg, fx; y; zg is a maximal simplex of D 2 .S/. Suppose that fy; zg is an edge of D 2 .S/ such that Lk.e/ D fŒX g. Since ŒX is the link of a simplex of D 2 .S /, it follows by Proposition 6.6, that X is elementary. Since e is an edge of D 2 .S/, S is neither a sphere with four holes nor a closed torus. Let D fx; y; zg. Then is a triangle of D 2 .S / and a maximal simplex of 2 x of D.S/ D .S /. Since D.S/ is finite dimensional, there exists a maximal simplex

containing . It follows by Proposition 5.12 that there exists a system of curves C on x D C . S such that

Let F D fYi j 1 i mg be a tiling representing C and fAj j 1 j ng be the set of ties of F . x which are not vertices of . Let B be the set of vertices of

Since is a maximal simplex of D 2 .S /, the vertices in B are represented by biperipheral pairs of pants on S . It follows that the vertices of B are nonannular vertices of the tiling fYi j 1 i mg. Hence, there exists a map f W B ! 0 defined by the rule ŒYi 7! ŒYk , where Yk is the unique annular tile tied to the nonannular tile Yi by an annular codomain of the tiling fYi j 1 i mg. Suppose that the map B ! 0 is not injective. Then there exist two disjoint biperipheral pairs of pants Yi and Yj and an annular tile Yk such that Yi is tied by an annular codomain Ap of fYi j 1 i mg to Yk and YK is tied by an annular codomain Aq of fYi j 1 i mg to Yj . It follows that S D Yi [ Ap [ Yk [ Aq [ Yj and hence, S is a sphere with four holes. This is a contradiction. Hence, the map B ! 0 is injective. Let r be the number of elements of B. Since is a triangle and the map B ! 0 is injective, it follows that r 3. Let X, Y , and Z be the tiles of fYi j 1 i mg which represent the vertices x, x y, and z of

Suppose that r D 3. In other words, suppose that the map B ! is bijective. Since x, y, and z are vertices of and the map B ! is bijective, there exist distinct tiles U , V , and W of fYi j 1 i mg and distinct annular codomains, A, B, and C of fYi j 1 i mg such that A joins U to X . B joins V to Y , and C joins W

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to Z. Let R D U [ A [ X [ V [ B [ Y [ W [ C [ Z and T be the closure of the complement of R in S . Note that T is a domain on S with at least three essential boundary components. Hence, T is neither an annulus nor a biperipheral pair of pants. Since T is not a biperipheral pair of pants on S , T represents a vertex t of D 2 .S /. Note that T is isotopic to a domain on S which is disjoint from X [ Y [ Z. Since T is not an annulus, T is not isotopic on S to X , Y , or Z. Hence, it follows that t ¤ x and t 2 Lk.fy; zg/ D fxg. This is a contradiction. Hence, r < 3. x D . Then is a maximal Suppose that r D 0. In other words, suppose that

simplex of D.S/. Hence, F D fX; Y; Zg. It follows that x 2 Lk.fy; zg; D.S //. Suppose, conversely, that w 2 Lk.fy; zg; D.S ///. It follows that W is represented by a domain contained in the complement of Y [ Z in S . Since fX; Y; Zg is a tiling of S with three tiles, the complement of Y [ Z in S is equal to the union of X with the ties of F . By Proposition 5.7, each tie of F either joins X to Y , X to Z, or Y to Z. The unique codomain X 0 of Y [ Z on S which contains X is equal to the union of X with those ties of F which join X to either Y or Z. Any remaining codomains of Y [ Z on S are ties of F joining Y to Z. Note that X 0 is isotopic to X on S. Since W is a domain on S contained in the complement of Y [ Z in S, either W is contained in X or W is contained in an annular codomain of X [ Y [ Z joining Y to Z. Suppose that W is contained in a tie A of F joining Y to Z. By Proposition 5.7, since F D fX; Y; Zg and A is a tie of F joining Y to Z, either Y or Z is an annulus isotopic to A on S . Since W is a domain on S contained in the annulus A on S , W is isotopic to A on S. It follows that W is isotopic to either Y or Z on S. Since w 2 Lk.fy; zg; D.S///, this is a contradiction. Hence, W is not contained in a tie of F joining Y to Z. It follows that W is contained in X 0 . Since X 0 is isotopic on S to X , we may assume that W is contained in X. Since X is elementary and W is a domain on S contained in X , W is isotopic to either X or a regular neighborhood of an essential boundary component of X on S. Suppose that W is isotopic to a regular neighborhood of an essential boundary component C of X on S . There exists a unique tie A of F having C as an essential boundary component. Note that A is an annulus joining X to either Y or Z. We may assume that A joins X to Y . By Proposition 5.7, since F D fX; Y; Zg and A is a tile of F joining X to Y , either X or Y is an annulus isotopic to A. Since C is an essential curve on S contained in the annulus A and W is a regular neighborhood of C on S , A is isotopic to W on S. Hence, W is isotopic to either X or Y on S. Suppose that W is isotopic to Y on S. Then w D y. Since w 2 Lk.fy; zg; D.S ///, this is a contradiction. Hence, W is isotopic to X; that is to say, w D x. x D fx; y; z; pg where p is Suppose that r D 1. In other words, suppose that

represented by a biperipheral pair of pants on S. Let P be the tile of F representing p. Then F D fX; Y; Z; P g.


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Let C be the unique essential boundary component of P . Let A be the unique tie of F containing C . Note that A is an annulus on S joining the nonannular tile P to an annular tile Q of F which is isotopic to A on S . It follows that Q is a regular neighborhood of a biperipheral curve on S. Since X is not a regular neighborhood of a biperipheral curve on S, it follows that Q is equal to either Y or Z. We may assume that Q is equal to Y . It follows that Y is an annular tile of F and P [ A [ Y is a domain P 0 on S which is isotopic to P on S. Let D be the unique essential boundary component of P 0 on S and let B be the unique tie of F containing D. Note that B is an annulus on S joining the annular tile Y to a nonannular tile R and that R is equal to either X or Z. Suppose that R is equal to Z. Then Z is not an annulus. Since X is a domain on S , X has an essential boundary component E on S. Let F be the unique tie of F containing E. Note that F is an annulus on S joining X to either Y or Z or P . On the other hand, no such tie can join X to either P or Y . Hence, F joins X to Z. Since Z is not an annulus and R is a tie of F joining Z to X, it follows that X is an annulus isotopic to F on S. Since X is an annular domain on S and E is an essential boundary component of X on S , there exists an essential boundary component G of X on S which is not equal to E. Let H be the unique tie of F containing G. As for the codomain F , it follows that H joins X to Z. Since F [ X [ H is an annulus on S joining two distinct boundary components of the domain Z of S and the interior of F [ X [ H is disjoint from Z, it follows that X is a regular neighborhood of a nonseparating curve on S . This is a contradiction. Hence, R is equal to X. Then X is not an annulus. Since Z is a domain on S, Z has an essential boundary component E on S. Let F be the unique tie of F containing E. Note that F is an annulus on S joining Z to either X or Y or P . On the other hand, no such tie can join Z to either P or Y . Hence, F joins Z to X. Since X is not an annulus and R is a tie of F joining X to Z, it follows that Z is an annulus isotopic to F on S . Since Z is an annular domain on S and E is an essential boundary component of Z on S , there exists an essential boundary component G of Z on S which is not equal to E. Let H be the unique tie of F containing G. As for the tie F , it follows that H joins Z to X. It follows that S D P [ A [ Y [ B [ X [ F [ Z [ H . Since X is elementary with at least 3 boundary components, one on each of the ties B, F , and H of F , it follows that X is a pair of pants. Hence, S is a torus with two holes and X is a nonperipheral pair of pants on S . x D fx; y; z; p; qg where p and Suppose that r D 2. In other words, suppose that

q are represented by disjoint biperipheral pairs of pants on S.

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Let P and Q be the tiles of fYi j 1 i mg representing p and q. Then F D fX; Y; Z; P; Qg. Let C and D be the unique essential boundary components of P and Q and let A be the unique tie of F containing C . Note that A is an annulus on S joining the nonannular tile P to an annular tile R of F which is isotopic to A on S . It follows that R is a regular neighborhood of a biperipheral curve on S . Since X is not a regular neighborhood of a biperipheral curve on S, it follows that R is equal to either Y or Z. We may assume that R is equal to Y . Let B be the unique tie of F containing D. Then B is an annulus on S joining the nonannular tile Q to an annular tile T of F which is isotopic to B on S . It follows that T is a regular neighborhood of a biperipheral curve on S . Since X is not a regular neighborhood of a biperipheral curve on S , it follows that T is equal to either Y or Z. Suppose that T is equal to Y . Then S D P [ A [ Y [ B [ Q. Since fX; Y; Z; P; Qg is a tiling of S, this is a contradiction. Hence, T is equal to Z. By arguments used in the proof for the case where r D 1, it follows that there exist distinct ties F and G of F joining Y to X and Z to X and, hence, S D P [ A [ Y [ F [ X [ G [ Z [ B [ Q. Since Y is an annulus and F is a tie of F joining Y to X , X is not an annulus. Since X is elementary it follows that X is a pair of pants. Since S D P [ A [ Y [ F [ X [ G [ Z [ B [ Q, this implies that S is a sphere with five holes and X is a monoperipheral pair of pants on S .

6.4 Recognizing annular vertices in D 2 .S / Proposition 6.10. Suppose that S is not a torus with one hole and let x 2 D02 .S /. Then the following are equivalent: (1) x is an annular vertex of D 2 .S/. (2) For each vertex y of D 2 .S/ which is not equal to x, St.x/ is not contained in St.y/. Proof. Since D 2 .S/ is a flag complex, requiring property (2) of a vertex x of D 2 .S / is equivalent to requiring that for each vertex y of D 2 .S / which is not equal to x, there exists a vertex z of D 2 .S/ such that fx; zg is a simplex of D 2 .S / and fy; zg is not a simplex of D 2 .S/. We begin by proving that (1) implies (2). To this end, let x be an annular vertex of D 2 .S / and y be a vertex of D 2 .S/ such that y ¤ x. We shall deduce (2) by contradiction. To this end, suppose the following: For every vertex z of D 2 .S / such that fx; zg is a simplex of D 2 .S/, fy; zg is a simplex of D 2 .S /.

./


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Since x is an annular vertex of D 2 .S/, there exists an essential curve ˛ on S such that x is represented by a regular neighborhood of ˛ on S . Choose a maximal system C of curves on S containing ˛. Let R be a regular neighborhood of the support jC j of C on S . For each curve ˇ in the system C, let Rˇ be the unique component of R which contains ˇ and xˇ D ŒRˇ 2 D 2 .S/. Let ˇ 2 C. Since x˛ D x, it follows that fx; xˇ g is a simplex of D 2 .S / and, hence, by condition ./, fy; xˇ g is a simplex of D 2 .S /. In particular, fy; xg D fy; x˛ g is a simplex of D 2 .S /. Since y 6D x, this implies that fy; xg is an edge of D 2 .S/. Hence, S is neither a sphere with at most four holes nor a closed torus. In particular, the Euler characteristic of S is negative. Hence, the maximal system C of curves on S is a pants decomposition of S. Let P be the collection of pairs of pants of C . Let Y be a domain on S representing y. Since fy; xˇ g is a simplex of D 2 .S/ for every curve ˇ in the pants decomposition C of S , it follows that Y is a domain on S which is isotopic on S either to Rˇ for some ˇ in C or to P for some pair of pants P in P . Suppose that Y is isotopic on S to Rˇ for some ˇ in C . Then xˇ D y ¤ x D x˛ and, hence, ˇ ¤ ˛. Since ˛ and ˇ are disjoint nonisotopic essential curves on S, it follows from Proposition 2.3 that there exists a curve on S such that i.˛; / D 0 and i.ˇ; / ¤ 0. Let Z be a regular neighborhood of on S and z D ŒZ 2 D 2 .S /. Since i.˛; / D 0, fx; zg is a simplex of D 2 .S /. Hence, by condition ./, fy; zg is a simplex of D 2 .S/. Since fy; zg is a simplex of D.S/ and z is an annular vertex, it follows that Y is isotopic on S to a domain which is disjoint from Z. Hence, i.ˇ; / D 0, which is a contradiction. Thus, Y is isotopic on S to some pair of pants P in P . Since P represents the vertex y of D 2 .S/, P is not a biperipheral pair of pants on S. Since S is not a torus with one hole and P is a domain on S which is a nonbiperipheral pair of pants on S, it follows that there exists a pair of distinct nonisotopic essential boundary components, and , of P . Since each essential boundary component of a pair of pants of a pants decomposition of S is isotopic to one of the curves of the pants decomposition, there exist distinct curves, ˇ and ı of C such that and are isotopic on S to ˇ and ı. Since ˇ ¤ ı, we may assume that ˛ ¤ ˇ. As before, it follows from Proposition 2.3 that there exists a curve on S such that i.˛; / D 0 and i.ˇ; / ¤ 0. Let Z be a regular neighborhood of on S and z D ŒZ 2 D 2 .S /. Since i.˛; / D 0, fx; zg is a simplex of D 2 .S /. Hence, by condition ./, fy; zg is a simplex of D 2 .S/. Since fy; zg is a simplex of D.S/ and z is an annular vertex, it follows that Y is isotopic on S to a domain which is disjoint from Z. Since ˇ is isotopic on S to and Y , it follows that i.ˇ; / D i.; / D 0, which is a contradiction.

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This shows that (1) implies (2). We shall now show that (2) implies (1). To this end, suppose that the vertex x of D 2 .S / is not an annular vertex of D 2 .S/. Let X be a domain on S representing x. Since x is not an annular vertex of D 2 .S /, X is not an annulus. Since X is a domain on S, X has an essential boundary component on S. Let Y be a regular neighborhood of an essential boundary component of X on S and y D ŒY 2 D 2 .S/. Then Y is isotopic on S to a domain on S which is disjoint from X . Since X is not an annulus and Y is an annulus, Y is not isotopic to X on S . It follows that y is not equal to x and fx; yg is an edge of D 2 .S /. Suppose that z is a vertex of D 2 .S/ such that fx; zg is a simplex of D 2 .S /; that is to say, suppose that either x D z or fx; zg is an edge of D 2 .S /. Suppose, on the one hand, that z D x. Then fy; zg is equal to the simplex fx; yg of D 2 .S /. Suppose, on the other hand, that fx; zg is an edge of D 2 .S /. Then z is represented by a domain Z on S which is disjoint from the domain X on S. Since Y is a regular neighborhood of an essential boundary component of X, it follows that Y is isotopic to a domain on S which is disjoint from Z. This implies that if Z is isotopic to Y , then fy; zg is equal to the simplex fyg of D 2 .S /, whereas, if Z is not isotopic to Y on S, then fy; zg is an edge of D 2 .S/. In any case, fy; zg is a simplex of D 2 .S /. This shows that (2) implies (1). Corollary 6.11. Suppose that S is not a torus with one hole. Then every simplicial automorphism of D 2 .S/ restricts to a simplicial automorphism of the subcomplex .C.S // of D 2 .S/ induced from the set of annular vertices of D 2 .S /. Proof. Let ' 2 Aut.D 2 .S//. Suppose that x is an annular vertex of D 2 .S / (i.e. a vertex of .C.S//). By Proposition 6.10, for each vertex y of D 2 .S / which is not equal to x, there exists a vertex z of D 2 .S / such that fx; zg is a simplex of D 2 .S / and fy; zg is not a simplex of D 2 .S/. Let u D '.x/. Since ' 2 Aut.D 2 .S//, it follows that for each vertex v of D 2 .S / which is not equal to u, there exists a vertex w of D 2 .S / such that fu; wg is a simplex of D 2 .S / and fv; wg is not a simplex of D 2 .S /. Hence, by Proposition 6.10, u is a vertex of .C.S//. This shows that ' maps the zero skeleton of .C.S // into the zero skeleton of .C.S//. Note that a simplex of D 2 .S/ is a simplex of .C.S // if and only if each of its vertices is a vertex of .C.S//. It follows that ' restricts to a simplicial map W .C.S // ! .C.S//. Likewise, the simplicial automorphism ' 1 W D 2 .S / ! D 2 .S / restricts to a simplicial map W .C.S // ! .C.S //. Note that the restrictions W .C.S// ! .C.S // and W .C.S // ! .C.S // of ' W D 2 .S / ! D 2 .S/ and ' 1 W D 2 .S/ ! D 2 .S / are inverse simplicial maps. Hence, W .C.S // ! .C.S// is a simplicial isomorphism. This proves that ' W D 2 .S / ! D 2 .S / restricts to a simplicial automorphism ' W .C.S // ! .C.S //.


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6.5 Recognizing biperipheral edges in D.S / In this section, we give a characterization biperipheral edges of D.S/ . This will be used in the proof of the rigidity result on the automorphism group of the complex of domains, proved in §8 below. Each biperipheral pair of pants on S has a unique biperipheral boundary component. It follows that there is a natural map from the set of vertices of D.S/ corresponding to biperipheral pairs of pants on S to the set of vertices of D.S/ corresponding to biperipheral curves on S . This map is a bijection if and only if S is not a sphere with four holes. Definition 6.12. Suppose that X and Y are domains on S such that X is a biperipheral pair of pants on S and Y is isotopic to a regular neighborhood of the unique essential boundary component of X on S. Then we say that fX; Y g is a biperipheral pair of domains on S and sometimes, a biperipheral pair, and the edge fŒX ; ŒY g of D.S/ is a biperipheral edge of D.S/. Suppose that fX; Y g is a biperipheral pair of domains on S . We may assume that X is a biperipheral pair of pants on S . Since Y is isotopic to a regular neighborhood of an essential boundary component of X on S, Y is isotopic to a domain Y1 on S which is disjoint from X. Since X is not an annulus and Y1 is an annulus, X and Y1 are not isotopic on S . It follows that fX; Y1 g is a system of domains on S and, hence, fŒX ; ŒY g D fŒX; ŒY1 g is, indeed, an edge of D.S/. Since fX; Y1 g is both a biperipheral pair of domains on S and a system of domains on S , we say that fX; Y1 g is a biperipheral system of domains on S. Proposition 6.13 (Vertices with nested stars in D.S/). Let x and y be distinct vertices of D.S /. Then the following are equivalent: (1) St.x; D.S// St.y; D.S//. (2) There exist disjoint domains, X and Y , on S representing x and y which belong to one of the following cases: (a) X is not an annulus and Y is an annulus on S which is joined to X by exactly one annular codomain of X [ Y on S . (b) X is not an annulus and Y is an annulus on S which is joined to X by exactly two annular codomains of X [ Y on S. (c) Y is a biperipheral pair of pants on S which is joined to X by exactly one annular codomain of X [ Y on S . (d) Y is a monoperipheral pair of pants on S which is joined to X by exactly two annular codomains of X [ Y on S . (e) Y is a nonperipheral pair of pants on S which is joined to X by exactly three annular codomains of X [ Y on S.

394


X

C1

B1

Y X

Y

B2

B1 X

C1

Y

Y

X

B2

B1 X

B2

Y

B3 Figure 21. The five cases of ordered pairs of disjoint domains .X; Y / representing ordered pairs of vertices .x; y/ that satisfy the equation St.x; D.S// St.y; D.S// with x ¤ y. See Proposition 6.13.

Proof. Suppose that St.x; D.S// St.y; D.S //. Since x 2 St.x; D.S//, it follows that x 2 St.y; D.S //. Since x ¤ y, this implies that fx; yg is an edge of D.S/; that is to say, x and y are represented by disjoint nonisotopic domains X and Y on S. Suppose that Y is not elementary. By Proposition 2.18, there exist curves ˛ and ˇ on S such that i.˛; ˇ/ ¤ 0 and ˛ and ˇ are contained in the interior of Y . Let W be a regular neighborhood of ˛ on S such that W is contained in the interior of Y . Since W is contained in Y and X is disjoint from Y , X is disjoint from W . This implies that fx; wg is a simplex of D.S/, where w is the vertex of D.S/ represented by W . It follows that w is a vertex of St.x; D.S// and, hence, w is a vertex of St.y; D.S //; that is to say, fy; wg is a simplex of D.S/. Since fy; wg is a simplex of D.S/, either y D w or fy; wg is an edge of D.S/. Since Y is not an annulus on S and W is an annulus on S, Y is not isotopic to W on S; that is to say, y ¤ w. Hence, fy; wg is an edge of D.S/. It follows that W is isotopic on S to a domain on S which is disjoint from Y . Since ˛ is contained in W , it follows that ˛ is isotopic on S to a curve ˛1 which is disjoint from Y and, hence, from ˇ. Since ˛ is isotopic on S to ˛1 and ˛1


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and ˇ are disjoint, it follows that i.˛; ˇ/ D i.˛1 ; ˇ/ D 0, which is a contradiction. Hence, Y is elementary. Suppose that there exists an essential boundary component ˛ of Y which is not isotopic to any essential boundary component of X . Since X and Y are disjoint, it follows from Proposition 2.3 that there exists an essential curve on S such that i.˛; / ¤ 0 and i.ˇ; / D 0 for every essential boundary component ˇ of X . We may assume that the collection C of curves on S consisting of ˛, , and the essential boundary components of X on S is in minimal position. It follows from the above constraints on geometric intersection numbers, that is disjoint from X. Hence, there exists a regular neighborhood Z of on S such that Z is disjoint from X. Since Z is disjoint from X, Z represents a vertex z of St.x; D.S // and, hence, of St.y; D.S//. Thus, fy; zg is a simplex of D.S/. Since fy; zg is a simplex of D.S/, either y D z or fy; zg is an edge of D.S/. Suppose that y D z; that is to say, suppose that Y is isotopic to Z on S. Since Z is an annulus on S , it follows that Y is an annulus on S . Thus Y is isotopic to a regular neighborhood of its essential boundary component ˛. Since Z is a regular neighborhood of and Y is isotopic to Z, it follows that ˛ is isotopic to . Hence, i.˛; / D i.˛; ˛/ D 0 which is a contradiction. Proposition 6.14 (Vertices with the same star in D.S/). Let x and y be distinct vertices of D.S/. Then the following are equivalent: (1) St.x; D.S// D St.y; D.S//. (2) There exist disjoint domains, X and Y , on S representing x and y which belong to one of the following cases: (a) X is a biperipheral pair of pants on S, Y is an annulus on S, and X [ Y has exactly two codomains, exactly one of which is an annulus joining X to Y . (b) Case (2a) with the roles of X and Y interchanged. (c) S is a sphere with four holes, X and Y are biperipheral pairs of pants on S, and X [ Y has exactly one codomain, an annulus joining X to Y . (d) S is a torus with two holes, X and Y are monoperipheral pairs of pants on S, and X [ Y has exactly two codomains, both of which are annuli joining X to Y . (e) S is a closed surface of genus two, X and Y are pairs of pants on S, and X [ Y has exactly three codomains, all of which are annuli joining X to Y . (f) S is a torus with one hole, X is a monoperipheral pair of pants on S, Y is an annulus on S , and X [ Y has exactly two codomains, both of which are annuli joining X to Y . (g) Case (2f) with the roles of X and Y interchanged.

396


Y

X

.X/

.Y /

X

X

Y

X

Y .X/

Y

Y

X .Y /

Figure 22. The seven topological types of ordered pairs of disjoint domains .X; Y / representing ordered pairs of vertices .x; y/ that satisfy the equation St.x; D.S// D St.y; D.S// with x ¤ y. See Proposition 6.14.

Proof. We begin by proving that (1) implies (2). To this, suppose that St.x; D.S // D St.y; D.S //. Since x ¤ y and St.x; D.S// St.y; D.S //, it follows from Proposition 6.13 that x and y are represented by disjoint nonisotopic domains X and Y satisfying one of the five cases, (2a), (2b), (2c), (2d), or (2e), of Proposition 6.13. Note that for such domains, X and Y , the ordered triple .S; X; Y / is uniquely determined up to isotopies on S . Since x ¤ y and St.y; D.S // St.x; D.S //, it follows that .Y; X/ satisfies one of the five cases obtained by interchanging the roles of X and Y in Proposition 6.13. Hence, X and Y are each either an annulus or a pair of pants. Moreover, if either is a pair of pants, all of its essential boundary components are joined to essential boundary components of the other by annular codomains of X [ Y and if either is an annulus, one or both of its essential boundary components is joined to the other by annular codomains of X [ Y . Since X is not isotopic to Y on S and X and Y are joined by at least one annular codomain of X [ Y on S, it follows that either X is a pair of pants on S or Y is a pair of pants on S . If X and Y are both pairs of pants, it follows that they have the same number n of essential boundary components, with 1 n 3. Hence, X and Y satisfy Case (2c), when n D 1; Case (2d), when n D 2; and Case (2e), when n D 3.


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If X is a pair of pants and Y is an annulus, it follows that X is either biperipheral when X is joined to Y by exactly one annular codomain of X [ Y on S or monoperipheral when X is joined to Y by exactly two annular codomains of X [ Y on S. Hence, X and Y satisfy Case (2a), when X is biperipheral; and Case (2f), when X is monoperipheral. Likewise, if X is an annulus and Y is a pair of pants, then X and Y satisfy Case (2b) or Case (2g), according as Y is either biperipheral or monoperipheral. This proves that (1) implies (2). We now prove that (2) implies (1). To this end, suppose that X and Y are disjoint domains as in (2). We must prove that St.x; D.S// D St.y; D.S//. Since D.S/ is a flag complex, it follows from Proposition 3.9 that it suffices to prove that St 0 .x; D.S // D St0 .y; D.S //. We shall give the arguments for Cases (2a) and (2c). The argument for Case (2b) is similar to that for Case (2a). The argument for each of Cases (2d), (2e), (2f), and (2g) is similar to that for Case (2c). First, consider Case (2a). Let X and Y be as in this case. Suppose, on the one hand, that w is a vertex of St.x; D.S //. In other words, suppose that fx; wg is a simplex of D.S/. Then either w D x or fx; wg is an edge of D.S /. If w D x, then fy; wg is the simplex fx; yg of D.S/. Suppose that fx; wg is an edge of D.S/. Then w is represented by a domain W on S which is disjoint from X. Since W is a domain on S disjoint from X and Y is an annulus on S which is joined to X along the unique essential boundary component of X by the unique annular codomain of X [ Y on S, it follows that Y is isotopic on S to a domain which is disjoint from W . It follows, in any case, that fy; wg is a simplex of D.S/; that is to say, w is a vertex of St.y; D.S//. Suppose, on the other hand, that w is a vertex of St.x; D.S //. In other words, suppose that fy; wg is a simplex of D.S/. Then either w D y or fy; wg is an edge of D.S /. If w D y, then fx; wg is the simplex fx; yg of D.S/. Suppose that fy; wg is an edge of D.S/. Then w is represented by a domain W on S which is disjoint from Y . Since W is a domain on S disjoint from Y and Y is an annulus on S which is joined to X along the unique essential boundary component of X by the unique annular codomain of X [ Y on S, it follows that W is isotopic on S to a domain on S which is contained in either X or the complement of X . If W is contained in the complement of X, then fx; wg is a simplex of D.S/. Suppose that W is contained in X. Since W is a domain on S which is contained in the biperipheral pair of pants X on S , either W is isotopic to X on S or W is isotopic to a regular neighborhood of the unique essential boundary component of X on S and, hence, to Y . Hence, fx; wg is equal to either the simplex fxg of D.S/ or the simplex fx; yg of D.S/. It follows, in any case, that fx; wg is a simplex of D.S/; that is to say, w is a vertex of St.x; D.S //. This proves that St0 .x; D.S// D St0 .y; D.S //. This completes the argument for Case (2a). Now consider Case (2c). Let X and Y be as in this case. Let Z be the unique codomain of X [ Y on S. Note that Z is an annulus on S joining the unique essential

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boundary component of X on S to the unique essential boundary component of Y on S . Hence, S D X [ Z [ Y . Since X and Y are biperipheral pairs of pants on S, it follows that X [ Z and Y [ Z are biperipheral pairs of pants on S which are isotopic to X and Y on S and Z is isotopic on S to regular neighborhoods on S of the unique essential boundary components of each of X and Y . Suppose, on the one hand, that w is a vertex of St.x; D.S //. In other words, suppose that fx; wg is a simplex of D.S/. Then either w D x or fx; wg is an edge of D.S /. If w D x, then fy; wg is the simplex fx; yg of D.S/. Suppose that fx; wg is an edge of D.S/. Then w is represented by a domain W on S which is disjoint from X. Since W is a domain on S disjoint from X , it follows that W is a domain on S contained in Y [ Z. Since Y [ Z is isotopic on S to Y , we may assume that W is contained in Y . Since Y is a biperipheral pair of pants on S , it follows that W is isotopic on S to either Y or the unique essential boundary component of Y on S and, hence, to Z. In other words, either w D y or w D z. Hence, fy; wg is equal to either the simplex fyg of D.S/ or the simplex fy; xg of D.S/. This shows, in any case, that fy; wg is a simplex of D.S/; that is to say, w is a vertex of St.y; D.S //. This proves that St 0 .x; D.S// St0 .y; D.S //. By interchanging the roles of X and Y , it follows that St0 .y; D.S// St0 .x; D.S //. Hence, St0 .x; D.S // D St 0 .y; D.S //. This completes the argument for Case (2c). Since, as indicated above, the remaining cases follow by similar arguments, it follows that (2) implies (1). Proposition 6.15. Suppose that S is not a sphere with four holes. Let fx; yg be a pair of distinct vertices of D.S/. Let ' 2 Aut.D.S //. Then fx; yg is a biperipheral edge if and only if f'.x/; '.y/g is a biperipheral edge. Proof. Suppose, on the one hand, that fx; yg is a biperipheral edge of D.S/. (Note that since fx; yg is a biperipheral edge of D.S/, S has at least two boundary components.) We may assume that x and y are represented by disjoint domains X and Y on S satisfying case (2a) of Proposition 6.14. Let A be the unique codomain of X [ Y which joins X to Y . Then X [ A [ Y is a biperipheral pair of pants P on S which is isotopic on S to X . Since S is not a sphere with four holes, the unique codomain Q of P on S is nonelementary. Since P represents the vertex x of D.S/, it follows that Lk.x; D.S // has infinitely many vertices. By Proposition 6.14, St.x; D.S// D St.y; D.S //. As ' W D.S/ ! D.S/ is an automorphism of D.S/, it follows that f'.x/; '.y/g is an edge of D.S/, Lk.'.x/; D.S // has infinitely many vertices, and St.'.x/; D.S // D St.'.y/; D.S //. Since '.x/ ¤ '.y/ and St.'.x/; D.S// D St.'.y/; D.S //, it follows from Proposition 6.14 that x and y are represented by disjoint domains X 0 and Y 0 satisfying one of the seven cases of Proposition 6.14. Suppose that f'.x/; '.y/g is not a biperipheral edge of D.S/. Then X 0 and Y 0 satisfy one of cases (2c), (2d), (2e), (2f), or (2g) of Proposition 6.14. Note that in any case, since X 0 represents the vertex '.x/ of D.S/, it follows that Lk.'.x/; D.S // has


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at most four vertices, which is a contradiction. (In fact, Lk.'.x/; D.S // has at most three vertices.) Hence, f'.x/; '.y/g is a biperipheral edge of D.S/. Suppose, on the other hand, that f'.x/; '.y/g is a biperipheral edge of D.S/. Then, since ' 1 W D.S/ ! D.S/ is an automorphism of D.S/, it follows from the above argument, that fx; yg is a biperipheral edge of D.S/. This completes the proof.

7 Automorphisms of the truncated complex of domains 7.1 Distinguishing vertices of D.S / via their annular links Definition 7.1. Let x be a vertex of D.S/. The annular link of x in D.S/ is the subcomplex Ann.x/ of D.S/ consisting of those simplices of Lk.x; D.S // all of whose vertices are annular. Proposition 7.2. Suppose that S is a sphere with four holes. Let x and y be vertices of D.S / and Ann.x/ and Ann.y/ be their annular links in D.S/. Then Ann.x/ D Ann.y/ if and only if one of the following holds: (1) x D y; (2) x and y are annular vertices of D.S/; (3) x and y are represented by the two pairs of pants of some pants decomposition of S . Proof. Note that there are two types of vertices of D.S/; those which are represented by a biperipheral pair of pants on S , and those which are represented by a regular neighborhood of a biperipheral curve on S . Let x 2 D0 .S / and X be a domain on S representing x. If X is a biperipheral pair of pants on S , then Ann.x/ is the vertex of D.S / represented by a regular neighborhood of the unique essential boundary component of X. If X is an annulus on S, then Ann.x/ D ;. This implies that Ann.x/ D Ann.y/ in each of the three cases: (1), (2), and (3). Conversely, suppose that Ann.x/ D Ann.y/. Let X and Y be domains on S representing x and y. We may assume that X is a biperipheral pair of pants on S . It follows that Ann.y/ D Ann.x/ D fwg, where w is represented by a regular neighborhood of the unique essential boundary component @ of X . Since Ann.y/ is nonempty, it follows that Y is also a biperipheral pair of pants on S. Since Ann.y/ D fwg, it follows that the unique essential boundary component of Y is isotopic to @. Hence, we may assume that X and Y are joined on S by an annulus A on S with boundary components @ and . Since X and Y are biperipheral pairs of pants on S, it follows that S D X [ A [ Y . Hence, X and Y are pairs of pants of a pants decomposition of S.

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Proposition 7.3. Suppose that S is a torus with one hole. Let x and y be vertices of D.S / and Ann.x/ and Ann.y/ be their annular links in D.S/. Then Ann.x/ D Ann.y/ if and only if one of the following holds: (1) x D y; (2) x and y are annular vertices of D.S/. Proof. The proof follows along the same lines as the proof of Proposition 7.2. In this case the two types of vertices of D.S/ are those which are represented by monoperipheral pairs of pants on S and those which are represented by annuli on S. Each monoperipheral pair of pants on S has two essential boundary components joined by an annulus on S. The corresponding vertex of D.S/ has the vertex of D.S/ represented by this annulus as its annular link. The annular vertices of D.S/ have empty annular links. Proposition 7.4. Suppose that S is a torus with two holes. Let x and y be vertices of D.S / and Ann.x/ and Ann.y/ be their annular links in D.S/. Then Ann.x/ D Ann.y/ if and only if one of the following holds: (1) x D y; (2) x and y are represented by the two pairs of pants of an embedded pants decomposition of S . Proof. Again, the proof follows along the same lines as the proof of Proposition 7.2. In this case, there are four types of vertices of D.S/: those which are represented by annuli on S; those which are represented by monoperipheral pairs of pants on S ; those which are represented by biperipheral pairs of pants on S; and those which are represented by tori with one hole on S. Suppose that X is an annulus. Then Ann.x/ is an infinite discrete set of vertices, corresponding to the curve complex of a four holed sphere if X is a regular neighborhood of a nonseparating curve on S and to the curve complex of a one-holed torus if X is a regular neighborhood of an essential separating curve on S . Suppose that X is a monoperipheral pair of pants on S. Then X has a unique pair of essential boundary component ˛ and ˇ on S and a unique codomain Y on S . Moreover, Y is a monoperipheral pair of pants on S. Let u and v be the annular vertices of D.S/ corresponding to regular neighborhoods of ˛ and ˇ on S. Then Ann.x/ and Ann.y/ are both equal to the edge fu; vg of D.S/. Suppose that X is a biperipheral pair of pants on S . Then X has a unique essential boundary component @ and a unique codomain Y on S. Moreover Y is a one-holed torus on S. Let w be the annular vertex of D.S/ corresponding to a regular neighborhood of @ on S . Then Ann.x/ is the join of w to the infinitely many annular vertices of D.S/ corresponding to C.Y /. In particular, Ann.x/ is an infinite connected subcomplex of D.S/. Suppose that X is a torus with one hole. Then X has a unique essential boundary component @ and a unique codomain Y on S. Moreover, Y is a biperipheral pair of


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pants on S. Let y be the nonannular vertex of D.S/ corresponding to Y and w be the annular vertex of D.S/ corresponding to a regular neighborhood W of @ on S . Then Ann.x/ D fwg and Lk.x/ D fy; wg. The result now follows by using the above descriptions of the annular links of the four types of vertices of D.S/ to compare Ann.x/ with Ann.y/. Proposition 7.5. Suppose that S is a closed surface of genus two. Let x and y be vertices of D.S/ and Ann.x/ and Ann.y/ be their annular links in D.S/. Then Ann.x/ D Ann.y/ if and only if one of the following holds: (1) x D y; (2) x and y are represented by the two pairs of pants of some embedded pants decomposition of S . Proof. Again, the proof follows along the same lines as the proof of Proposition 7.2. Proposition 7.6. Suppose that S is neither a sphere with four holes nor a torus with at most one hole. Let x and y be vertices of D.S/. Suppose that x is annular. Then the following are equivalent: (1) Ann.x/ Ann.y/; (2) x D y or there exist disjoint domains X and Y on S representing x and y such that X is an annulus on S, Y is a biperipheral pair of pants on S , and X [ Y has exactly two codomains, exactly one of which is an annulus joining X to Y . Proof. We begin by proving that (1) implies (2). Suppose that Ann.x/ Ann.y/. Since x is annular, x is represented by a regular neighborhood X of an essential curve ˛ on X. Since S has an essential curve ˛, S is not a sphere with at most three holes. Since S is also not a closed torus, there exists a pants decomposition C of S containing ˛. Let R be a regular neighborhood of the support jC j of C on S and P be the collection of codomains of R on S. We may assume that X is the unique component of R which contains the element ˛ of C. Note that each element of P is a pair of pants on S. Let ˇ be an element of C which is not equal to ˛. Then a regular neighborhood Z of ˇ on S represents a vertex z of Ann.x/ and, hence, of Ann.y/. It follows that y is represented by a domain Y on S which is disjoint from and not isotopic to each of the components of a regular neighborhood W of the union of all the elements of C which are not equal to ˛. Since Y is connected, Y is contained in a codomain of W on S . Note that the unique codomain V of W on S which contains X is equal to the union of X with those elements of P which share at least one common essential boundary component with X. If there is exactly one such element of P , then W is a torus with one hole. Otherwise, there are exactly two such elements of P and W is a sphere with four holes.

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Every other codomain U of W on S is a pair of pants on S all of whose essential boundary components are isotopic to elements of C which are not equal to ˛. Suppose that Y is contained in one of these other codomains U of W on S . Since U is a pair of pants and Y is a domain on S contained in U , Y is isotopic on S to a domain Y1 on S such that Y1 is equal to U or Y1 is a regular neighborhood of an essential boundary component of U on S. Note, in any case, that an essential boundary component ı of U is contained in Y1 . By assumption, ı is isotopic to an element ˇ of C which is not equal to ˛. Since ˛ and ˇ are distinct elements of the pants decomposition C , ˛ and ˇ are disjoint nonisotopic essential curves on S . It follows from Proposition 2.3 that there exists a curve on S such that i.; ˛/ D 0 and i.; ˇ/ ¤ 0. Since ı is isotopic to ˇ, i.; ı/ D i.; ˇ/. Hence, i.; ı/ ¤ 0. It follows that a regular neighborhood Z of on S represents a vertex z of Ann.x/ and, hence, of Ann.y/. Since Y1 represents the vertex y of D.S/ and Z represents the vertex z of D.S/, it follows that Z is isotopic on S to a domain Z1 on S which is disjoint from Y1 . Thus, is isotopic on S to a curve 1 on S which is disjoint from ı. Thus, i.; ı/ D i.1 ; ı/ D 0, which is a contradiction. Hence, Y is not contained in one of these other codomains U of W on S. It follows that Y is not isotopic on S to a domain which is contained in one of the other codomains U of W on S. It follows that Y is contained in the unique codomain V of W on S which contains X . Since S is not a sphere with four holes nor a torus with one hole, V has an essential boundary component ı on S. Note that ı is isotopic on S to an element ˇ of C which is not equal to ˛. As before, it follows from Proposition 2.3 that there exists a curve on S such that i.; ˛/ D 0 and i.; ˇ/ ¤ 0. Since i.; ˛/ D 0, we may assume that is disjoint from ˛. Since ı is isotopic to ˇ, i.; ı/ D i.; ˇ/. Hence, i.; ı/ ¤ 0. It follows that a regular neighborhood Z of on S represents a vertex z of Ann.x/ and, hence, of Ann.y/. Hence, Y is isotopic on X to a domain Y1 on X which is disjoint from Z and, hence, from . Note that X and Y1 are both domains on S which are contained in V and are disjoint from . We may assume that the number of points of intersection of with each essential boundary component of V is equal i.; /. Since i.; ı/ ¤ 0, it follows that \ V is a nonempty disjoint union of properly embedded essential arcs on V . Suppose, on the one hand, that V is a torus with one hole. Then, since X and Y1 are both domains on S contained in V and disjoint from a properly embedded essential arc on V , it follows that X and Y1 are isotopic annuli on S and, hence, x D y. Suppose, on the other hand, that V is a sphere with four holes. Then, since X and Y1 are both domains on S contained in V and disjoint from a properly embedded essential arc on V , it follows that Y1 is isotopic to a domain Y2 on S which is contained in one of the two elements P of P which share an essential boundary component with the annulus X.


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Suppose that Y2 is isotopic to a regular neighborhood of an essential boundary component of P . Note that is isotopic to an element ˇ of P . Since Y2 is not isotopic to a regular neighborhood of any element of C which is not equal to ˛, it follows that ˇ is equal to ˛. This implies that X is isotopic to Y2 on S and, hence, x D y. Hence, we may assume that Y2 is not isotopic to a regular neighborhood of any essential boundary component of P . Since Y2 is a domain on S contained in the pair of pants P , it follows that Y2 is isotopic to P on S . Suppose that there exists an essential boundary component of P such that is not isotopic to ˛ on S. Then is isotopic to an element ˇ of C which is not equal to ˛. As before, it follows from Proposition 2.3 that there exists a curve on S such that i.; ˛/ D 0 and i.; ˇ/ ¤ 0. It follows that a regular neighborhood Z of on S represents a vertex z of Ann.x/ and, hence, of Ann.y/. Since P represents y, it follows that Z is isotopic to a domain on S which is disjoint from P . Hence, is isotopic to a curve 1 on S which is disjoint from . This implies that i.; ˇ/ D i.1 ; / D 0, which is a contradiction. Hence, each essential boundary component of P is isotopic to ˛ on S. It follows that P is either a monoperipheral pair of pants sharing both of its essential boundary components with X or a biperipheral pair of pants sharing its unique essential boundary component with X. In the former case, it follows that S is a torus with one hole, which is a contradiction. Hence, the latter case holds. Since Y2 is a nonannular domain on S contained in the biperipheral pair of pants P on S , it follows that Y2 is a biperipheral pair of pants on S whose unique codomain on P is an annulus on S joining X to Y2 . This completes the proof that (1) implies (2). It remains to prove that (2) implies (1). If x D y, then Ann.x/ D Ann.y/ and, hence, Ann.x/ Ann.y/. Suppose that x and y are represented by disjoint domains X and Y on S such that X is an annulus on S , Y is a biperipheral pair of pants on S , and X [ Y has exactly two codomains, exactly one of which is an annulus joining X to Y . Let P be the unique codomain of X on S such that Y is contained in P . Note that P is a biperipheral pair of pants on S . Suppose that z is an element of Ann.x/. Then z is represented by an annulus Z on S which is disjoint from and not isotopic to X. Since Z is connected and disjoint from X, Z is contained in a codomain Q of X on S . Suppose that Q is equal to P . Then since Z is an annular domain on S and P is a biperipheral pair of pants on S, it follows that Z is isotopic to a regular neighborhood of the unique essential boundary component of P on S. Since every essential boundary component of a codomain of X on S is an essential boundary component of the annulus X , it follows that Z is isotopic to X on S , which is a contradiction. Hence, Q is not equal to P .

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Since any two distinct codomains of X on S are disjoint, it follows that Z is disjoint from P and, hence, from Y . Note that the annulus Z is not isotopic on S to the pair of pants Y . Hence, the vertex z of D.S/ represented by the annulus Z is an element of Ann.y/. This proves that (2) implies (1), completing the proof. Proposition 7.7. Suppose that S is neither a sphere with four holes nor a torus with at most one hole. Let x and y be vertices of D.S/. Suppose that x is annular. Then Ann.x/ D Ann.y/ if and only if x D y. Proof. It suffices to prove that Ann.x/ D Ann.y/ implies x D y. To this end, suppose that Ann.x/ D Ann.y/. Then x is annular and Ann.x/ Ann.y/. It follows from Proposition 7.6 that either (i) x D y or (ii) x and y are represented by an annulus X on S and a biperipheral pair of pants Y on S such that X [ Y has exactly two codomains, exactly one of which is an annulus joining X to Y . Suppose that (ii) holds. Then x is a vertex of Ann.y/; that is to say, since Ann.x/ D Ann.y/, x is a vertex of Ann.x/. Since Ann.x/ is a subcomplex of Lk.x; D.S //, it follows that x is a vertex of Lk.x; D.S//, which is a contradiction. Hence, (ii) does not hold. It follows that (i) holds; that is to say, it follows that x D y, completing the proof. Remarks 7.8. If S is a sphere with at most three holes, then D.S/, D 2 .S /, and C.S / are empty. Hence, Proposition 7.7 is vacuously true in that case. If S is a closed torus, then C.S/ is a countably infinite set of vertices. Each vertex of D.S / is an annular vertex of D.S/ and, hence, D 2 .S / is equal to the image of C.S / in D.S/ under the natural inclusion W C.S / ,! D.S/, and D.S/ D D 2 .S /. Moreover, each vertex of D.S/ has empty annular link. Hence, Proposition 7.7 is false for the closed torus. Proposition 7.9. Let x and y be vertices of D.S/. Suppose that fx; yg is a simplex of D.S /. Then Ann.x/ Ann.y/ if and only if either x D y or x and y are represented by disjoint domains X and Y on S such that Y is a pair of pants with each of its essential boundary components on S joined to X by annuli. Proof. Suppose, on the one hand, that Ann.x/ Ann.y/. We may assume that x is not equal to y. Then, since fx; yg is a simplex of D.S/, fx; yg is an edge of D.S/. It follows that x and y are represented by disjoint domains X and Y on S which are not isotopic to one another on S. Suppose that Y is a nonelementary domain on S . It follows from Proposition 2.18 that there exist curves ˛ and ˇ on S such that i.˛; ˇ/ ¤ 0 and ˛ and ˇ are contained in the interior of Y . Let U and V be regular neighborhoods of ˛ and ˇ in the interior of Y . Suppose that V is isotopic to X on S. Then ˇ is isotopic on S to a curve ˇ1 which is contained


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in the interior of X and, hence, is disjoint from Y . Since ˇ is isotopic to ˇ1 on S, i.˛; ˇ/ D i.˛; ˇ1 /. Since ˛ is contained in Y and ˇ1 is disjoint from Y , it follows that ˛ and ˇ1 are disjoint and, hence, i.˛; ˇ1 / D 0. We conclude that i.˛; ˇ/ D 0 which is a contradiction. Hence, V is not isotopic to X on S. Since V is contained in Y and X and Y are disjoint, X and V are disjoint domains on S . It follows that V represents an annular vertex v of Lk.x; D.S//. This implies that v is a vertex of Ann.x/ and, hence, of Ann.y/. Since V represents v and Y represents y, it follows that V is isotopic on S to a domain on S which is disjoint from Y . Since ˇ is contained in V , it follows that ˇ is isotopic on S to a curve ˇ2 which is disjoint from Y . Again, this implies that i.˛; ˇ/ D i.˛; ˇ2 / D 0, which is a contradiction. It follows that Y is an elementary domain on S . Suppose that Y is an annulus. Since X and Y are disjoint nonisotopic domains on S , it follows that the annular vertex y of D.S/ represented by Y is a vertex of Ann.x/ and, hence of Ann.y/. Since Ann.y/ is a subcomplex of Lk.y; D.S //, it follows that y is a vertex of Lk.y; D.S//, which is a contradiction. Hence, Y is not an annulus. Since Y is an elementary domain on S, it follows that Y is a pair of pants. Let ˇ be an essential boundary component of Y on S . Suppose that ˇ is not isotopic to any essential boundary component of X on S. Then, by Proposition 2.3, there exists an essential curve on S such that i.; ˛/ D 0 for every essential boundary component ˛ of X on S and i.; ˇ/ ¤ 0. Since Y is disjoint from X, it follows that a regular neighborhood W of on S represents a vertex w of Ann.x/ and, hence, of Ann.y/. It follows that is isotopic on S to a curve 1 on S which is disjoint from Y . It follows that i.; ˇ/ D i.1 ; ˇ/ D 0, which is a contradiction. Hence, the essential boundary component ˇ of Y on S is isotopic on S to some essential boundary component ˛ of X on S . Since X and Y are disjoint, it follows that there is an annulus A on S whose boundary components are ˛ and ˇ. Since Y is a pair of pants, it follows that A \ Y D ˇ. Moreover, either A \ X D ˛ or X A. In the former case, A is an annulus joining ˇ to X . Suppose X A. Then X is an annulus contained in A. It follows that A D X [B, where B is an annulus joining ˇ to X. In any case, ˇ is joined to X by an annulus. This proves the “only if” direction. It remains to prove the “if” direction. If x D y, then Ann.x/ D Ann.y/ and, hence, Ann.x/ Ann.y/. Suppose that x and y are represented by disjoint domains X and Y on S such that Y is a pair of pants with each of its essential boundary components on S joined to X by annuli. Since Y is disjoint from X on S and each of the essential boundary components of Y on S is joined to X by an annulus, it follows that Y is isotopic on S to the unique codomain Y1 of X on S which contains Y .

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Let z be an element of Ann.x/. Then z is represented by an annulus Z on S which is disjoint from X and, hence, is contained in a codomain W of X on S . Suppose that W is not equal to Y1 . Then Z is disjoint from Y1 and, hence, from Y . Since Y is a pair of pants, it follows that the annulus Z is not isotopic to Y on S . Hence, the annular vertex z of D.S/ represented by Z is a vertex of Ann.y/. Suppose that W is equal to Y1 . Since Z is an annular domain on S contained in the pair of pants Y1 on S, it follows that Z is isotopic on S to an annulus Z1 on S which is disjoint from Y1 and, hence, from Y . Note that the annulus Z1 is not isotopic to the pair of pants Y on S . Hence, the vertex z of D.S/ represented by Z1 is a vertex of Ann.y/. In any case, z is an element of Ann.y/. Again, we conclude that Ann.x/ Ann.y/. In any case, Ann.x/ Ann.y/. This proves the “if” direction, completing the proof. Proposition 7.10. Suppose that S is neither a sphere with four holes nor a torus with at most one hole. Let x and y be vertices of D.S/. Suppose that fx; yg is not a simplex of D.S /. Then Ann.x/ Ann.y/ if and only if x and y are represented by domains X and Y on S such that Y is a domain on X . Proof. Since fx; yg is not a simplex of D.S /, x ¤ y. Suppose that Ann.x/ Ann.y/. Suppose that x is an annular vertex of D.S/. Since S is neither a sphere with four holes nor a torus with at most one hole and x ¤ y, it follows from Proposition 7.6 that x and y are represented by disjoint domains X and Y on S. Hence, fx; yg is a simplex of D.S/, which is a contradiction. Therefore, x is not an annular vertex of D.S /. It follows that a regular neighborhood Z of any essential boundary component ˛ of X represents a vertex z of Ann.x/ and, hence, of Ann.y/. It follows that y is represented by a domain Y on S which is not isotopic to a regular neighborhood of any essential boundary component of X on S and is disjoint from a regular neighborhood of the union of the essential boundary components of X on S. Since Y is disjoint from a regular neighborhood of the union of the essential boundary components of X on S, either Y is disjoint from X or Y is contained in X. Since fx; yg is not a simplex of D.S/, Y is not disjoint from X. Hence, Y is contained in X. Since Y is a domain on S contained in the domain X on S, it follows from Proposition 2.13, that Y is isotopic on S to either X , or a domain on X , or a regular neighborhood of an essential boundary component of X . Since x ¤ y, Y is not isotopic on S to X . Since Y is not isotopic on S to a regular neighborhood of any essential boundary component of X on S, we conclude that Y is isotopic to a domain Y1 on X . Hence, x and y are represented by domains X and Y1 on S such that Y1 is a domain on X.


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This proves the “only if” direction. It remains to prove the “if” direction. Suppose that x and y are represented by domains X and Y on S such that Y is a domain on X . Let z be a vertex of Ann.x/. Then z is represented by an annulus Z on S which is disjoint from X . Since Y is contained in X, it follows that Z is disjoint from Y . Suppose that Y is isotopic to Z on S. Then Y is an annulus on S. Hence, Y is a regular neighborhood of an essential curve ˛ on S . Since Y is a domain on X , ˛ is an essential curve on X. It follows from Proposition 2.10, that there exists an essential curve ˇ on X such that the geometric intersection number of ˛ and ˇ on S is not equal to zero. Since Z is an annular domain on S, Z is a regular neighborhood of an essential curve on S . Since ˇ is contained in X and Z is disjoint from X , it follows that ˇ is disjoint from and, hence, i.; ˇ/ D 0. Since i.˛; ˇ/ ¤ 0, it follows that ˛ is not isotopic to on S . This implies that Y is not isotopic to Z. Since Z is an annulus disjoint from Y and not isotopic to Y on S, it follows that the vertex z of D.S / represented by Z is a vertex of Ann.y/. This proves the “if” direction, completing the proof. Proposition 7.11. Suppose that S is neither a sphere with four holes nor a torus with at most one hole. Let x and y be vertices of D.S/. Then the following are equivalent: (1) Ann.x/ D Ann.y/. (2) x D y or there exist disjoint domains X and Y on S, representing x and y, which belong to one of the following cases: (a) S is a torus with two holes, X and Y are monoperipheral pairs of pants on S , and X [ Y has exactly two codomains, both of which are annuli joining X to Y ; (b) S is a closed surface of genus two, X and Y are pairs of pants on S, and X [ Y has exactly three codomains, all of which are annuli joining X to Y . Proof. We begin by proving the “only if” direction. Suppose that Ann.x/ D Ann.y/. If x is annular, then, since Ann.x/ D Ann.y/, it follows from Proposition 7.7 that x D y. Likewise, if y is annular, then, since Ann.y/ D Ann.x/, it follows from Proposition 7.7 that y D x. Hence, if either x or y is annular, then x D y. Thus, we may assume that neither x nor y is annular. We may assume that x ¤ y. Suppose that fx; yg is not a simplex. Then, since Ann.x/ Ann.y/, it follows from Proposition 7.10, that x and y are represented by domains X and Y on S such that Y is a domain on X. Likewise, since Ann.y/ Ann.x/, it follows from Proposition 7.10, that y and x are represented by domains Y1 and X1 on S such that Y1 is a domain on X1 . Thus X is isotopic to a domain on Y and Y is isotopic to a domain on X , which contradicts Proposition 2.11. Hence, fx; yg is a simplex of D.S/. Since Ann.x/ Ann.y/ and x ¤ y, it follows from Proposition 7.9 that x and y are represented by disjoint domains X and Y on

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S such that Y is a pair of pants with each of its essential boundary components on S joined to X by annuli. This implies that the number of essential boundary components of Y on S is less than or equal to the number of essential boundary components of X on S . Likewise, since Ann.y/ Ann.x/ and y ¤ x, it follows from Proposition 7.9 that y and x are represented by disjoint domains Y1 and X1 on S such that X1 is a pair of pants with each of its essential boundary components on S joined to Y1 by annuli. Again, this implies that the number of essential boundary components of X1 on S is less than or equal to the number of essential boundary components of Y1 on S. Since X and X1 both represent x, X is isotopic to the pair of pants X1 on S. This implies that X is a pair of pants on S with the same number of essential boundary components on S as X1 . Likewise, Y1 is a pair of pants on S with the same number of essential boundary components on S as Y . Since the number of essential boundary components of X1 on S is less than or equal to the number of essential boundary components of Y1 on S, it follows that the number of essential boundary components of X on S is less than or equal to the number of essential boundary components of Y on S . Since the number of essential boundary components of Y on S is less than or equal to the number of essential boundary components of X on S, we conclude that X is a pair of pants on S with the same number of essential boundary components on S as the pair of pants Y on S. Thus, X and Y are disjoint pairs of pants on S with the same number n of essential boundary components on S and the n essential boundary components of X on S are joined by disjoint annuli to the n essential boundary components of Y on S. Since X is a pair of pants domain on S , 1 n 3. If n D 1, then S is a sphere with four holes, which is a contradiction. Hence, 2 n 3. If n D 2, then X and Y satisfy case (2a). If n D 3, then X and Y satisfy case (2b). This completes the proof of the “only if” direction. It remains to prove the “if” direction. If x D y, then Ann.x/ D Ann.y/. Suppose that X and Y are as in case (2a). Note that the two codomains of X [ Y on S are annuli which are disjoint from and not isotopic on S to the pairs of pants X and Y on S . Hence, they represent vertices of Ann.x/ and Ann.y/. Suppose that z is a vertex of Ann.x/. Then z is represented by an annulus on S which is contained in the unique codomain Y1 of X on S. Note that Y1 is a pair of pants on S which is isotopic to Y on S. It follows that Z is isotopic to a regular neighborhood of an essential boundary component ˛ of Y1 on S . Since Y1 is a codomain of X on S , ˛ is an essential boundary component of X on S . It follows that Z is isotopic to one of the two codomains of X [ Y on S . This proves that Ann.x/ is the edge of D.S / whose vertices are represented by the two codomains of X [ Y on S . Likewise, Ann.y/ is equal to this edge and, hence, Ann.x/ D Ann.y/. Similarly, if X and Y are as in case (2b), then Ann.x/ D Ann.y/. In any case, Ann.x/ D Ann.y/. This proves the “if” direction, completing the proof.


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7.2 Automorphisms of D 2 .S / are geometric In this section, we prove that if S is not a sphere with at most four holes, a torus with at most two holes, or a closed surface of genus two, then each automorphism of D 2 .S / is induced by a self-homeomorphism of S which is uniquely defined up to isotopy on S . This will imply that, under the same hypothesis on S, Aut.D 2 .S // ' .S / and, if b 1, Aut.D.S// ' .S/. Lemma 7.12. Suppose that S is not a torus with one hole. Let i W C.S / ! D 2 .S / be the natural inclusion corresponding to forming regular neighborhoods of essential curves on S. Let ' W D 2 .S/ ! D 2 .S/ be an automorphism of D 2 .S /. Then there exists an automorphism W C.S/ ! C.S/ such that ' B i D i B . Proof. Let a be a vertex of C.S/, x D i.a/, and u D '.x/. Note that x is an annular vertex of D 2 .S/. Since ' 2 Aut.D 2 .S//, it follows by Corollary 6.11, that u is an annular vertex of D 2 .S/. Hence, there exists a vertex b of C.S / such that i.b/ D u. Since i W C.S/ ! D.S/ is injective, such a vertex is unique. It follows that the correspondence a 7! b yields a well-defined function W C0 .S / ! C0 .S / such that '.i.a// D i..a// for every vertex a of C.S /. Since curves on S are disjoint if and only if they have disjoint regular neighborhoods, it follows that W C0 .S / ! C0 .S / extends to a simplicial map W C.S/ ! C.S /. Since '.i.a// D i. .a// for every vertex a of C.S/, it follows that ' B i D i B W C.S / ! D 2 .S /. This shows that there exists a simplicial map W C.S/ ! C.S/ such that ' B i D i B W C.S / ! D 2 .S /. Likewise, there exists a simplicial map W C.S / ! C.S / such that ' 1 B i D i B W C.S / ! D 2 .S/. Since i is injective, it follows that is an inverse for . Hence, W C.S / ! C.S/ is an automorphism of C.S /. Theorem 7.13. Suppose that S is not a sphere with four holes, a torus with at most two holes, or a closed surface of genus two. Then the natural homomorphism W .S / ! Aut.D 2 .S // corresponding to the action of .S / on D 2 .S / is an isomorphism. Proof. We begin by showing that is surjective. To this end, we let ' 2 Aut.D 2 .S // and show that there exists a homeomorphism H W S ! S such that ' D H W D 2 .S / ! D 2 .S /. To simplify the exposition, we identify C.S /, via i W C.S / ! D.S/, with its image in D 2 .S /. Since S is not a torus with one hole, using this identification, we may restate Lemma 7.12 as saying that ' restricts to an element of Aut.C.S //. Since S is neither a sphere with at most four holes nor a torus with at most two holes, it follows from Theorem 1 of Ivanov [53] and Theorem 1 of Korkmaz [74] (see Luo [82] for a different proof) that there exists a homeomorphism H W S ! S such that D H W C.S/ ! C.S/. Let D H1 B ' W D 2 .S / ! D 2 .S /. Note that fixes every vertex of C.S/. We shall now show that is equal to the identity map of D 2 .S /; that is to say, we shall show that ' D H W D 2 .S/ ! D 2 .S/.

410


Let v 2 D 2 .S/. Since is an automorphism of D 2 .S / preserving C.S /, .Ann.v// D Ann. .v//. On the other hand, since Ann.v/ is a subcomplex of C.S / and fixes each vertex of C.S/, .Ann.v// D Ann.v/. Hence, Ann. .v// D Ann.v/. Since S is not a sphere with four holes, a torus with at most two holes, or a closed surface of genus two, it follows from Proposition 7.11 that .v/ D v. This proves that ' D H W D 2 .S/ ! D 2 .S / and, hence, the natural homomorphism W .S / ! Aut.D 2 .S// is surjective. It remains to show that W .S/ ! Aut.D 2 .S // is injective. To this end, suppose that h is an element of the kernel of . Let H W S ! S be a homeomorphism representing h. Since h 2 ker./, H induces the trivial automorphism of D 2 .S /. Let ˛ be an essential curve on S and X be a regular neighborhood of ˛ on S. It follows that ŒX D H ŒX D ŒH.X/ and, hence, H.X / is isotopic to X on S . This implies that H.˛/ is isotopic to ˛ on S . Thus H W S ! S preserves the isotopy class of every essential curve on S. In other words, h is in the kernel of the action of .S / on C.S /. Since S is not a sphere with at most three holes, it follows from Proposition 2.6 that H W S ! S is orientation-preserving. This implies that h is in the kernel of the action of .S/ on C.S/. Since S is not a sphere with at most four holes, a torus with at most two holes, or a closed surface of genus two, it follows from [60], Lemma 5.1 and Theorem 5.3, that h is equal to the identity element of .S /. This proves that W .S/ ! Aut.D 2 .S // is injective, completing the proof.

8 Automorphisms of the complex of domains 8.1 Exchange automorphisms of D.S / Throughout the rest of this chapter, let E denote the set of biperipheral edges of D.S/. Proposition 8.1. Suppose that S is not a sphere with four holes. Then there exists a monomorphism ˆ W B.E/ ! Aut.D.S// from the Boolean algebra B.E/ of all subsets of E to Aut.D.S// such that for each collection F of biperipheral edges of D.S /, ˆ.F / D 'F exchanges the two vertices of each biperipheral edge in F and fixes every vertex of D.S/ which is not a vertex of some biperipheral edge in F . Proof. It follows from Propositions 6.14 and 3.28 that E is a collection of exchangeable edges of D.S/. Since S is not a sphere with four holes, no two distinct edges in E have a common vertex. Hence, the result follows from Proposition 3.31. Following the terminology of Definition 3.32, we call the image of the Boolean algebra B.E/ under the monomorphism ˆ of Proposition 8.1 the Boolean subgroup BE of D.S /. In particular, the Boolean subgroup BE is naturally isomorphic to the Boolean algebra B.E/.


411

Proposition 8.2. Let ' 2 Aut.D.S//, F E and G D '.F /. Then G E and ' B 'F B ' 1 D 'G . Proof. This is an immediate consequence of Propositions 6.15 and 3.33. Proposition 8.3. BE is a normal subgroup of Aut.D.S //. Proof. This is an immediate consequence of Proposition 8.2. Proposition 8.4. The monomorphism ˆ W B.E/ ! Aut.D.S // is natural with respect to the action of the extended mapping class group .S / on D.S/. More precisely, if h 2 .S / and F E, then ˆ.h .F // D h B ˆ.F / B h1 . Proof. This is an immediate consequence of Propositions 8.2 and 8.3. Proposition 8.5. There is a natural monomorphism: W B.E/ Ì .S/ ! Aut.D.S // corresponding to the action of .S/ on D.S/ and the induced action on the set E of biperipheral edges of D.S/. Proof. Since, by Proposition 8.4, the monomorphism ˆ W B.E/ ! BE is natural, there exists a natural homomorphism W B.E/ Ì .S / ! Aut.D.S //. Since a pair of pants is not homeomorphic to an annulus, a geometric automorphism of D.S/ cannot exchange the vertices of any biperipheral edge of D.S/. It follows that the image of the extended mapping class group .S / in Aut.D.S // under the natural homomorphism W .S/ ! Aut.D.S// corresponding to the action of .S / on D.S / has trivial intersection with the Boolean subgroup BE of Aut.D.S //. Since the natural homomorphism ˆ W B.E/ ! BE is injective, it remains only to show that W .S / ! Aut.D.S// is injective. To this end, suppose that h 2 .S / is in the kernel of . Since D 2 .S/ is a subcomplex of D.S/, it follows that h induces the trivial automorphism of D 2 .S/. Since S is not a sphere with four holes, a torus with at most two holes, or a closed surface of genus two, it follows from Theorem 7.13 that h is equal to the identity element of .S/. This proves that W .S / ! Aut.D.S // is injective, completing the proof.

8.2 Automorphisms of D.S / Proposition 8.6. Suppose that S is not a sphere with four holes. Let W D.S/ ! D 2 .S / be the natural projection from D.S/ to D 2 .S / sending each vertex of D.S/ corresponding to a biperipheral pair of pants on S to the annular vertex of D.S/ corresponding to its unique essential boundary component on S . If ' 2 Aut.D.S //, then there exists a unique simplicial automorphism ' W D 2 .S / ! D 2 .S / such that ' B D B ' W D.S/ ! D 2 .S/ .

412


Proof. Let i W D 2 .S/ ! D 2 .S/ denote the inclusion map of the subcomplex D 2 .S / of D.S / into D.S/ and ' D B ' B i W D 2 .S / ! D 2 .S /. Note that ' W D 2 .S / ! D 2 .S / is a simplicial map from D 2 .S/ to D 2 .S /. We shall prove that ' B D B ' W D.S/ ! D 2 .S /. To this end, let x be a vertex of D.S/. Suppose, on the one hand, that x 2 D 2 .S /. Then, by the definition of W D.S/ ! 2 D .S /, .x/ D x and, hence, .' B /.x/ D ' ..x// D ' .x/ D . B ' B i /.x/ D .'.i.x/// D .'.x// D . B '/.x/. Suppose, on the other hand, that x is not in D 2 .S /. Since x 2 D.S/ and x is not in D 2 .S/, x is represented by a biperipheral pair of pants X on S. Let Y be a regular neighborhood of the unique essential boundary component of X on S and y be the vertex of D.S/ represented by Y . Then fx; yg is a biperipheral edge of D.S /. It follows from the definition of W D.S/ ! D.S/, that .x/ D y D .y/. Moreover, it follows from Proposition 6.15 that f'.x/; '.y/g is a biperipheral edge of D.S /. Hence, either '.x/ is represented by a biperipheral pair of pants on S or '.y/ is represented by a biperipheral pair of pants on S. In the former case, it follows from the definition of W D.S/ ! D 2 .S/, that .'.x// D '.y/ D .'.y//. In the latter case, it follows from the definition of W D.S/ ! D 2 .S /, that .'.x// D '.x/ D .'.y//. Hence, in any case, .'.x// D .'.y//. It follows that .' B /.x/ D ' ..x// D .'.i..x/// D .'..x// D .'.y// D .'.x// D . B '/.x/. This shows, in any case, that .' B /.x/ D . B '/.x/ and, hence, ' B D B ' W D.S / ! D 2 .S/. Suppose that ˇ W D 2 .S/ ! D 2 .S/ is a simplicial map such that ˇ B D B ' W D.S / ! D 2 .S/. Then ˇ B D ' B W D.S/ ! D 2 .S /. Since W D.S/ ! D 2 .S / is surjective, it follows that ˇ D ' W D.S/ ! D 2 .S /. This proves that there exists a unique simplicial map ' W D 2 .S / ! D 2 .S / such that ' B D B ' W D.S / ! D 2 .S/. It remains only to prove that ' W D 2 .S/ ! D 2 .S / is a simplicial automorphism of 2 D .S /. To this end, consider the simplicial automorphism D ' 1 W D.S/ ! D.S/. of D.S /. Repeating the above argument, we conclude that there exists a unique simplicial map W D 2 .S/ ! D 2 .S/ such that B D B W D.S/ ! D 2 .S /. It follows that .' B / B D ' B . B / D ' B . B / D .' B / B D . B '/ B D B .' B / D W D.S/ ! D 2 .S /. Since W D.S/ ! D 2 .S / is surjective, it follows that ' B W D 2 .S/ ! D 2 .S / is the identity map of D 2 .S /. Likewise, we conclude that B ' W D 2 .S/ ! D 2 .S / is the identity map of D 2 .S /. Hence, ' W D 2 .S/ ! D 2 .S/ and W D 2 .S / ! D 2 .S / are inverse simplicial maps. This shows that ' W D 2 .S/ ! D 2 .S / is a simplicial automorphism of D 2 .S /, completing the proof. Proposition 8.7. Suppose that S is not a sphere with four holes. Let W D.S/ ! D 2 .S / be the natural projection from D.S/ to D 2 .S / sending each vertex of D.S/ corresponding to a biperipheral pair of pants on S to the annular vertex of D.S/ corresponding to its unique essential boundary component on S. Then there exists a unique


413

homomorphism W Aut.D.S// ! Aut.D 2 .S // such that for each automorphism ' 2 Aut.D.S//, .'/ is the unique simplicial automorphism ' W D 2 .S / ! D 2 .S / such that ' B D B ' W D.S/ ! D 2 .S /. Moreover, there exists a natural exact sequence 1 ! BE ! Aut.D.S // ! Aut.D 2 .S //: Proof. By Proposition 8.6, there is a map W Aut.D.S // ! Aut.D 2 .S // such that for each automorphism ' 2 Aut.D.S//, .'/ is the unique simplicial automorphism ' W D 2 .S / ! D 2 .S/ such that ' B D B ' W D.S/ ! D 2 .S /. Suppose that ' W D.S/ ! D.S/ and W D.S/ ! D.S/ are elements of Aut.D.S //. Since ' W D 2 .S / ! D 2 .S/ and W D 2 .S/ ! D 2 .S / are automorphisms of D 2 .S /, D . B '/ B .' B / B D ' B . B / D ' B . B / D .' B / B D B .' B /. It follows from the uniqueness clause of Proposition 8.6 that .' B / D ' B W D 2 .S/ ! D 2 .S/ and, hence, W Aut.D.S // ! Aut.D 2 .S // is a homomorphism. This proves the existence and uniqueness of such a homomorphism W Aut.D.S// ! Aut.D 2 .S//. Since BE is by definition a subgroup of Aut.D.S //, the natural homomorphism BE ! Aut.D.S// is injective. Suppose that ' 2 BE . By the definition of BE , there exists a unique subset F of the collection E such that ' exchanges the two vertices of each pair of distinct vertices of D.S / in the collection F and fixes every vertex of D.S/ which is not one of the two vertices of some pair of distinct vertices of D.S/ in the collection F . Suppose that z is a vertex of D 2 .S/. Since W D.S/ ! D 2 .S / is a surjective simplicial map, there exists a vertex x of D.S/ such that .x/ D z. Suppose, on the one hand, that x is one of the two vertices of some distinct pair of vertices fx; yg of D.S/ in the collection F . Since fx; yg is in F , ' interchanges x and y and, hence, '.x/ D y. Since F is a subset of E, it follows from the definition of W D.S / ! D 2 .S/ that .y/ D .x/. Hence, ' .z/ D ' ..x// D .'.x// D .y/ D .x/ D z. Suppose, on the other hand, that x is not one of the two vertices of any distinct pair of vertices of D.S/ in the collection F . Then, '.x/ D x. Hence, ' .z/ D ' ..x// D .'.x// D .x/ D z. In any case, it follows that the simplicial automorphism ' W D 2 .S / ! D 2 .S / of 2 D .S / fixes every vertex of D 2 .S/ and is, hence, the identity map of D 2 .S /. This proves that the image of the natural homomorphism BE ! Aut.D.S // is in the kernel of W Aut.D.S// ! Aut.D 2 .S//. Conversely, suppose that ' W D.S/ ! D.S/ is in the kernel of W Aut.D.S // ! Aut.D 2 .S //. Then ' W D 2 .S/ ! D 2 .S/ is the identity map of D 2 .S /. Let x be a vertex of D.S/, y D '.x/, and z D .x/. Since z is a vertex of D 2 .S /, it follows that z D ' .z/ D ' ..x// D .'.x// D .y/. Hence, x and y are in the same fiber of W D.S/ ! D 2 .S /. It follows from the definition of W D.S/ ! D 2 .S/ that either y D x or fx; yg is a pair of distinct vertices of D.S/ in the collection E.

414


Suppose that y is not equal to x. Then fx; yg is a pair of distinct vertices of D.S/ in the collection E. Let w D '.y/. Repeating the previous argument, with .y; w; z/ rather than .x; y; z/. we conclude that either w D y or fy; wg is a pair of distinct vertices of D.S/ in the collection E. Suppose that w D y. Then '.y/ D w D y D '.x/. Since ' W D.S/ ! D.S/ is a simplicial automorphism and, hence, injective, it follows that y D x, which is a contradiction. It follows that w is not equal to y and, hence, fy; wg is a pair of distinct vertices of D.S/ in the collection E. Since fx; yg and fy; wg are both pairs of distinct vertices of D.S/ in the collection E having at least one common vertex y and no two distinct pairs of vertices in the collection E have a common vertex, it follows that fx; yg D fy; wg. Since w is not equal to y, it follows that w D x; that is to say, '.y/ D x. This proves that for each vertex x of D.S /, either '.x/ D x or x is one of the two vertices of a pair fx; yg of distinct vertices of D.S/ in E and ' exchanges x and y. Let F be the subset of E consisting of all pairs of distinct vertices of D.S/ in E which are exchanged by '. It follows that ' W D.S/ ! D.S/ is equal to the generalized exchange 'F W D.S/ ! D.S/ of D.S/ associated to F . By the definition of BE , ' is an element of BE . Hence, the kernel of W Aut.D.S // ! Aut.D 2 .S // is in the image of the natural homomorphism BE ! Aut.D.S //. This proves that the image of the natural homomorphism BE ! Aut.D.S // is equal to the kernel of W Aut.D.S// ! Aut.D 2 .S//. Theorem 8.8. Suppose that S is not a sphere with at most four holes, a torus with at most two holes, or a closed surface of genus two. Every automorphism of D.S/ is a composition of an exchange automorphism of D.S/ with a geometric automorphism of D.S /. More precisely, let E be the collection of biperipheral edges of D.S/. Let ' 2 Aut.D.S //. Then there exists a unique subset F of E and a unique element h of .S / such that ' is equal to the composition 'F B h of the exchange automorphism 'F of D.S / corresponding to F and the geometric automorphism h of D.S/ induced by h. Proof. Let ' 2 Aut.D.S//. We begin by proving the existence of such a factorization of '. Since S is not a sphere with four holes, it follows from Proposition 8.6 that there exists a unique simplicial automorphism W D 2 .S / ! D 2 .S / such that B D B ' W D.S / ! D 2 .S/. Since S is not a sphere with at most four holes, a torus with at most two holes, or a closed surface of genus two, it follows from Theorem 7.13 that there exists a homeomorphism H W S ! S such that .ŒX / D ŒH.X / for every domain X on S which is not a biperipheral pair of pants. Let G D H 1 W S ! S and G W D.S/ ! D.S/ be the geometric automorphism of D.S / defined by the rule G .ŒX/ D ŒG.X / for every domain X on S. Note that G B ' W D.S/ ! D.S/ is an automorphism of D.S/. We shall now show that G B ' is an exchange automorphism.


415

Since S is not a sphere with four holes and ' 2 Aut.D.S //, it follows from Proposition 6.15 that '.E/ D E/. Suppose that X is a domain on S which is not a biperipheral pair of pants or a regular neighborhood of a biperipheral curve. Note that ŒX is not a vertex of an edge in E. Since ' is an automorphism of D.S/ and '.E/ D E, it follows that '.ŒX / is not a vertex of an edge in E. Hence, '.ŒX/ D ŒY where Y is a domain on S which is not a biperipheral pair of pants or a regular neighborhood of a biperipheral curve. By the definition of the natural projection W D.S/ ! D 2 .S /, .ŒX / D ŒX and .ŒY / D ŒY . Hence, .'.ŒX/ D '.ŒX/. Therefore, '.ŒX/ D .'.ŒX// D

..ŒX / D

.ŒX / D ŒH.X /;

and it follows that .' B G /.ŒX/ D 'ŒG.X/ D ŒH.G.X // D ŒX : This shows that ' B G fixes every vertex of D.S/ which is not a vertex of an edge in E. By a similar argument, it follows that if X is a domain on S which is a biperipheral pair of pants and Y is a regular neighborhood of the corresponding biperipheral curve, then either (i) .'BG /.ŒX/ D ŒX and .'BG /.ŒY / D ŒY or (ii) .'BG /.ŒX / D ŒY and .' B G /.ŒY / D ŒX. Let i D ' B G . It follows that i is an exchange automorphism of D.S/. Since i D ' B G , we conclude that ' D 'F B h where F is the subcollection of E consisting of all biperipheral edges of D.S/ whose vertices are exchanged by i , and h is the isotopy class of H W S ! S. This proves the existence of such a factorization of '. Suppose that 'F B h D 'P B q . Then 'P 4F D .g B h1 / . Since an automorphism of D.S/ which is induced by a homeomorphism of S cannot exchange an annular vertex with a nonannular vertex of D.S/, it follows that P 4F D ;. In other words, F D P . Since 'P 4F D .g B h1 / and F D P , it follows that .g B h1 / is the trivial automorphism id W D.S/ ! D.S/ of D.S/. In other words, g B h1 acts trivially on D.S /. Since D 2 .S/ is a subcomplex of D.S/, it follows that g B h1 acts trivially on 2 D .S /. Since S is not a sphere with four holes, a torus with at most two holes, or a closed surface of genus two, it follows from Theorem 7.13 that g B h1 is equal to the identity element of .S/. In other words, g is equal to h. This proves the uniqueness of such a factorization of ', completing the proof. We can summarize the preceding results as follows. Theorem 8.9. Suppose that S is not a sphere with at most four holes, a torus with at most two holes, or a closed surface of genus two. Then we have a natural commutative

416


diagram of exact sequences: 1

/ B.E/ '

1

/ B.E/ Ì .S / '

/ BE

/ Aut.D.S//

/ .S /

/1

'

/ Aut.D 2 .S //

/1

Proof. The exactness of the first row of the above diagram follows immediately from the definition of a semi-direct product. The commutativity of the left-hand square follows from Propositions 8.1 and 8.5. The commutativity of the right-hand square follows from Propositions 8.5 and 8.6. ' The isomorphism B.E/ ! BE follows from Proposition 8.1 and the definition of the Boolean subgroup BE of Aut.D.S//. Since S is not a sphere with at most four holes, a torus with at most two holes, or a closed surface of genus two, it follows from Theorem 7.13 that the natural homo' morphism .S/ ! Aut.D 2 .S// is an isomorphism. Since the natural homomorphisms B.E/ Ì .S / ! .S / and .S / ! Aut.D 2 .S // are both surjective, it follows from the commutativity of the right-hand square that the natural homomorphisms Aut.D.S // ! Aut.D 2 .S // is also surjective. Hence, since S is not a sphere with four holes, the exactness of the second row of the above diagram follows from Proposition 8.7. This shows that the diagram is a commutative diagram of exact sequences. Since the vertical arrows on the left and right are both isomorphisms, it follows from standard results that the natural monomorphism B.E/Ì .S / ! Aut.D.S // of Proposition 8.5 is an isomorphism, completing the proof.

9 Directions for further work The following directions for further work naturally arise after this overview: Study the various subcomplexes of the complex of domains that are mentioned in §5, in particular their automorphism group, their large-scale geometry and their boundary structure. Classify their isometries. Study axes of actions of pseudo-Anosov elements. Find relations between these complexes and Teichmüller spaces and their boundaries. Discover to what extent combinatorics implies topology; that is, from the combinatorial information at the vertices, recover as much as possible the topological objects on the surface that are represented by these vertices, as we did here in Section 6. Study natural maps between the various complexes. Use these complexes to get cohomological properties of mapping class groups. Study analogous complexes associated to non-orientable surfaces and to surfaces of infinite topological type.


417

References [1]

J. Aramayona, Simplicial embeddings between pants graphs. Geom. Dedicata 144 (2010), 115–128. 350

[2]

J. Aramayona, , H. Parlier and K. J. Shackleton, Totally geodesic subgraphs of the pants complex. Math. Res. Lett. 15 (2008), no. 2–3, 309–320. 350

[3]

J. Aramayona, H. Parlier, and K. J. Shackleton, Constructing convex planes in the pants complex. Proc. Am. Math. Soc. 137 (2009), no. 10, 3523–3531. 350

[4]

J. Behrstock, B. Kleiner, Y. Minsky, and L. Mosher, Geometry and rigidity of mapping class groups. Preprint, first version 2008, arXiv:0801.2006v4 [math.GT]. 340

[5]

J. Behrstock and D. Margalit, Curve complexes and finite index subgroups of mapping class groups. Geom. Dedicata 118 (2006), 71–85. 340

[6]

G. C. Bell and K. Fujiwara, The asymptotic dimension of a curve graph is finite. J. Lond. Math. Soc. (2) 77 (2008), no. 1, 33–50. 339

[7]

S. Benvenuti, Finite presentations for the mapping class group via the ordered complex of curves. Adv. Geom. 1 (2001), No.3, 291–321. 342

[8]

L. Bers, An inequality for Riemann surfaces. In Differential geometry and complex analysis, Springer-Verlag, Berlin 1985, 87–93.

[9]

J. Behrstock and D. Margalit, Curve complexes and finite index subgroups of mapping class groups. Geom. Dedicata 118, no. 1, (2006), 71–85.

[10]

J. S. Birman, Mapping class groups and their relationship to braid groups. Comm. Pure. Appl. Math. 22 (1969), 213–238.

[11]

J. S. Birman. On Siegel’s modular group. Math. Ann. 191 (1971), 59–68. 360

[12]

J. S. Birman, Braids, links and mapping class groups. Ann. of Math. Stud. 82, Princeton University Press, Princeton, N.J., 1974. 336

[13]

J. S. Birman and H. M. Hilden, On isotopies of homeomorphisms of Riemann surfaces. Ann. of Math. 97 (1973), 424–439.

[14]

J. S. Birman, A. Lubotzky, and J. McCarthy, Abelian and solvable subgroups of the mapping class group. Duke Math. J. 50 (1983), 1107–1120.

[15]

B. Bowditch, Hyperbolic 3-manifolds and the geometry of the curve complex. In Proceedings of the 4th European congress of mathematics (Stockholm, 2004), Europ. Math. Soc., Zurich 2005, 103–115. 339

[16]

B. Bowditch, Intersection numbers and the hyperbolicity of the curve complex. J. Reine Angew. Math. 598 (2006), 105–129. 338

[17]

B. Bowditch, Tight geodesics in the curve complex. Invent. Math. 171 (2008), no. 2, 281–300. 338

[18]

J. Brock and B. Farb, Curvature and rank of Teichmüller space. Am. J. Math. 128 (2006), no. 1, 1–22. 350

[19]

T. E. Brendle and D. Margalit, Commensurations of the Johnson kernel. Geom. Topol. 8 (2004), 1361–1384. 359, 360, 361

[20]

T. E. Brendle and D. Margalit, Addendum to: Commensurations of the Johnson kernel. Geom. Topol. 12 (2008), 97–101. 361

418


[21]

J. Brock, The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex cores. J. Amer. Math. Soc. 16 (2003), 495–535. 350

[22]

J. Brock and D. Margalit, Weil–Petersson isometries via the pants complex. Proc. Amer. Math. Soc. 135 (2007), no. 3, 795–803 350

[23]

C. Charitos, I. Papadoperakis, and A. Papadopoulos, On the homeomorphisms of the space of geodesic laminations on a hyperbolic surface. Preprint, arXiv:1112.1935v2 [math.GT]. 301

[24]

G. Daskalopoulos and R. Wentworth, Classification of Weil–Petersson isometries. Amer. J. Math. 125 (2003), no. 4, 941–975. 351

[25]

M. Dehn, Die Gruppe der Abbildungsklassen. Acta Math. 69 (1938), 135–206. 309

[26]

V. Disarlo, On the coarse geometry of the complex of domains. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2012, 425–439. 302

[27]

D. B. A. Epstein, Curves on two-manifolds and isotopies. Acta Math. 115 (1965), 83–107. 304, 307

[28]

B. Farb, Automorphisms of the Torelli group. To appear, preliminary report, AMS Sectional Meeting, Ann Arbor, MI., 2003.

[29]

B. Farb and N. V. Ivanov, The Torelli geometry and its applications: research announcement. Math. Res. Lett. 12 (2005), no. 2–3, 293–301. 359, 360, 361

[30]

D. Gabai, Almost filling laminations and the connectivity of ending lamination space. Geom. Topol. 13 (2009), 1017–1041. 339

[31]

R. Guo and F. Luo, Cell decompositions of Teichmüller spaces of surfaces with boundary. Preprint 2010. 345

[32]

L. Funar, C. Kapoudjian, and V. Sergiescu, Asymptotically rigid mapping class groups and Thompson’s groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2011, 595–664. 349

[33]

U. Hamenstädt, Geometry of the complex of curves and of Teichmüller space. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, EMS Publishing House, Zurich 2007, 447–467. 339, 342

[34]

U. Hamenstädt, Stability of quasigeodesics in Teichmüller space. Geom. Dedicata 146 (2010), 101–116. 340

[35]

U. Hamenstädt, Train tracks and the Gromov boundary of the complex of curves. In Spaces of Kleinian groups (Y. Minsky et al., eds.), London Math. Soc. Lecture Note Ser. 329, Cambridge University Press, Cambridge 2006, 187–207. 339, 341

[36]

U. Hamenstädt, Train tracks and the Gromov boundary of the complex of curves. In Spaces of Kleinian groups (Y. Minsky et al., eds.), London Math. Soc. Lecture Note Ser. 329, Cambridge University Press, Cambridge 2006, 187–207. 339, 341

[37]

K. Hartshorn, Heegaard splittings of Haken manifolds have bounded distance. Pacific. J. Math. 204 (2002), no. 1, 61–75. 341

[38]

W. J. Harvey, Geometric structure of surface mapping class groups. In Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser. 36, Cambridge University Press, Cambridge 1979, 255–269. 334


419

[39]

J. L. Harer, Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2) 121 (l985), 215–249. 301, 334, 351

[40]

J. L. Harer, The second homology group of the mapping class group of an orientable surface. Invent. Math. 72 (1983), no. 2, 221–239. 358

[41]

J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84 (1986), 157–176. 338, 342

[42]

J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves. Invent. Math. 85 (1986), 457–485. 345

[43]

A. E. Hatcher, On triangulations of surfaces. Topology Appl. (1991), 40, no. 2, 189–194. A revised and expanded version is available on the author’s webpage, http://www.math.cornell.edu/~hatcher/Papers/TriangSurf.pdf 346, 351

[44]

A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface. Topology 19 (1980), 221–237. 298, 299, 349, 356, 357

[45]

J. Hempel, 3-manifolds as viewed from the curve complex. Topology 40 (2001), no. 3, 631–657. 341

[46]

S. Hirose, A complex of curves and a presentation for the mapping class group of a surface. Osaka J. Math. 39 (2002), no. 4, 795–820. 342

[47]

E. Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups. Topology 43 (2004), no. 3, 513–541. 337, 338

[48]

E. Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups. II. Topology Appl. 153 (2006), no. 8, 1309–1340. 337, 338

[49]

E. Irmak, Complexes of nonseparating curves and mapping class groups. Michigan Math. J. 54 (2006), no. 1, 81–110. 337, 338, 356, 358

[50]

E. Irmak and J. D. McCarthy, Injective simplicial maps of the arc complex. Turkish J. of Math. 33 (2009), 1–16. 343, 344

[51]

E. Irmak, Injective simplicial maps of the complex of arcs on nonorientable surfaces. Algebr. Geom. Topol. 9 (2009), no. 4, 2055–2077. 344

[52]

E. Irmak and M. Korkmaz, Automorphisms of the Hatcher–Thurston complex. Isr. J. Math. 162 (2007), 183–196. 299, 357, 358

[53]

N. V. Ivanov, Automorphisms of Teichmüller modular groups. Lecture Notes in Math. 1346, Springer-Verlag, Berlin 1988, 199–270. 335, 409

[54]

N. V. Ivanov, Subgroups of Teichmüller modular groups. Transl. Math. Monogr. 115, Amer. Math. Soc., Providence, R.I., 1992.

[55]

N. V. Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces. Preprint IHES/M/89/60, 1989; also in Progress in knot theory and related topics, Travaux en Cours, V. 56, Hermann, Paris 1997, 113–120.

[56]

N. V. Ivanov, Permutation representations of braid groups of surfaces. Mat. Sb. 181 (1990), no. 11, 1464–1474; English transl. Math. USSR-Sb. 71 (1992), no. 2, 309–318).

[57]

N. V. Ivanov,Automorphisms of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices 1997 (1997), no. 14, 651–666.

420


[58]

N. V. Ivanov, Isometries of Teichmüller spaces from the point of view of Mostow rigidity. In Topology, ergodic theory, real algebraic geometry (V. Turaev and A. Vershik, eds.), Amer. Math. Soc. Transl. Ser. 2 202, Amer. Math. Soc., R.I., 2001, 131–149. 338

[59]

N. V. Ivanov, Mapping class groups. Handbook of geometric topology, North-Holland, Amsterdam 2002, 523–633. 334

[60]

N. V. Ivanov and J. D. McCarthy, On injective homomorphisms between Teichmüller modular groups, I. Invent. Math. 135 (1999), 425–486. 309, 410

[61]

N. V. Ivanov and L. Ji, Infinite topology of curve complexes and non-Poincaré duality of Teichmüller modular groups. Enseign. Math. (2) 54 (2008), no. 3–4, 338

[62]

L. Ji, Curve complexes versus Tits buildings: structures and applications. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zurich, to appear. 301, 334, 342

[63]

D. L. Johnson, Homeomorphisms of a surface which act trivially on homology. Proc. Amer. Math. Soc. 75 (1979), 119–125. 309

[64]

D. Johnson, A survey of the Torelli group. Contemp. Math. 20 (1983), 165–179. 360

[65]

R. M. Kaufmann, Arc geometry and algebra: foliations, moduli spaces, string topology and field theory. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zurich, to appear. 346

[66]

R. M. Kaufmann, M. Livernet, and R. C.Penner, Arc operads and arc algebras. Geom. Topol. 7 (2003), 511–568. 346

[67]

Y. Kida, Automorphisms of the Torelli complex and the complex of separating curves. J. Math. Soc. Japan 63 (2011), no. 2, 363–417. 359, 360, 361

[68]

Y. Kida, The co-Hopfian property of the Johnson kernel and the Torelli group. Osaka J. Math., to appear. 359, 361

[69]

Y. Kida, Injections of the complex of separating curves into the Torelli complex. Preprint, arXiv:0911.3926v3 [math.GR]. 361

[70]

Y. Kida and S. Yamagata, Automorphisms of the Torelli complex for the one-holed genus two surface. Preprint, arXiv:1009.0568v2 [math.GR]. 361

[71]

Y. D. Kim, The Thurston boundary of Teichmüller space and complex of curves. Topology Appl. 154 (2007), no. 3, 675–682. 339

[72]

E. Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space. Unpublished manuscript, 1999. 339

[73]

M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147 (1992), 1–23. 345

[74]

M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topology Appl. 95 (1999), no. 2, 85–111. 335, 409

[75]

M. Korkmaz and A. Papadopoulos, On the arc and curve complex of a surface. Math. Proc. Cambridge Philos. Soc. 148 (2010), no. 3, 473–483. 346, 347

[76]

M. Korkmaz and A. Papadopoulos, On the ideal triangulation graph of a punctured surface. Ann. Inst. Fourier, to appear. 351, 352, 353

[77]

M. Korkmaz and S. Schleimer. Automorphisms of the disk complex. Preprint, arXiv:0910.2038v1 [math.GT]. 333


421

[78]

C. Leininger and S. Schleimer, Connectivity of the space of ending laminations. Duke Math. J. 150 (2009), no. 3, 533–575. 339

[79]

W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds. Ann. of Math. (2) 76 (1962), 531–540. 334

[80]

E. Looijenga, Cellular decompositions of compactified moduli spaces of pointed curves. In The moduli space of curves (R. H. Dijkgraaf et al., eds.), Progr. Math. 129, Birkhäuser, Basel 1995, 369–400. 345

[81]

E. Looijenga, Connectivity of complexes of separating curves. Preprint, arXiv:1001.0823v2 [math.GT]. 359

[82]

F. Luo, Automorphisms of the complex of curves. Topology 39 (2000), no. 2, 283–298. 336, 337, 409

[83]

F. Luo, Automorphisms of Thurston’s space of measured laminations. In In the tradition of Ahlfors and Bers, Proceedings of the first Ahlfors-Bers colloquium (I. Kra et al., eds.), Contemp. Math. 256, Amer. Math. Soc., Providence, R.I., 2000, 221–225. 300, 337

[84]

F. Luo, Grothendieck’s reconstruction principle and 2-dimensional topology and geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 733–765. 337

[85]

J. Maher, Linear progress in the complex of curves. Trans. Amer. Math. Soc. 362 (2010), no. 6, 2963–2991, 341

[86]

D. Margalit, Automorphisms of the pants complex. Duke Math. J. 121 (2004), no. 3, 457–479. 299, 349

[87]

H. Masur and M. Wolf, The Weil–Petersson isometry group, Geom. Dedicata 93 (2002), 177–190. 300, 339, 350

[88]

H. A. Masur and Y. N. Minsky, Geometry of the complex of curves I: Hyperbolicity. Invent. Math. 138 (1999), 103–149. 334, 338, 341

[89]

H. A. Masur and Y. N. Minsky, Geometry of the complex of curves II: Hierarchical structure. Geom. Funct. Anal. 10 (2000), no.4, 902–974. 339

[90]

H. A. Masur and Y. N. Minsky, Quasiconvexity in the curve complex. In In the tradition of Ahlfors and Bers III (W. Abikoff et al. eds.), Proceedings of the 3rd Ahlfors-Bers colloquium, Contemp. Math. 355, Amer. Math. Soc., Providence, R.I., 2004, , 309–320. 341

[91]

H. Masur and S. Schleimer, The pants complex has only one end. In Spaces of Kleinian groups (Y. Minsky et al., eds.), London Math. Soc. Lecture Note Ser. 329, Cambridge University Press, Cambridge 2006, 209–218. 350, 359

[92]

H. Masur and S. Schleimer, The geometry of the disc complex. Preprint, arXiv:1010.3174v1 [math.GT]. 333, 341

[93]

M. Mj, Mapping class groups and interpolating complexes: rank. J. Ramanujan Math. Soc. 24 (2009), no. 4, 341–357. 350

[94]

J. D. McCarthy, Normalizers and centralizers of pseudo-Anosov mapping classes. Preprint, 1982.

[95]

J. D. McCarthy, A “Tits-alternative” for subgroups of surface mapping class groups. Trans. Amer. Math. Soc. 291 (1985), no. 2, 583–612.

422


[96]

J. D. McCarthy, Automorphisms of surface mapping class groups. A recent theorem of N. Ivanov, Invent. Math. 84 (1986), no. 1, 49–71. 310, 311

[97]

J. D. McCarthy and W. R. Vautaw, Automorphisms of the Torelli Group in genus > 2. Preliminary report, AMS Sectional Meeting, Baton Rouge, Louisiana, 2003.

[98]

J. D. McCarthy and W. R. Vautaw, Automorphisms of the complex of separating curves in genus 3. Preliminary report, Braids, links, and mapping class groups, a conference in honor of Professor Joan Birman, Columbia University, New York 2005.

[99]

J. D. McCarthy and W. R. Vautaw, Automorphisms of the Torelli group. Preprint, arXiv:math/0311250v1 [math.GT]. 359

[100] T. N. Minsky, Kleinian groups and the complex of curves. Geom. Topol. 4 (2000), 117– 148. 342 x g;n are tautological. Internat. Math. Res. [101] G. Mondello, Combinatorial classes on M Notices 2004 (2004), no. 44, 2329–2390. 345 [102] G. Mondello, Riemann surfaces, arc systems and Weil–Petersson form. Boll. Unione Mat. Ital. (9) 1 (2008), no. 3, 751–766. 345 [103] G. Mondello, Riemann surfaces, ribbon graphs and combinatorial classes. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 151–215. 345 [104] G. Mondello, Triangulated Riemann surfaces with boundary and the Weil–Petersson Poisson structure. J. Differential Geom. 81 (2009), no. 2, 391–436. 345 [105] J. R. Munkres, Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, Calif., 1984. 317 [106] K. Ohshika, Reduced Bers boundaries of Teichmüller spaces. Preprint, arXiv:1103.4680v2 [math.GT]. 300, 337 [107] K. Ohshika, A note on the rigidity of unmeasured lamination spaces. Proc. Amer. Math. Soc., to appear. 300 [108] A. Papadopoulos, A rigidity theorem for the mapping class group action on the space of unmeasured foliations on a surface. Proc. Am. Math. Soc. 136 (2008), no. 12, 4453–4460. 300, 337 [109] A. Papadopoulos, Measured foliations and mapping class groups of surfaces. Balkan J. Geom. Appl. 13 (2008), no. 1, 93–106. 300, 337 [110] R. C. Penner, Perturbative series and the moduli space of Riemann surfaces. J. Differential Geom. 27 (1988), no. 1, 35–53. 345 [111] R. C. Penner, The simplicial compactification of Riemann’s moduli space. In Topology and Teichmüller spaces, Proceedings of the 37th Taniguchi symposium (S. Kojima et al., ed.)„ World Scientific, Singapore 1996, 237–252. 345 [112] R. C. Penner, Decorated Teichmüller theory of bordered surfaces. Commun. Anal. Geom. 12 (2004), no. 4, 793–820. 345 [113] R. C. Penner, The structure and singularities of quotient arc complexes. J. Topology 1 (2008), no. 3, 527–550. 345 [114] J. Powell, Two theorems on the mapping class group of a surface. Proc. Amer. Math. Soc. 68 (3) (1978), 347–350. 360


423

[115] A. Putman, A note on the connectivity of certain complexes associated to surfaces. Enseign. Math. (2) 54 (2008), no. 3–4, 287–301. 334, 349, 356, 357, 359 [116] K. Rafi and S. Schleimer, Covers and the curve complex. Geom. Topol. 13 (2009), no. 4, 2141–2162. 340 [117] K. Rafi and S. Schleimer, Curve complexes are rigid. Duke Math. J. 158 (2011), no. 2, 225–246. 340 [118] H. L. Royden, Automorphisms and isometries of Teichmüller space. In Advances in the theory of Riemann surfaces, Ann. of Math. Stud. 66, Princeton University Press, Princeton, N.J., 1971, 369–383. 300 [119] T. Saito and R. Yamamoto, Complexity of open book decompositions via arc complex. J. Knot Theory Ramifications 19 (2010), no. 1, 55–69. 346 [120] R. Shackleton, Combinatorial rigidity in curve complexes and mapping class groups. Pacific J. Math. 230 (2007), no. 1, 217–232. 337 [121] R. Shackleton, Geodesic axes in the pants complex of the five-holed sphere. Topology Appl. 158 (2011), no. 13, 1640–1654. 350, 351 [122] M. Scharlemann, The complex of curves on non-orientable surfaces. J. London Math. Soc. (2) 25 (1982), no. 1. 171–184. 342 [123] M. Scharlemann, Proximity in the curve complex: boundary reduction and bicompressible surfaces. Pacific J. Math. 228 (2006), no. 2, 325–348. 341 [124] S. Schleimer, Notes on the complex of curves. Available on the web, http://www.warwick.ac.uk/ masgar/math.html. 334 [125] P. Schmutz Schaller, Mapping class groups of hyperbolic surfaces and automorphism groups of graphs. Compos. Math. 122 (2000), 243–260. 354, 355 [126] B. Szepietowski, A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves. Osaka J. Math. 45 (2008), no. 2, 283–326. 342 [127] A. Ushijima, A canonical cellular decomposition of the Teichmüller space of compact surfaces with boundary. Comm. Math. Phys. 201 (1999), no. 2, 305–326. 345 [128] W. van der Kallen and E. Looijenga, Spherical complexes attached to symplectic lattices. Geom. Dedicata 152 (2011), 197–211. 359 [129] O. I. Viro, Links, two-sheeted branched coverings and braids. Math. Sb. (NS) 87 (129) (1972), 216–228. 336 [130] C. Walsh, The horoboundary and isometry group of Thurston’s Lipschitz metric. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume IV, EMS Publishing House, Zurich, to appear. 300, 337 [131] S. Yamada, Weil–Petersson completion of Teichmüller spaces and mapping class group actions. Preprint. 351

Chapter 6

On the coarse geometry of the complex of domains Valentina Disarlo

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . The complex of domains D.S/ and its subcomplexes . . 2.1 Subcomplexes of D.S/ containing C.S / . . . . . . 3 The arc complex A.S / as a coarse subcomplex of D.S / 3.1 The boundary graph complex AB .S / . . . . . . . . 3.2 AB .S / is quasi-isometric to P@ .S / . . . . . . . . . 4 Application: the arc and curve complex . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Let S D Sg;b be a compact orientable surface of genus g with b boundary components. A simple closed curve on S is essential if it is not null-homotopic or homotopic to a boundary component. An essential annulus on S is a regular neighbourhood of an essential curve. The well-known complex of curves is a simplicial complex whose simplices are defined in the following way: for k 0 any collection of k C 1 distinct isotopy classes of essential annuli, which can be realized disjointly on S , spans a k-simplex. This complex has been introduced by Harvey in [1]. Ivanov, Korkmaz and Luo proved that except for a few surfaces the action of the extended mapping class group of S on C.S/ is rigid, i.e. the automorphism group of C.S / is the extended mapping class group of S (see [3], [4], [6]). Like any other simplicial complex here mentioned, the curve complex can be equipped with a metric such that each simplex is a Euclidean simplex with sides of length 1. This metric is natural in the sense that the simplicial automorphism group acts by isometries with respect to it. As a metric space, C.S / is quasi-isometric to its 1-skeleton. Masur and Minsky proved that the complex of curves C.S/ is Gromov hyperbolic (see [7]). In this chapter, we shall deal with the coarse geometry of some sort of “generalized” curve complex, the so-called complex of domains D.S/. A domain D on S is a proper connected subsurface of S such that each boundary component of @D is either a boundary component of S or an essential curve on S . Pairs of pants of S and essential annuli are the simplest examples of domains of S . The complex of domains D.S/,

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introduced by McCarthy and Papadopoulos (see [9]), is defined as follows: for k 0, its k-simplices are the collections of k C 1 distinct isotopy classes of domains which can be realized disjointly on S . It has a very rich simplicial structure, which encodes a large amount of information about the structure of S . There is a simplicial inclusion C.S / ! D.S/. It is then natural to ask whether this and other “natural” simplicial maps between subcomplexes of D.S/ (see [9]) have any interesting property when regarded as maps between metric spaces, and we shall deal with this question. We shall also deal with the relation between D.S/ and the similarly defined arc complex A.S /. The k-simplices of A.S/ are the collections of k C 1 isotopy classes of disjoint essential arcs on S for every k 0. Although this definition is very close to the one of D.S /, there is no “natural” simplicial inclusion A.S / ! D.S/. The chapter is organized as follows: Section 2 contains some generalities about the complex of domains and the proof that any simplicial inclusion of C.S / in a subcomplex .S/ of D.S/ induces a quasi-isometry C.S / ! .S/. In Section 3 we describe a quasi-isometry between A.S/ and the subcomplex P@ .S / of D.S/ spanned by peripheral pairs of pants, and we prove that the simplicial inclusion P@ .S / ! D.S/ is a quasi-isometric embedding if and only if S has genus 0. In Section 4 we combine the previous results to prove that the arc and curve complex AC.S / is quasi-isometric to C.S / and to show that the simplicial inclusion A.S / ! AC.S / is a quasi-isometric embedding if and only if S has genus 0. Acknowledgments. I would like to thank Athanase Papadopoulos, Mustafa Korkmaz and Daniele Alessandrini for helpful remarks.

2 The complex of domains D.S / and its subcomplexes Let S D Sg;b be a connected, orientable surface of genus g and b boundary components. Let c.S/ D 3g C b 4 be the complexity of the surface. Recall that c.S / > 0 if and only if S is different from a sphere with at most four holes or a torus with at most one hole. In the extended cases C.S/ is either empty or not connected. We shall then always assume c.S/ > 0. We recall a few facts about the complex of domains. For further details, the reader can consult [9]. A domain X in S is a proper connected subsurface of S such that every boundary component of X is either a boundary component of S or an essential curve on S. Definition 2.1. The complex of domains D.S/ is the simplicial complex such that for all k 0 its k-simplices are the collections of k C 1 distinct isotopy classes of disjoint domains on S. Proposition 2.2. If c.S/ > 0, then D.S/ is connected and dimD.S/ D 5g C 2b 6. There is a natural simplicial inclusion C.S / ! D.S/ which sends every vertex of C.S / to the isotopy class of an essential annulus representing it. One simplicial

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difference between the curve complex C.S/ or the arc complex A.S / and the complex of domains D.S/ is that maximal (in the sense of inclusion) simplices of D.S/ are not necessarily top-dimensional simplices. In the complex of domains, we can find maximal simplices of all dimensions between 1 and dim D.S/. The automorphism group of the complex of domains is closely related to the group of isotopy classes of homeomorphisms of S, the so-called extended mapping class group of S, though in general these two groups do not coincide (for further details see [9]).

2.1 Subcomplexes of D.S / containing C.S / We prefer to work in the setting of length spaces rather than the more general setting of metric spaces. Recall that for h; k 2 RC an .h; k/-quasi-isometric embedding between two length spaces .X; dX / and .Y; dY / is a map f W X ! Y such that for every x; y 2 X the following holds: 1 dX .x; y/ k dY .f .x/; f .y// hdX .x; y/ C k: h A bilipschitz equivalence is an .h; 0/-quasi-isometric embedding. Recall also that a quasi-isometric embedding is a quasi-isometry if Y is c-dense for some c > 0, that is, if every point in Y is at distance less or equal to c from some point in f .X/. Since we shall be dealing with length metrics on simplicial complexes, when we refer to an arbitrary subcomplex we shall always assume that the subcomplex is connected. Moreover, since the dimension of any complex we consider is bounded, we shall always identify the complexes with their 1-skeleta when we refer to their large scale geometry. In this section we discuss metric properties of some natural maps between subcomplexes of D.S/. We shall use the same notation for geometric objects on the surface (curves, pairs of pants, domains) and their isotopy classes as vertices of D.S/. As a first result, we have: Theorem 2.3. Let .S/ be a subcomplex of D.S/ that contains C.S / as a subcomplex. Then the natural simplicial inclusion i W C.S / ! .S/ is an isometric embedding and a quasi-isometry. Proof. Let us first prove the theorem in the case .S/ D D.S/. For every c1 and c2 in C.S/, we have dD.S/ .i.c1 /; i.c2 // dC.S / .c1 ; c2 /. Indeed, let be a geodesic path on C.S/ joining c1 and c2 , namely is given by an edge path c1 D x0 xn D c2 such that dC.S/ .xi ; xiC1 / D 1. By definition xi and xiC1 are represented by two disjoint non-homotopic annuli on S , hence dD.S/ .i.xi /; i.xiC1 // D 1, and i. / is a path on D.S/ with the same length as . We thus see that dD.S/ .i.c1 /; i.c2 // LengthD.S/ .i.// D LengthC.S / . / D dC.S / .c1 ; c2 /: Let us now prove the reverse inequality. Let † be a geodesic segment in D.S/ joining i.c1 / and i.c2 /, namely † is given by the edge path i.c1 / D X0 XkC1 D i.c2 /

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with dD.S/ .Xi ; XiC1 / D 1 for every 0 i k. Choose for every vertex Xi a curve xib among the essential boundary components of Xi . The condition Xi \ XiC1 D ¿ b implies that either xib is homotopic to xibC1 , or xib and xiC1 are disjoint. In the first b b case xi and xi C1 are represented by the same vertex in the curve complex C.S /, in the second case these xib and xibC1 are represented by two different vertices joined by an edge in C.S/. Then, we can consider the path in C.S / given by the xib ’s, namely †b W c1 x1b xkb c2 , and notice that its length is not greater than the length of † in D.S /. We conclude that dC.S / .c1 ; c2 / LengthC.S/ .†b / LengthD.S/ .†/ D dD.S/ .i.c1 /; i.c2 //: Now we notice that for an arbitrary .S/, by the above case, for every pair of vertices c1 ; c2 2 C.S/ the following holds: dC.S / .c1 ; c2 / D dD.S/ .i.c1 /; i.c2 // d.S/ .i.c1 /; i.c2 // dC.S / .c1 ; c2 /: The image of i is 1-dense in .S/: every domain X in .S/ admits an essential boundary component x b , which actually determines an element of i.C.S// at distance 1. Hence, i is a quasi-isometry. We remark that the composition of simplicial inclusions C.S / ! .S/ ! D.S/ is the natural inclusion C.S/ ! D.S/. By the above theorem, we have: Corollary 2.4. (1) Let .S/ be a subcomplex of D.S/ that contains C.S / as a simplicial subcomplex. Then, the natural simplicial inclusion .S/ ! D.S/ is a quasi-isometry. (2) Let ƒ.S/ be a subcomplex of D.S/ and ƒC.S / be the subcomplex of D.S/ spanned by the vertices of ƒ.S/ and the vertices of C.S /. Then, the natural simplicial inclusion ƒ.S/ ! ƒC.S/ is a quasi-isometric embedding if and only if the natural simplicial inclusion ƒ.S/ ! D.S/ is a quasi-isometric embedding. If i W C.S/ ! .S/ is the above mentioned natural inclusion, we can exhibit an uncountable family of right inverse maps to i which are quasi-isometries between C.S / and .S/. For every domain X, let us choose one of its essential boundary components, say x b . Given any such choice, we now define a coarse projection W .S/ ! C.S / as the map such that .X/ is the vertex in C.S/ given by x b . Of course, by our definition, we have B i D idC.S/ , and there exist infinitely many such coarse projections. We also notice that for every coarse projection and for every X 2 .S/, we have d.S / .i B .X/; X/ 1. Theorem 2.5. The following statements hold: (1) Let 1 ; 2 W .S/ ! C.S/ be coarse projections. For every X; Y 2 .S/ we have dC.S / .2 .X/; 2 .Y //2 dC.S/ .1 .X /; 1 .Y // dC.S / .2 .X /; 2 .Y //C2:


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(2) Let W .S/ ! C.S/ be a coarse projection. Then is a .1; 2/-quasi-isometric embedding and a quasi-isometry. Proof. Let us prove (1). We notice that if 1 .X / ¤ 2 .X /, there is an edge in C.S / connecting them, for they are different boundary components of the same domain X. We get a path in C.S/ of vertices 1 .X/2 .X /2 .Y /1 .Y /, and we can conclude that dC.S / .1 .X/; 1 .Y // dC.S/ .1 .X /; 2 .X // C dC.S / .2 .X /; 2 .Y // C dC.S/ .2 .Y /; 1 .Y // D dC.S/ .2 .X /; 2 .Y // C 2 and dC.S / .2 .X/; 2 .Y // dC.S/ .2 .X /; 1 .X // C dC.S / .1 .X /; 1 .Y // C dC.S/ .1 .Y /; 2 .Y // D dC.S/ .1 .X /; 1 .Y // C 2: Now we prove (2). Consider the path given by the edge path i..X //X Y i..Y // in .S / and remark that d.S/ .i..X//; X /; d.S/ .i..Y //; Y / are at most 1. By Theorem 2.3, the simplicial inclusion i W C.S / ! .S/ is an isometric embedding; then dC.S / ..X/; .Y // D d.S/ .i..X //; i..Y /// d.S/ .X; Y / C 2 and d.S / .X; Y / d.S/ .i..X//; i..Y /// C 2 D dC.S / ..X /; .Y // C 2:

3 The arc complex A.S / as a coarse subcomplex of D.S / Recall that an essential arc on S D Sg;b is an embedded arc whose endpoints are on the boundary of S such that it is not isotopic to a piece of boundary of S . The arc complex A.S/ is the simplicial complex whose k-simplices for k 0 are collections of k C 1 distinct isotopy classes of essential arcs on S which can be realized disjointly. Recall that A.S/ is not a natural subcomplex of D.S/, unlike C.S /. The aim of this section is to prove that if b 3 and S ¤ S0;4 (or, equivalently, b 3 and c.S / > 0), then the arc complex A.S/ is quasi-isometric to the subcomplex of D.S/ whose vertices are pairs of pants in S having at least one boundary component on @S. Under the hypothesis on S , A.S/ is a locally infinite complex with infinitely many vertices, each maximal simplex corresponds to a decomposition of S as union of hexagons, and we have dimA.S/ D 6g C 3b 7 (for further details see [9]). In [2] Irmak and McCarthy prove that, except for a few cases, the group of simplicial automorphisms of A.S/ is the extended mapping class group of S.

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We regard A.S/ as a metric space with the natural length metric such that every simplex is Euclidean, with edges of length 1. Like the above mentioned complexes, A.S / is quasi-isometric to its 1-skeleton.

3.1 The boundary graph complex AB .S / Given an essential arc ˛ on S , its boundary graph G˛ is the graph obtained as the union of ˛ and the boundary components of S that contain its endpoints (see [9]). Definition 3.1. The complex of boundary graphs AB .S / is the simplicial complex whose k-simplices, for each k 0, are collections of k C 1 distinct isotopy classes of disjoint boundary graphs on S. We shall always assume S D Sg;b with b 3 and S ¤ S0;4 , for in the excluded cases either AB .S/ or C.S/ is not arcwise connected. By identifying G˛ with ˛, we find that AB .S / and A.S / have the same set of vertices, but in general AB .S/ has fewer simplices than A.S /, for disjoint arcs with endpoints on the same boundary components are joined by an edge in A.S / but not in AB .S /. There is a natural simplicial inclusion between the complex of boundary graphs and the arc complex AB .S/ ! A.S /: We will always regard AB .S/ and A.S/ as metric spaces with their natural simplicial metrics dAB .S/ and dA.S/ . We shall use the same notation for arcs on the surfaces and their isotopy classes on A.S/ or AB .S/. Lemma 3.2. Let a, b be two vertices of A.S / that are joined by an edge in that complex. Consider now a and b as vertices of AB .S /. Then dAB .S / .a; b/ 4: Proof. By our assumption on S, either the genus g of S is 0 and S has at least 5 boundary components, or the genus of S is at least 1 and S has at least 3 boundary components. Now, let a, b be different vertices in A.S /, and assume they are not connected by an edge in AB .S/. In the case g D 0, since b 5, for every pair of vertices a; b 2 A.S / there exists a connected component of S n a [ b which contains at least two different boundary components of S, and we can find a boundary graph that is disjoint from a and b. Hence, the distance in AB .S/ between a and b is 2. For g 1, we will give a detailed proof only in the case where S has exactly 3 boundary components, for this is the most restrictive case. We have different situations depending on the union of a and b as arcs intersecting minimally in the surface (here and in the rest of this proof, we shall consider the boundary components as marked points. This will simplify the figures below): (1) a [ b is a simple closed curve. It can bound a disc or not. In both cases, we have dAB .S / .a; b/ 4 (see Figures 1 and 2, subcases (a) and (b)).

Chapter 6. On the coarse geometry of the complex of domains u

b

a

431

w

z

a

w

u

z

b

Figure 1. The case where a [ b is a simple closed curve, subcase (a). u

b

a

a

u

b

Figure 2. The case where a [ b is a simple closed curve, subcase (b).

(2) a [ b is a simple arc with two different endpoints. In this case, we have dAB .S/ .a; b/ 3 (see Figure 3). u a w

b

a

u

w

b

Figure 3. The case where a [ b is an open arc.

(3) a bounds a disc, and b is not a closed curve. In this case, we have dAB .S/ .a; b/ 4 (see Figure 4, 5, 6, subcases (a), (b), (c)). (4) Both a and b are simple closed curves, and a bounds a disc. In this case, we have dAB .S/ .a; b/ 4 (see Figures 7, 8, 9).

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u

b

w

a

w

u

b

Figure 4. The case where a bounds a disc and b is an open arc, subcase (a).

w

u

b

a z

a

w

u

z

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Figure 5. The case where a bounds a disc and b is an open arc, subcase (b).

a

b u

a

u

b

Figure 6. The case where a bounds a disc and b is an open arc, subcase (c).

(5) a and b are closed curves, but none of them bounds a disc. Recall that a and b pass through the same boundary component of @S, and a [ b can disconnect S in at most 3 distinct connected subsurfaces. Either one of them contains a simple arc joining the two remaining boundary components of S , or there is a non-disc component of S n a [ b which contains an essential arc with both endpoints on the same boundary component of @S, disjoint from a and b (see Figure 10). In this case, we get dAB .S / .a; b/ D 2. (6) a is a closed curve which does not bound a disc and b is not a closed curve. We can use the same argument as in the previous case.


433

u a w

b

a

w

u

b

Figure 7. The case where both a and b are closed curves, a bounds a disc, subcase (a). u a w

b z

a

w

u

z

b

Figure 8. The case where both a and b are closed curves, a bounds a disc, subcase (b). w u z

b

a

w

u

a

z

b

Figure 9. The case where both a and b are closed curves, a bounds a disc, subcase (c).

At this point, we conclude with the following result: Proposition 3.3. The simplicial inclusion j W AB .S / ! A.S / induces a bilipschitz equivalence between .AB .S/; dAB .S/ / and .A.S /; dA.S / /. Proof. First, let us notice that for every pair of vertices a1 , a2 in A.S /, we have dA.S/ .a1 ; a2 / dAB .S/ .a1 ; a2 /:

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Valentina Disarlo b

a

a

u

b

Figure 10. The case where both a and b are closed curves, but none of them bounds a disc.

Let us consider a geodesic path in A.S / with endpoints a1 ; a2 ; say is given by a D x0 x1 xn xnC1 D a2 , where x0 ; : : : ; xnC1 are vertices in A.S /. Consider the curve ] in AB .S/ obtained by concatenating the geodesic segments Œxi ; xiC1 in AB .S /, say ] W Œx0 ; x1 Œxn ; xnC1 . By Lemma 3.2, we have LAB .S/ . ] / 4LA.S / . /: Of course, we also have dAB .S/ .a1 ; a2 / LAB .S/ . ] / 4LA.S / . / D 4dA.S / .a1 ; a2 /: Hence, we get the following bilipschitz equivalence between the two distances: dA.S/ .a1 ; a2 / dAB .S/ .a1 ; a2 / 4dA.S / .a1 ; a2 /:

3.2 AB .S / is quasi-isometric to P@ .S / A peripheral pair of pants on S is a pair of pants with at least one boundary component lying on @S . We define the complex of peripheral pair of pants P@ .S / as the subcomplex of D.S / induced by the vertices that are peripheral pairs of pants. A peripheral pair of pants is monoperipheral if it has exactly one of its boundary components belonging to @S , otherwise it is biperipheral (see [9]). Any regular neighborhood of a boundary graph is a peripheral pair of pants. Let us choose for every peripheral pair of pants P an essential arc whose boundary graph has a regular neigbourhood isotopic to P . Any such choice determines a simplicial inclusion map i W P@ .S/ ! AB .S /. Of course, there are infinitely many such maps. If P is a monoperipheral pair of pants in S, there exists only one essential arc in P whose boundary graph has a regular neighborhood isotopic to P . If P is biperipheral, we can find 3 such essential arcs (see Figure 11). The path determined by the concatenation of these vertices has length 2 in A.S / (see Figure 11) and length at most 8 in AB .S/ (by Lemma 3.2).


a

435

u b

a

u

b

Figure 11. Disjoint essential arcs with isotopic regular neighborhoods.

Proposition 3.4. The following statements hold: (1) Let i1 ; i2 W P@ .S/ ! AB .S/ be two simplicial inclusions. For every a; b 2 P@ .S /, the following inequalities hold dAB .S/ .i2 .a/; i2 .b// 16 dAB .S/ .i1 .a/; i1 .b// dAB .S / .i2 .a/; i2 .b// C 16: (2) Any simplicial map i W P@ .S/ ! AB .S / is an isometric embedding and a quasiisometry. Proof. Let us prove (1). Let a, b be two peripheral pairs of pants. By Lemma 3.2 we have the bounds dAB .S/ .i1 .a/; i2 .a// 8 and dAB .S / .i1 .b/; i2 .b// 8. Looking at the quadrilateral with vertices i2 .a/i1 .a/i1 .b/i2 .b/, we get the following dAB .S/ .i2 .a/; i2 .b// 16 dAB .S/ .i1 .a/; i1 .b// dAB .S / .i2 .a/; i2 .b// C 16: Let us prove (2). From our definition, i is surjective. We prove that it is an isometric embedding. If P1 , P2 are disjoint peripheral pairs of pants, then their images are disjoint boundary graphs, and we have dAB .S / .i.P1 /; i.P2 // dP@ .S / .P1 ; P2 /: Now, let us consider the geodesic in AB .S/ given by the edge path W i.P1 / D b0 bn D i.P2 /. In a similar fashion, the condition that bi is disjoint from biC1 implies that the arcs have regular neighbourghoods that are disjoint from each other. Hence, the geodesic projects to a curve ] on AB .S / given by isotopy classes of regular neighborhoods of the bi ’s, whose endpoints are P1 and P2 . We get dP@ .S/ .P1 ; P2 / L. ] / L. / D dAB .S / .i.P1 /; i.P2 //: Let us now consider the natural surjective map W AB .S / ! P@ .S /, which assigns to a boundary graph a in AB .S/ the isotopy class of the peripheral pair of pants given by a regular neighbourhood of a in S. We notice that the map is not injective: any two vertices as in Figure 11 have the same image. In general, if dP@ .S / .b1 ; b2 / D 0, then dAB .S/ .b1 ; b2 / 8 (by Lemma 3.2). For every simplicial inclusion map i as in Proposition 3.4, we have B i D idP@ .S / and dAB .S / .i B .x/; x/ 4dA.S / .i B .x/; x/ 8. We have the following:

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Proposition 3.5. The natural surjective map W AB .S / ! P@ .S / is a .1; 8/ quasiisometric embedding and a quasi-isometry. Proof. If b1 and b2 are disjoint boundary graphs, then their regular neighborhoods are disjoint peripheral pairs of pants. Furthermore, if P1 , P2 are disjoint pairs of pants, then one can realize disjointly every pair of boundary graphs b1 , b2 , whose regular neighbourhoods are P1 , P2 . Thus, we have the first inequality dP@ .S / .b1 ; b2 / dAB .S/ .b1 ; b2 /. If b1 ¤ b2 , let be a geodesic in P@ .S / defined by the concatenation of vertices W b1 D P1 Pn D b2 , with Pi \ PiC1 D ¿. Choosing for every Pi a boundary graph bi , we get a curve ] in AB .S/ given by the edge path ] W b1 bn , with dAB .S/ .bi ; biC1 / D 1. Hence, we get dAB .S / .b1 ; bn / L. ] / D dP@ .S / .b1 ; b2 /: Finally, we get dP@ .S/ .b1 ; b2 / 8 dP@ .S/ .b1 ; b2 / dP@ .S / .b1 ; b2 / C 8: Using all the results proved in this section, we have: Theorem 3.6. If S D Sg;b such that b 3 and S ¤ S0;4 , the following holds: (1) The complex of arcs A.S/ is quasi-isometric to the subcomplex P@ .S / of D.S/. (2) If g D 0, then the natural simplicial inclusion P@ .S / ! D.S/ is an isometric embedding and a quasi-isometry. (3) If g 1, the image of the natural simplicial inclusion k W P@ .S / ! D.S/ is 2-dense in D.S/, but the map k is not a quasi-isometric embedding. Proof. The proof of (1) follows from the consideration that both compositions j B i , j B W A.S / ! P@ .S/ of the maps in Lemma 3.2, Proposition 3.4 and Proposition 3.5 are quasi-isometries. Let us prove (2). Let X be a domain. As a vertex of D.S/, X is at distance 1 from each of the vertices representing its essential boundary components, and each essential boundary component of X is at distance 1 from a pair of pants in P@ .S /. This shows that the image of the inclusion P@ .S / ! D.S/ is 2-dense. Recall that by our hypothesis on S, if S D S0;b , then b 5. Since the genus of S is 0, each domain X of S is a sphere with holes, each simple closed curve on S disconnects the surface into two connected components, and each of them has at least one boundary component lying on @S. Let P1 , P2 be peripheral pairs of pants of S , and be a geodesic in D.S/ joining them, say W P1 X1 Xn1 P2 . Let W D.S/ ! C.S/ be a coarse projection and i W C.S / ! D.S/ the natural simplicial inclusion. Notice that if X1 is not a curve then i..X1 // is a curve and it is disjoint from both P1 and X2 . Moreover, i..X1 // is also distinct from X2 (otherwise we could shorten ). Up to substituting X1 with i..X1 // and Xn1 with i..Xn1 //, we can assume that both X1 and Xn1 are represented by simple closed curves. Moreover, up to substituting the segment of given by X1 Xn1 with the


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geodesic in C.S/ which joins X1 and Xn1 (see Theorem 2.3), we can assume that each Xi is a curve, say Ci . If has length 2, it is represented by a path P1 CP2 . By geodesity, P1 and P2 belong to the same connected component of S n C . Hence, there is a peripheral pair of pants P ? in the other connected component, and we find a new geodesic of length 2 connecting P1 ; P2 contained in P@ .S/, namely P1 P ? P2 . If has length greater than 2, let us focus on the initial segment of given by P1 C1 C2 . With the same argument, we can find a peripheral pair of pants P c1 disjoint from both P1 and C2 , which can substitute C1 in . Notice that the curve P1 P c1 C2 Cn1 P2 has the same length as , hence is a geodesic. Continuing this way, we get a path ? W P1 P c1 P cn1 P2 with all vertices in P@ .S / with the same length as . Hence, we have dD.S/ .P1 ; P2 / [email protected] / .P1 ; P2 / [email protected]/ . ? / D LD.S/ . ? / D dD.S/ .P1 ; P2 /: Let us prove (3). As in the previous case, it is easy to see that the image of k is 2-dense. Let c be a simple loop on S wrapping around all the boundary components of S as in Figure 12. Since S is not a sphere, c is essential, and disconnects the surface in two domains: one of them is the sphere B D B0;bC1 S which contains all the boundary components of S , the other is the subsurface C D Cg;1 which has c as unique boundary component (see Figure 12). We claim that the inclusion P@ .B/ ! P@ .S / is

B c C

Figure 12. The subsurface C D Cg;1 .

an isometric embedding. Let us prove that for every pair of peripheral pairs of pants P1 , P2 we have dP@ .S/ .P1 ; P2 / dP@ .B/ .P1 ; P2 /. Let be a geodesic in P@ .S / joining P1 , P2 . If Q is a vertex of , but not a peripheral pair of pants of B, then Q crosses transversely a regular neighborhood of the curve c and Q \ C is a strip. We can then replace this strip with one of the two strips of the neighborhood of c in order to get a new peripheral pair of pants Q0 2 P@ .B/, which can substitute Q in . The curve obtained from after all these substitutions may be shorter than , but all its vertices belong to P@ .B/. Hence, we have dP@ .S / .P1 ; P2 / dP@ .B/ .P1 ; P2 /. Now, by the hypothesis B has at least 4 boundary components, hence by statement (2) we have diam P@ .B/ D diam D.B/ D C1. Hence, there exist peripheral pairs of pants

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P0 , Pn on B such that dP@ .B/ .P0 ; Pn / n. Without loss of generality, we can also assume that all the boundary components of P0 , Pn are boundary components of S . By our claim, we have dA@ .S/ .P0 ; Pn / n. Moreover, since both P0 and Pn are disjoint from C , we have dD.S/ .P0 ; Pn / D 2, and this concludes the proof. As an immediate consequence of Theorem 3.6 and Theorem 2.3, we have: Corollary 3.7. If S D S0;b and b 5, then A.S / is quasi-isometric to C.S /.

4 Application: the arc and curve complex The arc and curve complex AC.S/ is the simplicial complex whose k-simplices are collections of k C 1 isotopy classes of essential arcs or curves on S . Its automorphism group is the extended mapping class group, and it is quasi-isometric to the curve complex (see [5] and see also [10]). As an application of Theorem 3.6 and Theorem 2.3, we give a simple proof of the quasi-isometry between C.S / and AC.S / stated in [5] in the case S D Sg;b with b 3 and S ¤ S0;4 . Moreover, we give necessary and sufficient conditions on S for the natural inclusion A.S / ! AC.S / to be a quasiisometry. The latter result is also stated in [8]. Theorem 4.1. If S D Sg;b with b 3 and S ¤ S0;4 , then the following holds: (1) The arc and curve complex AC.S/ is quasi-isometric to C.S /. (2) If S is a sphere with boundary, the simplicial inclusion A.S / ! AC.S / is a quasi-isometry. In the other cases, the simplicial inclusion is not a quasiisometric embedding. Proof. Let us prove (1). We consider the subcomplex of D.S/ spanned by the vertices of C.S / and P@ .S/, say P@ C.S/, and the subcomplex of AC.S / spanned by the vertices of AB .S/ and C.S/, say AB C.S/. We remark that: i. AC.S / is quasi-isometric to AB C.S/, by a quasi-isometry given by the inclusion of AB .S/ in A.S/ as in Lemma 3.2; ii. AB C.S/ is quasi-isometric to P@ C.S/, by a quasi-isometry induced by the natural isometric embedding i W P@ .S/ ! AB .S / as in Proposition 3.4. By Theorem 2.3 the complex P@ C.S/ is quasi-isometric to C.S /, and we conclude. Let us prove (2). By Corollary 2.4 the simplicial inclusion P@ C.S / ! D.S/ is a quasi-isometry. In the diagram below the horizontal rows are the natural simplicial inclusions, and the vertical rows are the above mentioned quasi-isometries. The diagram commutes, hence the natural inclusion A.S/ ! AC.S/ is a quasi-isometric embedding if and only if the natural inclusion P@ .S/ ! P@ C.S/ is a quasi-isometric embedding. Moreover,


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the latter holds if and only if the resulting inclusion map P@ .S / ! D.S/ is a quasiisometric embedding. We then conclude using Theorem 3.6: A.S/ O q:i:

AB .S/ O q:i:

P@ .S/

/ AC.S / O q:i:

/ AB C.S / O q:i:

/ P@ C.S / o

q:i:

/ D.S/.

References [1]

W. J. Harvey, Geometric structure of surface mapping class groups. In Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser. 36, Cambridge University Press, Cambridge 1979, 255–269. 425

[2]

E. Irmak and J. D. McCarthy, Injective simplicial maps of the arc complex. Turk. J. Math. 33 (2009), 1–16. 429

[3]

N. Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices 14 (1997), 651–666. 425

[4]

M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori. Topology Appl. 95 (1999), no. 2, 85–111 . 425

[5]

M. Korkmaz and A. Papadopoulos, On the arc and curve complex of a surface. Math. Proc. Cambridge Philos. Soc. 148 (2010), no. 3, 473–483. 438

[6]

F. Luo, Automorphisms of the complex of curves. Topology 39 (2000), no. 2, 283–298. 425

[7]

H. A. Masur and Y. N. Minsky, Geometry of the curve complex I: Hyperbolicity. Invent. Math. 138 (1999), 103–149. 425

[8]

H. Masur and S. Schleimer, The geometry of the disc complex. Preprint, arXiv:1010.3174v1 [math.GT]. 438

[9]

J. D. McCarthy and A. Papadopoulos, Simplicial actions of mapping class groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zürich 2012, 297–423. 426, 427, 429, 430, 434

[10] S. Schleimer, Notes on the curve complex. www.math.rutgers.edu/~saulsch/Maths/notes.pdf 438

Chapter 7

Minimal generating sets for the mapping class group of a surface Mustafa Korkmaz

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . Mapping class groups and relations among Dehn twists 2.1 Dehn twists . . . . . . . . . . . . . . . . . . . . 2.2 The commutativity relation . . . . . . . . . . . . 2.3 The braid relation . . . . . . . . . . . . . . . . . 2.4 The two-holed torus relation . . . . . . . . . . . 2.5 The chain relation . . . . . . . . . . . . . . . . . 2.6 The lantern relation . . . . . . . . . . . . . . . . 3 Generating sets of the mapping class group . . . . . . 3.1 Dehn twist generators . . . . . . . . . . . . . . . 3.2 Minimal number of generators . . . . . . . . . . 3.2.1 Proof of Theorem 3.10 . . . . . . . . . . 3.3 Torsion generators . . . . . . . . . . . . . . . . 3.4 Involution generators . . . . . . . . . . . . . . . 4 Other related results . . . . . . . . . . . . . . . . . . 4.1 Extended mapping class groups . . . . . . . . . 4.2 Hyperelliptic mapping class groups . . . . . . . 4.3 Mapping class groups of nonorientable surfaces . 5 Problems . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction The group of self-homeomorphisms of a compact connected surface is a topological group with the compact open topology, and it is far from being finitely generated. It is an infinite dimensional topological group. When we consider homeomorphisms up to isotopy, we get a finitely generated (discrete) group, called the mapping class group of the surface. The algebraic properties of the mapping class group are of interest in low dimensional topology, specifically in the theory of the topology of 2-, 3- and 4-manifolds, and also in geometric group theory and in Teichmüller theory.

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The mapping class group shares many properties with the automorphism group and the outer automorphism group of a free group. Max Dehn [6] was the first to investigate the algebraic structure of the mapping class group. He proved that the mapping class group of a closed orientable surface is generated by finitely many twists, now called Dehn twists. Since then, many algebraic properties of the mapping class group have been discovered. In particular, it was shown that this group is generated by finitely many Dehn twists [6], [18], by two elements [32], by two torsion elements [13], and by a small number of involutions [20], [4]. In the present chapter, we will focus on the various sets of generators of the mapping class group of a closed connected orientable surface, and some related groups, such as extended mapping class groups, hyperelliptic mapping class groups, and mapping class groups of nonorientable surfaces. The generating sets we consider will consist of Dehn twists, or torsion elements, or involutions. We will give the known minimal number of these types of elements generating the mapping class group. When the surface has boundary components, the mapping class group may be defined in various ways: The homeomorphisms and the isotopies may fix all points on some boundary components while fixing some boundary components setwise and permuting the others. Each of these groups is useful in some context. The algebraic properties of these groups are also interesting. We will focus on closed surfaces, and will occasionally mention surfaces with boundary. The only new result in this survey is in Theorem 4.3, where we observe that the mapping class group of a closed nonorientable surface of genus three is generated by two elements, and also by three involutions. This observation is extracted from the known presentation of the group. We will state the results without proofs. We will give a partial proof, or even a full proof, whenever it is not too technical. Sometimes, we will not state a result in full generality, but will mention it as a remark. For example, it is known that the mapping class group of a closed orientable surface can be generated by four involutions, with some exceptions. But we will only prove that it is generated by seven involutions, in order not to make this chapter too technical. In the following, † will always denote an orientable surface equipped with an orientation and N a nonorientable surface. All surfaces we consider are connected. Simple close curves and homeomorphisms are considered up to isotopy. Arcs are considered up to isotopy relative to their endpoints. For the composition of functions, we use the usual notation: FH means that H is applied first.

2 Mapping class groups and relations among Dehn twists Let † be a closed orientable surface of genus g 0. The mapping class group Mod.†/ is defined as the group of isotopy classes of orientation-preserving homeomorphisms † ! †. If we include orientation-reversing homeomorphisms too into the definition, the group we get is called the extended mapping class group.

Chapter 7. Minimal generating sets for the mapping class group

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The simplest homeomorphism of † is obtained by cutting † along a simple closed curve and gluing it back after twisting one of the resulting boundary components by 360 degrees. It is called a Dehn twist. Dehn twists are the building blocks of the theory of the mapping class group. The precise definition and various properties of Dehn twists are given below. There are topologically four types of relations among Dehn twists given below. It is known that these relations are enough to give a finite presentation of the mapping class group [31], which we will not mention. For more properties of the mapping class group the reader is referred to Nikolai Ivanov’s article [9], to Shigeyuki Morita’s survey [23] and to the book of Benson Farb and Dan Margalit [7]. We start with a basic lemma, which will be used throughout without referring to it. Lemma 2.1 (Alexander Lemma). Let D be a 2-disk and let F W D ! D be a homeomorphism which restricts to the identity on the boundary on D. Then F is isotopic to the identity relative to the boundary of D. Proof. We can assume that D is the unit disk fx 2 R2 W jxj 1g. Define ´ x / if 0 jxj 1 t ; .1 t /F . 1t F t .x/ D x if 1 t jxj 1: F t is an isotopy between F0 D F and F1 , which is the identity. Using Alexander’s Lemma, in order to show that two self-homeomorphisms of a surface † are isotopic, it suffices to show that their actions on a set consisting of simple closed curves and arcs whose complement in S is a disk are equal up to isotopy. We will use this fact to prove the relations among Dehn twists. In the case g D 0, the surface † is the sphere. As an application of the Alexander Lemma, one can easily show that every orientation-preserving homeomorphism of the sphere † is isotopic to the identity. Hence, the mapping class group Mod.†/ is the trivial group, and the extended mapping class group is a cyclic group of order two. From now on, we will always assume that g 1.

2.1 Dehn twists Let U denote the cylinder Œ0; 1 S 1 with coordinates t and e . Define a map W U ! U by .t; e i / D .t; e i. C2t / /: The map is a self-homeomorphism of U and its restriction to the boundary of U is the identity. Let a be a simple closed curve on † such that a regular neighborhood of a is an annulus. Let M be a closed neighborhood of a homeomorphic to U and choose an orientation-preserving embedding Ka W U ! † such that Ka .U / D M . Define a

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Figure 1. The twist on the annulus U .

homeomorphism Ta W † ! † by ´ Ka Ka1 .x/ if x is in Ka .U /; Ta .x/ D x if x is not in Ka .U /: The effect of the homeomorphism Ta is to cut † along a, to twist one of the sides by 360 degrees to the right on the positive face of † and to glue it back (cf. Figure 2). We will show below that if a is isotopic to b, then the homeomorphisms Ta and Tb are isotopic, i.e. homotopic through homeomorphisms of †. We denote by A the isotopy class of Ta . We call A the right (or positive) Dehn twist about the simple closed curve a. In particular, A does not depend on the choice of the embedding of U .

K 1

K

a Ta

Figure 2. The Dehn twist Ta on the surface.

In general, a homeomorphism and its isotopy class will be denoted by the same letter. Simple closed curves on † are denoted by the lower case letters a; b; c, and the Dehn twists about them by the corresponding capital letters A, B, C . The letter A will also denote any homeomorphism representing this class. It is sometimes convenient for us to denote the Dehn twist A by ta . Accordingly, a simple closed curve and its isotopy class will be denoted by the same letter. Lemma 2.2. Let a be a simple closed curve on an oriented surface † and let F W † ! † be an orientation-preserving homeomorphism. Then the homeomorphism F Ta F 1


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is isotopic to the Dehn twist TF .a/ , i.e. F Ta F 1 D TF .a/

(2.1)

in the group Mod.†/. Proof. The annulus F .Ka .U // D KF .a/ .U / is a closed neighborhood of F .a/ and we can use it to define TF .a/ : ´ KF .a/ KF1.a/ .x/ if x is in KF .a/ .U /; TF .a/ .x/ D x if x is not in KF .a/ .U /: On the other hand, if x 2 KF .a/ .U / then F 1 .x/ 2 Ka .U /, and hence F .Ta .F 1 .x/// D F .Ka Ka1 .F 1 .x/// D .FKa /.FKa /1 .x/ D KF .a/ KF1.a/ .x/: If x … KF .a/ .U / then F 1 .x/ … Ka .U /, and hence F .Ta .F 1 .x/// D F .F 1 .x// D .x/: Hence we have the equality F Ta F 1 D TF .a/ . Lemma 2.3. Let a and b be two simple closed curves on an oriented surface †. If a is isotopic to b, then the corresponding Dehn twists Ta and Tb are isotopic, i.e. Ta D Tb in the group Mod.†/. Proof. Let H t W † ! † be an isotopy such that H0 is the identity and H1 .a/ D b. Then, Tb D H1 Ta H11 and H t Ta H t1 is an isotopy between the homeomorphisms Ta and Tb . From now on we consider self-homeomorphisms of a surface up to isotopy.

2.2 The commutativity relation Suppose that a and b are two disjoint simple closed curves on an oriented surface †. Since the support of the Dehn twist Ta can be chosen to be disjoint from b, the Dehn twist Ta fixes the points on the curve b. In particular, we have Ta .b/ D b. Thus by (2.1), we get AB D Ta Tb D Ta Tb Ta1 Ta D TTa .b/ Ta D Tb Ta D BA: Lemma 2.4 (Commutativity relation). If a and b are disjoint simple closed curves on an oriented surface †, then the Dehn twists A and B commute: AB D BA:

(2.2)

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2.3 The braid relation Suppose that two simple closed curves a and b on † intersect transversely at only one point. It can easily be shown that ta tb .a/ D b (cf. Figure 3). Hence, ta tb ta D ta tb ta tb1 ta1 ta tb D t ta tb .a/ ta tb D tb ta tb . That is, ABA D BAB, which is called the braid relation. Lemma 2.5 (Braid relation). If a and b are two simple closed curves on an oriented surface † intersecting transversely at only one point, then the Dehn twists A and B satisfy the relation ABA D BAB:

b

Tb .a/

(2.3)

Ta Tb .a/

b

a

Figure 3. The proof of Ta Tb .a/ D b.

2.4 The two-holed torus relation Suppose that a, b, c are three nonseparating simple closed curves on an oriented surface † such that a is disjoint from c, and b intersects a and c transversely at one point (cf. Figure 4 (a)). A regular neighborhood of a [ b [ c is a torus with two boundary components, say d and e. Then the Dehn twists about these simple closed curves satisfy the relation .ABC /4 D DE: Lemma 2.6 (Two-holed torus relation). If a, b, c, d , e are simple closed curves on an oriented surface † as in Figure 4 (a), then the corresponding Dehn twists satisfy the relation .ABC /4 D DE:

(2.4)

This relation, supported on a two-holed torus, is called the two-holed torus relation. In fact, it follows from the braid relations that three Dehn twists A; B; C on the left hand side of this relation can be written in any order. In order to prove the two-holed torus relation, by the Alexander Lemma, it suffices to prove that the action of the mapping classes on both sides of the relation are the same on the arcs and curves in Figure 4 (b), whose complement is a disk. The proof


a

b c

d

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F

e ( a)

F

F F (c)

( b)

F

F F F

(d )

Figure 4. The two-holed torus relation .ABC /4 D DE and its proof. Here, F denotes ABC .

of this fact can easily be deduced from Figure 4 (c) and (d), where the action of ABC on the curves in .b/ is illustrated. If the simple closed curve e bounds a disk, then one gets the one-holed torus relation .ABA/4 D D, which turns into the relation .AB/6 D D by using the braid relation between A and B.

2.5 The chain relation The two-holed torus relation and the one-holed torus relation are the special cases of a more general relation, called the chain relation. Let c1 ; c2 ; : : : ; ck be a sequence of simple closed curves on † such that ci intersects ciC1 transversally at one point for each 1 i k 1 and that ci and cj are disjoint if ji j j 2. Suppose first that k D 2h is even. Then a tubular neighborhood of the union c1 [c2 [ [c2h is an orientable subsurface of genus h with one boundary component,

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say d . The corresponding Dehn twists satisfy the relation .C1 C2 : : : C2h /4hC2 D D: Note that the special case h D 1 gives the one-holed torus relation. Suppose now that k D 2h C 1 is odd. Then a tubular neighborhood of c1 [ c2 [ [ c2hC1 is an orientable subsurface of genus h with two boundary components, say d1 and d2 . The corresponding Dehn twists satisfy the relation .C1 C2 : : : C2hC1 /2hC2 D D1 D2 : Note that the case h D 1 gives the two-holed torus relation.

2.6 The lantern relation Let Y be a 2-disk with boundary component d0 . In the interior of Y , choose a triangle with vertices V1 , V2 and V3 . Let eij be the edge of the triangle connecting Vi to Vj for 1 i < j 3. Assume that a traveler starting at the vertex V1 going in the counterclockwise direction along the triangle reaches the vertex V2 first and then V3 . In the interior of the disk Y , choose three disjoint disks Y1 , Y2 and Y3 with centers V1 , V2 and V3 , respectively, and let di be the boundary component of Yi . Let dij be the boundary component of a regular neighborhood of the arc eij so that it encircles Yi and Yj . Let X be the sphere with four boundary components d0 , d1 , d2 and d3 obtained from Y by deleting the interiors of the disks Y1 , Y2 and Y3 (cf. Figure 5 (a)). Suppose that X is embedded in an oriented surface † via an orientation-preserving map. Lemma 2.7 (Lantern relation). If the sphere X with four boundary components d0 , d1 , d2 , d3 is embedded in an oriented surface †, then the Dehn twists D0 , D1 , D2 , D3 , D12 , D13 and D23 satisfy the relation D0 D1 D2 D3 D D12 D23 D13 :

(2.5)

This relation is called the lantern relation. It was first discovered by Dehn [6], and rediscovered and made popular by Johnson [10]. Since then it has been used in many instances. The lantern relation can be proved by using the Alexander Lemma: Choose a set of arcs dividing the supporting subsurface into a disk and show that the actions on this set of arcs of the homeomorphisms on the two sides of the equality .2.5/ are equal (up to homotopy) (cf. Figure 5 (b)).


d12 d2

d1

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d13 d3

d 23

d0

( a)

a

b

c

( b)

Figure 5. The action of D12 D23 D13 on the arcs a, b and c is the same as that of D0 D1 D2 D3 .

3 Generating sets of the mapping class group Let † be a closed oriented surface of genus of g embedded in R3 illustrated in Figure 6. Sometimes we will write †g for † in order to stress the genus. In this section we will give various sets of finite generating sets for the mapping class group Mod.†/. We will also discuss the minimal size of such generating sets.

3.1 Dehn twist generators The problem of finding a finite set of generators was first considered by Dehn in [6], where he showed that Mod.†/ is generated by finitely many twists, now called Dehn twists. This result was rediscovered by Lickorish in [16], [18]. He proved that the mapping class group is generated by 3g 1 Dehn twists. We recall this set of Dehn twist generators.

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For this purpose, let us consider the curves a1 ; a2 ; : : : ; a2g ; b D b2 ; b3 ; : : : ; bg on † as in Figure 6. Throughout this chapter, ai and bi will always denote these simple closed curves. Theorem 3.1 (Dehn–Lickorish generators). The mapping class group Mod.†/ is generated by the Dehn twists A1 ; A2 ; : : : ; A2g ; B D B2 ; B3 ; : : : ; Bg about the curves a1 ; a2 ; : : : ; a2g ; b D b2 ; b3 ; : : : ; bg , respectively, shown in Figure 6. 1 D b1 b D b2 b3

bg1

bg d

3 2

4

5

6

2g 2

2g 1

y

2g

Figure 6. The curve labeled i is ai . A disk is glued along d if the surface is closed.

By Theorem 3.1, the mapping class group Mod.†/ of a closed orientable surface † is generated by 3g 1 Dehn twists. But this number is not minimal: Humphries [8] showed that the Dehn twists A1 ; A2 ; : : : ; A2g ; B are enough to generate Mod.†/. We prove this by using his idea. Humphries also showed that the number 2g C 1 is minimal; the group Mod.†/ cannot be generated by 2g (or less) Dehn twists. Lemma 3.2 (Humphries Lemma). For each 1 i 8 let Ci denote the Dehn twist about the simple closed curve ci in Figure 7. Then the Dehn twist C8 is contained in the subgroup of Mod.†/ generated by C1 ; C2 ; : : : ; C7 . Similarly, C6 is contained in the subgroup generated by C1 ; C2 ; : : : ; C5 ; C7 ; C8 . Proof. It is shown in Figure 7 that the homeomorphism C D C1 C2 C3 C4 C7 C3 C2 C1 C5 C4 C3 C7 C2 C3 C4 C5 maps the simple closed curve c8 to c6 , i.e. C.c8 / D c6 . Now it follows from Lemma 2.2 that C C8 C 1 D C6 . Both claims in the lemma follow from this. From Lemma 3.2, Theorem 3.1 and the fact that the mapping class group cannot be generated by 2g or fewer Dehn twists, we now derive the following theorem. Theorem 3.3 (Dehn–Lickorish–Humphries generators). The mapping class group Mod.†/ is generated by the Dehn twists B; A1 ; A2 ; : : : ; A2g about the simple closed curves b; a1 ; a2 ; : : : ; a2g . Furthermore, no subset of the group Mod.†/ consisting of 2g or less Dehn twists generates Mod.†/. Remark 3.4. By using Lemma 3.2 and the Dehn–Lickorish generators in Theorem 3.1, we observe that the mapping class group Mod.†/ is also generated by the Dehn

Chapter 7. Minimal generating sets for the mapping class group c6

c7

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c8

c1 c2 c3 c4 c5

Figure 7. The proof that the simple closed curve c8 is mapped to c6 by the homeomorphism C1 C2 C3 C4 C7 C3 C2 C1 C5 C4 C3 C7 C2 C3 C4 C5 .

twists about the 2g C 1 simple closed curves a2 ; a3 ; : : : ; a2g ; bj ; bj C1 for any j D 1; 2; : : : ; g 1. Remark 3.5. If the surface has one boundary component, it was shown by Johnson in [11] that the mapping class group is generated by the 2g C 1 Dehn twists about the simple closed curves b, a1 ; a2 ; : : : ; a2g in Figure 6.

3.2 Minimal number of generators Once we know the Dehn–Lickorish–Humphries generators of the mapping class group given in Theorem 3.3, it is easy to show that the group Mod.†/ is generated by three elements. Consider the model for † shown in Figure 8. Let R denote the counter. Note that R is an element of order g. We also have clockwise rotation of † by 2 g k R.a1 / D b, R .a2 / D a2kC2 for each 1 k g 1, and Rl .a3 / D a2lC3 for each 1 l g 2. It now follows from Theorem 3.3 that the elements R, A1 , A2 ,

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R

3

5 6

R1

4

1 2 2g 2g1

2g2

Figure 8. The curve labeled i is ai . R is the 180-degree rotation about the indicated lines.

360 -degree g

rotation, and R1 and R2 are the

A3 generate the mapping class group. Since .A1 A2 A3 /A1 .A1 A2 A3 /1 D A2 and .A1 A2 A3 /A2 .A1 A2 A3 /1 D A3 , the group Mod.†/ can also be generated by three elements, R, A1 , A1 A2 A3 . Let us now consider the element S D A2g A2g1 : : : A2 A1 , which will play an important role in the subsection on torsion generators. Since S 1 .ai / D aiC1 for each 1 i 2g 1, we have S 1 Ai S D AiC1 . Again it follows from this and Theorem 3.3 that the mapping class group Mod.†/ is also generated by the three elements S , B, A1 . We will show below that one can also omit A1 , so that the mapping class group is generated by the two elements S and B. Theorem 3.6. The mapping class group Mod.†/ of the closed oriented surface † is generated by each of the following sets: (i) fR; A1 ; A2 ; A3 g, (ii) fR; A1 ; A1 A2 A3 g, (iii) fS; B; A1 g. Remark 3.7. Note that the generator A2 (resp. A3 ) in (i) may be replaced by any A2k (resp. A2lC1 ) for 1 k g (resp. 1 l g 1). In (iii), the Dehn twist A1 may be replaced by any Ai for 1 i 2g. Since the group Mod.†/ is not cyclic, it cannot be generated by one element. We now show that it is possible to generate it with two elements, which is the smallest number of generating elements. This fact was first proved by Wajnryb in [32], which is stated below as Theorem 3.9. Let S denote the product A2g A2g1 : : : A2 A1 , as above. It can be shown that the order of S in Mod.†/ is 4g C 2. Lemma 3.8. The order of S D A2g A2g1 : : : A2 A1 in the mapping class group Mod.†/ of the closed oriented surface † of genus g is 4g C 2.


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Proof. It follows from the chain relation that S 4gC2 D 1. Hence, the order of S is at most 4g C 2. On the other hand, the action of S k on the first homology of † is nontrivial for all 1 k 4g 1, showing that the order of S is at least 4g C 2. Theorem 3.9 (Wajnryb). For g 3, the mapping class group Mod.†/ of the closed oriented surface † is generated by S and Bg1 Bg1 . We note that Theorem 3.9 holds also if the surface has one boundary component. If g D 1 (i.e. † is a torus), then the group Mod.†/ is isomorphic to SL.2; Z/ and is generated by the Dehn twists A1 and A2 . If g D 2 then the mapping class group is generated by A1 and A5 A4 A3 A2 A1 . Hence the mapping class group is generated by two elements in these cases too. Theorem 3.9 was improved in [13]; it was shown that the second generator Bg1Bg1 can be replaced by the Dehn twist B .D B2 /. Theorem 3.10 ([13]). Suppose that g 2 and † is a closed oriented surface of genus g. The mapping class group Mod.†/ of † is generated by S and B. We give a proof of Theorem 3.10 below. If is a subgroup of Mod.†/, it acts on the set of simple closed curves (considered up to isotopy). For a simple closed curve c on †, let us denote by c the -orbit of c; c D f F .c/ W F 2 Gg: Let us now denote by the subgroup of Mod.†/ generated by S and B. We prove that D Mod.†/. The main idea of the proof is to show that the -orbit b of the curve b contains the simple closed curves a1 ; a2 ; : : : ; a2g . Lemma 3.11. The Dehn twists C1 ; C2 ; : : : ; Cg1 and D1 ; D2 ; : : : ; Dg2 about the curves c1 ; c2 ; : : : ; cg1 and d1 ; d2 ; : : : ; dg2 shown in Figure 9 are contained in . Proof. It can easily be shown that S 1 .b/ D c1 , S 1 .ci / D di and S 1 .di / D ciC1 . Hence, S 1 BS D C1 , S 1 Ci S D Di and S 1 Di S D CiC1 . The lemma now follows. dg2

d1

c1

c2

cg2

Figure 9. The curves ci and di .

cg1

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3.2.1 Proof of Theorem 3.10. Since S.ai / D ai1 , we have SAi S 1 D Ai1 . Recall that a1 ; a2 ; : : : ; a2g and b are the simple closed curves in Figure 6. It follows that Ai 2 if and only if Aj 2 for any 1 i; j 2g. Since the mapping class group Mod.†/ is generated by the Dehn twists A1 ; A2 ; : : : ; A2g and B by Theorem 3.3, in order to finish the proof, it suffices to show that Aj 2 for some 1 j 2g. Suppose that the curve aj 2 b for some j , so that F .b/ D aj for some F 2 . Then, FBF 1 D Aj . Since B is contained in , we get that Aj is also contained in . Therefore, in order to finish the proof, it suffices to show that aj is contained in the orbit b for some j with 1 j 2g. Suppose first that g is odd. It is easy to see that the homeomorphism 1 1 C2 D1 BD31 C41 D51 C61 : : : Dg2 Cg1 S

maps the curve b to a3 (cf. Figure 10). Since all Ci and Di are contained in by Lemma 3.11, we have a3 2 b, so that A3 2 . 1 1 Dg2 Cg1

S.b/

1 1 D31 C41 : : : Dg4 Cg3

B

D1

C2

Figure 10. The case g is odd.

Suppose now that g is even. One can easily show that if g 4 the homeomorphism 1 1 BD21 C31 D41 C51 : : : Dg2 Cg1 S

maps the curve b to a4 . In the case g D 2, BS maps b to a4 . Again, since all Ci and Di are contained in , we conclude that a4 2 b, so that A4 2 . This concludes the proof of Theorem 3.10. Remark 3.12. The conclusion of Theorem 3.10 and its proof hold if the surface has one boundary component (cf. [13]).


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3.3 Torsion generators We recall that a torsion element in a group is an element of finite order and an involution is an element of order two. As we have seen in Theorem 3.10, if † is a closed orientable surface then the mapping class group Mod.†/ is generated by two elements; the torsion element S of order 4gC2 and the Dehn twist B, which is of infinite order. The mapping class group can also be generated by torsion elements. This fact was first proved by Maclachlan in [21], who showed that the mapping class group is generated by the conjugates of two torsion elements (See also [20]). He used this to prove that the moduli space is simply-connected. It is also possible to generate the mapping class group Mod.†/ by two torsion elements, which was proved in [13]. This is the minimum number of torsion generators. We give an outline of the proof of this result. If g D 1 then Mod.†/ is generated by the torsion elements A1 A2 A1 and A1 A2 , which have orders 4 and 6, respectively. If g D 2 then Mod.†/ is generated by the torsion elements S D A1 A2 A3 A4 and S 0 D A1 A2 A3 A4 B. The order of S is 10 by Lemma 3.8 and that of S 0 is 6. Hence, Mod.†/ is generated by two torsion elements. Suppose that g 3. Let S denote the product A2g A2g1 : : : A2 A1 as in the previous section. Recall that S is of order 4g C 2 by Lemma 3.8. Theorem 3.13 ([13]). Let g 3 and let † be a closed connected orientable surface of genus g. The mapping class group Mod.†/ is generated by the two torsion elements S and BSB 1 . We now give an outline of the proof this theorem. For the details of the proof we refer the reader to [13], Section 5. Let us denote by the subgroup of Mod.†/ generated by the two torsion elements S and BSB 1 . The proof uses the lantern relation in an essential way. Consider the curves a1 , a3 , a5 , b, b3 , d1 and e in Figure 11. The curves a1 , a3 , a5 , b3 bound a sphere with four a1

b

b3

e

d1 a3

a5

Figure 11. The curves in the lantern relation A1 A3 A5 B3 D BD1 E.

holes on † and the Dehn twists about these seven curves satisfy the lantern relation A1 A3 A5 B3 D BD1 E:

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Note that every Dehn twist on the left hand side of this lantern relation commutes with all other Dehn twists. We rewrite the relation as 1 1 A1 D .BA1 3 /.D1 A5 /.EB3 /:

The strategy of the proof is to show that each factor is contained in the subgroup . Once this is proved, we have A1 2 . The rest of the proof is easy, because then we have AkC1 D S k A1 S k 2 and B D .BA1 3 /A3 2 . We refer the reader to [13] for the proof that the element inside each parenthesis is contained in .

3.4 Involution generators Suppose that † is a closed connected orientable surface of genus g. A generating set consisting of involutions for the mapping class groups was first found by McCarthy and Papadopoulos in [22]. They proved, among other things, that for each g 3, the group Mod.†/ can be generated by infinitely many conjugates of a certain involution. In [20], Luo showed how to write a Dehn twist about a nonseparating curve as a product of six involutions for g 3. It then follows from Theorem 3.3 that Mod.†/ is generated by 12g C 6 involutions. In [4], Brendle and Farb proved that Mod.†/ is generated by six involutions for g 3, giving a universal bound for all g. In [12], Kassabov improved this upper bound, showing that Mod.†/ can be generated by four involutions if g 7 and by five involutions if g 5. We show below that Mod.†/ is generated by seven involutions. If g D 1 then Mod.†/ is isomorphic to SL.2; Z/ and its first homology H1 .Mod.†// is isomorphic to Z12 . Therefore, the group Mod.†/ cannot be generated by involutions. In fact, the only involution in the group SL.2; Z/ is I , so that the subgroup of Mod.†/ generated by the involutions is a cyclic group of order two, which is the center of Mod.†/. In the case g D 2, it is well known that H1 .Mod.†// is isomorphic to Z10 . This result is due to Mumford [24]. Hence, the group Mod.†/ cannot be generated by involutions in this case either. The subgroup of Mod.†/ generated by the involutions is a subgroup of index five (cf. [29]). In fact, it is equal to the kernel of the composition of the natural epimorphisms Mod.†/ ! H1 .Mod.†// ! Z5 . From now on we will concentrate to the case g 3. Consider the rotation R of order g given in Figure 8. The element R is a product of two involutions, R D R1 R2 , where R1 and R2 are the involutions given in Figure 8. Below we give a set of generators for Mod.†/ consisting of seven involutions. Most of the material below is adapted from [4]. Lemma 3.14. Let g 2 and let c and d be two disjoint nonseparating simple closed curves on †. Then there exists an involution I of † such that I.c/ D d . Proof. We recall that we consider simple closed curves up to isotopy. Suppose that c and d are not homologous. By the classification of surfaces, there is a self-


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homeomorphism F of † such that F .c/ D a2 and F .d / D a4 . Since R2 .a2 / D a4 the involution I D F 1 R2 F maps c to d , where R2 is the involution given in Figure 8. If c and d are homologous, then one may find such an involution similarly. Lemma 3.15. Let g 1 and let c and d be two simple closed curves on † such that I.c/ D d for some involution I of †. Let C and D denote the Dehn twists about c and d . Then the product CD 1 is a product of two involutions. Proof. Since I.c/ D d , we have ICI D D. Hence, CD 1 D C.IC 1 I / D .CIC 1 /I is a product of the involutions CIC 1 and I . Lemma 3.16 (Brendle–Farb Lemma). Let g 3 and let c be a nonseparating simple closed curve on †. The Dehn twist C about c is contained in a subgroup of the mapping class group generated by four involutions. Proof. Since any two Dehn twists about nonseparating simple closed curves are conjugate and a conjugate of an involution is again an involution, we may assume without loss of generality that c D a1 and show that A1 is a product of four involutions. We embed the surface obtained from † by cutting along a1 in R3 as in Figure 12 in such a way that it is invariant under the 180-degree-rotation J1 about the line l. The rotation J1 induces an involution of †, which is still denoted by J1 . Note that y

a1 z

a3

d1

a1 l

d x

a6

J1

Figure 12. The surface † cut along a1 is embedded in R3 , and the involution J1 is obtained by 180-degree rotation of † about l.

J1 .a3 / D a6 and J1 .x/ D y, so that A3 D J1 A6 J1 and X D J1 Y J1 . In particular, 1 we have XA1 3 D J1 .YA6 /J1 . Note that there is an orientation-preserving self-homeomorphism F of † such that F maps the triple .x; y; z/ to .y; z; x/ and the triple .a3 ; a6 ; d / to .a6 ; d; a3 /. The map F is obtained by permuting the three tubes containing a3 ; a6 and d , so that F 3 D A1 D11 . Let J2 be the involution FJ1 F 1 . Then J2 .a6 / D d and J2 .y/ D z, so that Z D J2 Y J2 and D D J2 A6 J2 . In particular, we have ZD 1 D J2 .YA1 6 /J2 . Let I be any involution with I.a6 / D y. So we have A6 D I Y I , and hence 1 /I . YA1 6 D .Y I Y

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By the lantern relation, we have A1 A3 A6 D D X Y Z; or 1 1 / A1 D .XA1 3 /.YA6 /.ZD 1 1 D J1 .YA6 /J1 .YA6 / J2 .YA1 6 /J2 D J1 .Y I Y 1 /I J1 .Y I Y 1 /I J2 .Y I Y 1 /I J2 :

Hence, A1 is a product of the four involutions I , Y I Y 1 , J1 and J2 . Theorem 3.17. Let g 3 and let † be a closed connected oriented surface of genus g. The mapping class group Mod.†/ is generated by seven involutions. Proof. By Remark 3.7, the group Mod.†/ is generated by the rotation R and the three Dehn twists A1 , A3 and A6 . Let J3 be an involution such that J3 .a1 / D a3 . Note that we have J1 .a6 / D a3 in the proof of Lemma 3.16. We collect the results we have so far as follows: • Mod.†/ is generated by R; A1 ; A3 and A6 ; • A1 is a product of the four involutions I , Y I Y 1 , J1 , J2 ; • A6 D J1 A3 J1 ; • A3 D J3 A1 J3 ; • R D R1 R2 . It follows that the mapping class group Mod.†/ is generated by the seven involutions R1 , R2 , I , Y I Y 1 , J1 , J2 , J3 .

4 Other related results In this section we give a few results on the minimal generating sets of some groups related to the mapping class group. These groups are the extended mapping class group, the hyperelliptic mapping class group and the mapping class group of a nonorientable surface.

4.1 Extended mapping class groups Let † be a closed connected oriented surface of genus g. The extended mapping class group Mod .†/ of † is the group of isotopy classes of all (including orientationreversing) self-homeomorphisms of †. With this definition, the mapping class group Mod.†/ is a normal subgroup of index two in Mod .†/. In fact, it is the only subgroup


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of index two if g 3, because Mod.†/ is perfect [25], so that the first homology group H1 .Mod .†/I Z/ is a cyclic group of order two. Let denote the reflection across the plane which contains the y-axis and perpendicular to the page in Figure 6. Let us denote by T the product A2g : : : A3 A2 A1 . It was shown in [13], Theorem 4.3, that the group Mod .†/ is generated by the two elements B and T , so that the minimal number of generators for Mod .†/ is also 2. The proof of this result follows the same lines as the proof of Theorem 3.10. It also follows from Theorem 3.13 that the group Mod .†/ is generated by the three torsion elements S; BSB 1 and . A result of Stukow [30] states that for each g 1, the extended mapping class group Mod .†/ is also generated by three orientationreversing elements of order two. Thus the minimal number of involution generators is three, as Mod .†/ cannot be generated by two involutions.

4.2 Hyperelliptic mapping class groups Let † be a closed connected oriented surface of genus g embedded in R3 as in Figure 6. Let H be the rotation by 180 degrees about the y-axis, which leaves † invariant so that it can be seen as an element of the mapping class group Mod.†/. The centralizer of the involution H in Mod.†/ is called the hyperelliptic mapping class group, denoted by HMod.†/: HMod.†/ D fF 2 Mod.†/ W FH D HF g: If g D 1 or g D 2 then every element of Mod.†/ commutes with H , so that HMod.†/ D Mod.†/. If g 3 then HMod.†/ is of infinite index in Mod.†/. Clearly, the Dehn twists A1 D B1 ; A2 ; A3 ; : : : ; A2g ; A2gC1 D Bg are contained in the hyperelliptic mapping class group HMod.†/. In fact, it was proved by Birman and Hilden in [3] that these 2g C 1 Dehn twists generate HMod.†/. Birman and Hilden also gave a presentation for HMod.†/. We now give this presentation. Theorem 4.1 ([3]). Let g 1 and let † be a closed connected oriented surface of genus g. The hyperelliptic mapping class group HMod.†/ admits a presentation with generators A1 ; A2 ; A3 ; : : : ; A2g ; A2gC1 , and with defining relations • Ai Aj D Aj Ai for ji j j 2, 1 i; j 2g C 1; • Ai Ai C1 Ai D AiC1 Ai AiC1 for 1 i 2g; • .A1 A2 : : : A2gC1 /2gC2 D 1; • .A1 A2 : : : A2g A2gC1 A2gC1 A2g : : : A2 A1 /2 D 1; • HA1 D A1 H , where H D A1 A2 : : : A2g A2gC1 A2gC1 A2g : : : A2 A1 . Let S D A2g A2g1 : : : A2 A1 and S 0 D A2gC1 A2g : : : A2 A1 . The orders of S and S 0 are 4g C 2 and 2g C 2, respectively. They also generate HMod.†/, so that the minimal number of (torsion) generators of HMod.†/ is two.

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By Theorem 4.1, H1 .HMod.†// is isomorphic to Z4gC2 (resp. Z8gC4 ) if g is even (resp. odd), so that HMod.†/ cannot be generated by involutions. However, it was shown by Stukow [29] that the subgroup of HMod.†/ generated by involutions contains the derived subgroup ŒHMod.†/; HMod.†/ as a subgroup of index two. In particular, the quotient of HMod.†/ by is cyclic of order 2g C 1 if g is even, and of order 4g C 2 if g is odd.

4.3 Mapping class groups of nonorientable surfaces Let N be a closed connected nonorientable surface of genus g. Recall that the genus of a nonorientable surface is defined to be the number of real projective planes in a connected sum decomposition. Equivalently, the genus (of any surface) is the maximum number of pairwise disjoint simple closed curves whose complement is connected. The mapping class group of N is defined to be the group of isotopy classes of all homeomorphisms of N , and we denote it by Mod.N /. If g D 1 then it is easy to see that the mapping class group of N is trivial. It is also well known that if g D 2 then Mod.N / is isomorphic to Klein’s four group, the direct sum of two cyclic groups of order two. Let us assume from now on in this subsection that g 3. A generating set for Mod.N / was first found by Lickorish [17], [19], who proved that Mod.N / is generated by Dehn twists about two-sided simple closed curves and a so-called crosscap slide (or a Y -homeomorphism). A simple closed curve is called two-sided if a regular neighborhood of it is homeomorphic to an annulus. It is called one-sided if a regular neighborhood of it is homeomorphic to a Möbius band. A crosscap slide can roughly be defined as the homeomorphism obtained by sliding a one-sided simple closed curve along another one-sided curve. It is supported on a Klein bottle with one boundary component. For the precise definition of a crosscap slide, we refer the reader to [17], where it is called a Y-homeomorphism. Lickorish also proved that Dehn twists generate a subgroup of index two. Chillingworth proved in [5] (see also [2]) that Mod.N / is generated by finitely many elements. Szepietowski showed in [27] that Mod.N / can be generated by involutions. He later showed in [28] that Mod.N / can be generated by three elements. As for the generating set consisting of involutions, he proved the following theorem. Theorem 4.2 ([28]). Let g 3 and let N be a connected nonorientable surface of genus g. The mapping class group Mod.N / is generated by three involutions if g D 3 and by four involutions if g 4. In the next theorem, we deduce, from the know presentation, the minimal number of generators and the minimal number of involution generators of the mapping class group in the case g D 3. Theorem 4.3. Let N be a connected nonorientable surface of genus three. The mapping class group Mod.N / is generated by two elements, and also by three involutions.


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Proof. It was shown by Birman and Chillingworth in [2] that Mod.N / admits a presentation with generators a, b and y, and with defining relations aba D bab, .aba/4 D 1; yay D a1 , yby D b 1 , y 2 D 1. It follows that Mod.N / is generated by the three involutions y, ay and by. We claim that Mod.N / can also be generated by the elements a and by. By an easy computation we get .by/ a1 .by/1 D bab 1 ; and a .bab 1 / a1 D b: It follows that a and by generate the group Mod.N /. Remark 4.4. From the presentation given in the above proof of Theorem 4.3 one can observe that the mapping class group of the closed nonorientable surface of genus three is isomorphic to the group GL.2; Z/. Since the first homology of Mod.N / is the direct sum of three cyclic groups of order two if g D 4 (cf. [14]), the minimal number of generators for Mod.N / in this case is three. It is not known whether Mod.N / can be generated by two elements for g 5.

5 Problems In this last section, we state a few problems concerning the generating sets of the mapping class group. Let † (resp. N ) denote a closed connected orientable (resp. nonorientable) surface of genus g. Problem 5.1. Is Mod.†/ generated by three involutions for g 3? It is generated by four involutions (for g 7), and it cannot be generated by two involutions since it is not a quotient of a dihedral group. Problem 5.2. The group Mod.†/ is generated by two elements of order 4g C 2. What are the other numbers k (less than 4g C 2) such that Mod.†/ is generated by two elements of order k? What is the smallest such k? Problem 5.3. Is the group Mod .†/ generated by two torsion elements? Problem 5.4. Let ng denote the smallest number of generators for Mod.N /. Then n1 D 0, n2 D 2, n3 D 2, n4 D 3 and 2 ng 3 for g 5. What is the exact value of ng for g 5?

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References [1]

J. S. Birman, Braids, links and mapping class groups. Ann of Math. Stud. 82, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo 1974.

[2]

J. S. Birman and D. R. J. Chillingworth, On the homeotopy group of a non-orientable surface. Proc. Cambridge Philos. Soc. 71 (1972), 437–448. 460, 461

[3]

J. S. Birman and H. M. Hilden, On the mapping class groups of closed surfaces as covering spaces. In Advances in the theory of Riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Stud. 66, Princeton University Press, Princeton, N.J., 1971, 81–115. 459

[4]

T. Brendle and B. Farb, Every mapping class group is generated by 6 involutions. J. Algebra 278 (2004), no. 1, 187–198. 442, 456

[5]

D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a nonorientable surface. Proc. Cambridge Philos. Soc. 65 (1969), 409–430. 460

[6]

M. Dehn, Die Gruppe der Abbildungsklassen. Acta Math. 69 (1938), 135–206. 442, 448, 449

[7]

B. Farb and D. Margalit, A primer on mapping class group. Princeton Math. Ser. 49, Princeton University Press, Princeton, N.J., 2011. 443

[8]

S. Humphries, Generators for the mapping class group. In Topology of low dimensional manifolds (R. Fenn, ed.), Lecture Notes in Math. 722, Springer-Verlag, Berlin 1979, 44–47. 450

[9]

N. V. Ivanov, Mapping class groups. In Handbook of geometric topology (R. Daverman and R. Sher, eds.), North-Holland, Amsterdam 2002, 523–633. 443

[10] D. L. Johnson, Homeomorphisms of a surface which act trivially on homology. Proc. Amer. Math. Soc. 75 (1979), no. 1, 119–125. 448 [11] D. L. Johnson, The structure of the Torelli group. I. A finite set of generators for «. Ann. of Math. (2) 118 (1983), no. 3, 423–442. 451 [12] M. Kassabov, Generating mapping class groups by involutions. arXiv:math/0311455v1 [math.GT]. 456 [13] M. Korkmaz, Generating the surface mapping class group by two elements. Trans. Amer. Math. Soc. 357 (2005), 3299–3310. 442, 453, 454, 455, 456, 459 [14] M. Korkmaz, First homology group of mapping class groups of nonorientable surfaces. Proc. Cambridge Philos. Soc. 123 no. 3 (1998), 487–499. 461 [15] M. Korkmaz, Mapping class groups of nonorientable surfaces. Geom. Dedicata 89 (2002), 109–133. [16] W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds. Ann. of Math. 76 (1962), 531–540. 449 [17] W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds. Proc. Cambridge Philos. Soc. 59 (1963), 307–317. 460 [18] W. B. R. Lickorish, A finite set of generators for the homeotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 60 (1964), 769–778. 442, 449 [19] W. B. R. Lickorish, On the homeomorphisms of a non-orientable surface. Proc. Cambridge Philos. Soc. 61 (1965), 61–64. 460


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[20] F. Luo, Torsion elements in the mapping class group of a surface. arXiv:math/0004048v1 [math.GT]. 442, 455, 456 [21] C. Maclachlan, Modulus space is simply-connected. Proc. Amer. Math. Soc. 29 (1971), 85–86. 455 [22] J. D. McCarthy and A. Papadopoulos, Involutions in surface mapping class groups. Enseign. Math. (2) 33 (1987), no. 3–4, 275–290. 456 [23] S. Morita, Introduction to mapping class groups of surfaces and related groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, EMS Publishing House, Zurich 2007, 353–386 443 [24] D. Mumford, Abelian quotients of the Teichmüller modular group. J. Analyse Math. 18 (1967), 227–244. 456 [25] J. Powell, Two theorems on the mapping class group of a surface. Proc. Amer. Math. Soc. 68 (1978), no. 3, 357–350. 459 [26] B. Szepietowski, Mapping class group of a non-orientable surface and moduli space of Klein surfaces. C. R. Math. Acad. Sci. Paris 335 (2002), no. 12, 1053–1056. [27] B. Szepietowski, Involutions in mapping class groups of non-orientable surfaces. Collect. Math. 55 (2004), no. 3, 253–260. 460 [28] B. Szepietowski, The mapping class group of a nonorientable surface is generated by three elements and by four involutions. Geom. Dedicata 117 (2006), 1–9. 460 [29] M. Stukow, Small torsion generating sets for hyperelliptic mapping class groups. Topology Appl. 145 (2004), 83–90. 456, 460 [30] M. Stukow, The extended mapping class group is generated by 3 symmetries. C. R. Acad. Sci. Paris Ser. I 338 (2004), 403–406. 459 [31] B. Wajnryb, A simple presentation for the mapping class group of an orientable surface. Israel J. Math. 45 (1983), 157–174. 443 [32] B. Wajnryb, Mapping class group of a surface is generated by two elements. Topology 35 (1996), 377–383. 442, 452 [33] B. Wajnryb, An elementary approach to the mapping class group of a surface. Geom. Topol. 3 (1999), 405–466.

Chapter 8

From mapping class groups to monoids of homology cobordisms: a survey Kazuo Habiro and Gwénaël Massuyeau

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Homology cobordisms and homology cylinders . . . . . . . . 2.1 Definition of the monoids . . . . . . . . . . . . . . . . . 2.2 The mapping class group and the Torelli group . . . . . 2.3 The closure construction . . . . . . . . . . . . . . . . . 2.4 The closed case and the bordered case . . . . . . . . . . 3 Johnson homomorphisms and Morita homomorphisms . . . . 3.1 The Dehn–Nielsen representation . . . . . . . . . . . . 3.2 Johnson homomorphisms . . . . . . . . . . . . . . . . . 3.3 Morita homomorphisms . . . . . . . . . . . . . . . . . 3.4 Infinitesimal versions of the Dehn–Nielsen representation 4 The LMO homomorphism . . . . . . . . . . . . . . . . . . . 4.1 The algebra of symplectic Jacobi diagrams . . . . . . . . 4.2 Definition of the LMO homomorphism . . . . . . . . . 4.3 The tree-reduction of the LMO homomorphism . . . . . 5 The Y -filtration on the monoid of homology cylinders . . . . 5.1 The Yk -equivalence relation . . . . . . . . . . . . . . . 5.2 Yk -equivalence and homology cylinders . . . . . . . . . 5.3 Yk -equivalence and the Torelli group . . . . . . . . . . . 6 The Lie ring of homology cylinders . . . . . . . . . . . . . . 6.1 Definition of the Lie ring of homology cylinders . . . . . 6.2 The Lie algebra of homology cylinders . . . . . . . . . . 6.3 The degree 1 part of the Lie ring of homology cylinders . 7 The homology cobordism group . . . . . . . . . . . . . . . . 7.1 Definition of the homology cobordism group . . . . . . 7.2 Representations of the homology cobordism group . . . 7.3 The Y -filtration on the homology cobordism group . . . 7.4 Other aspects of the homology cobordism group . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction It is well known that 3-manifolds can be presented in terms of self-homeomorphisms of surfaces. In particular, any closed oriented 3-manifold M can be presented by a Heegaard splitting: then M is obtained by gluing two copies of the same handlebody H with a self-homeomorphism of the closed surface @H . In that way, the study of mapping class groups interplays with 3-dimensional topology. For example, Lickorish used Heegaard splittings to show that any closed oriented 3-manifold can be obtained from S 3 by surgery [63] – a theorem deduced from the fact that the mapping class group is generated by Dehn twists. This is a contribution of mapping class groups to 3-dimensional topology. In the other direction and in a more recent context, the TQFT approach to 3-manifolds has produced a variety of representations for mapping class groups: see Masbaum’s survey [66]. As an alternative to Heegaard splittings, open book decompositions with connected binding can also be used to present all closed oriented 3-manifolds: one then needs to consider surfaces with connected nonempty boundary instead of closed surfaces. Adding homological considerations to 3-manifolds, 3-dimensional topology becomes intertwined with the study of the Torelli group. Let † be a compact connected oriented surface with at most one boundary component. Then, the Torelli group .†/ of the surface † is defined as the subgroup of the mapping class group M.†/ acting trivially on the homology of †. The study of the Torelli group, from a topological point of view, started with works of Birman [10] and was followed by Johnson through a series of paper, which notably resulted in a finite generating set for .†/ [48] and an explicit computation of its abelianization [49]. The reader is referred to Johnson’s survey [47] for an account of his work on .†/. A key ingredient in his paper [49] is the use of the Birman–Craggs homomorphisms [12], whose definition involves the Rochlin invariant of closed spin 3-manifolds and can be stated in a simple way using open book decompositions (with connected binding). The study of the structure of the Torelli group has been pursued by Morita. He discovered a deep relation with the Casson invariant of homology 3-spheres [72], [73], [77] and he reinforced the use of algebraico-topological methods in that study (via the action of .†/ on 1 .†/) [75], [76]. We refer to his survey [74] and to [78] for more recent developments. Morita’s work on the Casson invariant prefigures the use of finite-type invariants of homology 3-spheres in the study of the Torelli group. This approach of .†/ has been developed in subsequent works of Garoufalidis and Levine [24], [23]. Works of Goussarov [28], [29], [22] and the first author [35] breathed new life into this approach to the Torelli group. In their works, new surgery techniques, known as calculus of claspers, are developed for the study of finite-type invariants. Besides, the study of the mapping class group is tied to 3-dimensional topology in the following way. Goussarov and the first author consider homology cobordisms of †, i:e: cobordisms C (with corners if @† ¤ ¿) such that both the “top” inclusion and the “bottom” inclusion † C induce isomorphisms at the level of homology. With the

Chapter 8. From mapping class groups to monoids of homology cobordisms: a survey 467

usual “stacking” operation, the set of homology cobordisms forms a monoid C .†/, into which the group M.†/ embeds by the mapping cylinder construction: c W M.†/ ! C .†/: Calculus of claspers applies to the monoid C .†/ of homology cylinders over †, i:e: cobordisms of † that cannot be distinguished from the trivial cylinder by homology. Since .†/ is mapped into C .†/ by c, homology cylinders have opened new perspectives for the study of the Torelli group. In this chapter, we shall survey recent developments in that direction. The monoid C .†/ of homology cobordisms is presented, with special attention given to the submonoid C .†/ of homology cylinders. We outline how the structure of the monoid C .†/ can be studied by means of finite-type invariants and claspers. At the same time, we explain how this study is related to more classical results and constructions for the Torelli group. We are mainly interested in compact connected oriented surfaces † without boundary (referred to, below, as the closed case) or with connected boundary (the bordered case). Indeed, when the surface † has more than one boundary component, the study of homology cobordisms of † by means of finite-type invariants is still possible, but it gets a little bit more complicated. Since closed oriented 3-manifolds can be presented by Heegaard splittings (or, alternatively, by open book decompositions with connected binding), the interactions between 3-dimensional topology and mapping class groups are somehow contained in the closed case (or in the bordered case). The survey is organized as follows. Section 2 introduces the monoids C .†/ and C .†/, starting with their precise definitions. We give a few elementary facts about these monoids. For instance, it is shown that the groups of invertible elements of C .†/ and C .†/ coincide with M.†/ and .†/, respectively. We also discuss the passage from the bordered case to the closed case. Section 3 reviews constructions based on the Dehn–Nielsen representation, which is given by the canonical action of M.†/ on the fundamental group 1 .†/. Thus, we survey Johnson’s homomorphisms (in the bordered and closed cases) and their extensions by Morita (in the bordered case). Originally defined on subgroups of the mapping class group, these homomorphisms encode the action of M.†/ on nilpotent quotients of 1 .†/. Since, by virtue of Stallings’theorem, a homology equivalence between groups induces isomorphisms at the level of their nilpotent quotients, Johnson’s homomorphisms have natural extensions to the monoid C .†/. By using the Malcev Lie algebra of 1 .†/, we define “infinitesimal” versions of the Dehn–Nielsen representation and we use them to reformulate Johnson’s and Morita’s homomorphisms. Section 4 introduces the LMO homomorphism, which is a diagrammatic representation of the monoid C .†/. Derived from the Le–Murakami–Ohtsuki invariant of closed 3-manifolds, this invariant of homology cylinders is universal among Qvalued finite-type invariants. Thus, the LMO homomorphism seems very appropriate to the study of the Torelli group from the point of view of finite-type invariants. The tree-reduction of the LMO homomorphism (where all looped diagrams are “killed”)

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encodes the action of C .†/ on the Malcev Lie algebra of 1 .†/. It follows that the LMO homomorphism is injective on the Torelli group, and that it determines the Johnson homomorphisms as well as the Morita homomorphisms. (The results of this section apply to the bordered and closed cases.) Section 5 gives an overview of claspers and their application to the study of the monoid C .†/. We recall the definition of the Yk -equivalence relations (k 1), which is based on clasper surgery, and a few of their properties. These relations define a filtration C .†/ D Y1 C .†/ Y2 C .†/ Y3 C .†/ of the monoid C .†/ by submonoids. Although C .†/ by itself is only a monoid, this filtration shares several properties with the lower central series of a group. In particular, there is a graded Lie ring Gr Y C .†/ associated with this filtration. When restricted to the Torelli group, the Y -filtration contains the lower central series, and the two filtrations are expected to coincide (in the stable genus). Section 6 is an exposition of results on the graded Lie ring Gr Y C .†/ for a bordered or a closed surface †. On one hand, the graded Lie algebra Gr Y C .†/ ˝ Q has an explicit diagrammatic description. The proof, which is only sketched here, is based on the LMO homomorphism and claspers. This result is connected to Hain’s “infinitesimal” presentation of the Torelli group. On the other hand, the abelian group C .†/=Y2 (i:e: the degree 1 part of Gr Y C .†/) is explicitly described using the first Johnson homomorphism and the Birman–Craggs homomorphism. This is similar to Johnson’s result on the abelianization of .†/, which was mentioned above. Finally, following Garoufalidis and Levine, we consider in Section 7 homology cobordisms up to the relation H of homology cobordism. The quotient H .†/ WD C.†/= H is a group, in which the mapping class group M.†/ still embeds. We outline the extent to which the previous constructions apply to the study of the group H .†/. This group has been the subject of recent works in other directions: those developments are only evoked here, while they are presented in more detail in Sakasai’s chapter in this volume [90]. Conventions. Homology groups are assumed to be with integer coefficients unless otherwise specified. On pictures, the blackboard framing convention is used to represent 1-dimensional objects with framing. Equivalence classes are denoted by curly brackets f ˘ g, except for homology/homotopy/isotopy classes which are denoted by square brackets Œ ˘ . We denote by the same symbol a graded vector space and its degree completion.


2 Homology cobordisms and homology cylinders In the sequel, we denote by †g;b a compact connected oriented surface of genus g with b boundary components. We define the monoid of cobordisms of †g;b , and we recall how the mapping class group of †g;b embeds into this monoid. At the end of this section, we restrict ourselves to the closed surface †g (b D 0) and to the bordered surface †g;1 (b D 1).

2.1 Definition of the monoids A cobordism of †g;b is a pair .M; m/ where M is a compact connected oriented 3-manifold and m W @ †g;b Œ1; 1 ! @M is an orientation-preserving homeomorphism. (We will usually denote the cobordism .M; m/ simply by M , the convention being that the boundary parameterization is denoted by the lower-case letter m.) Two cobordisms M; M 0 are homeomorphic if there is an orientation-preserving homeomorphism f W M ! M 0 such that f j@M B m D m0 . The inclusion †g;b ! M defined by s 7! m.s; ˙1/ is denoted by m˙ . We call mC .†g;b / the top surface of M and m .†g;b / the bottom surface. Two cobordisms M; M 0 of †g;b can be composed by gluing the bottom of M 0 to the top of M : M B M 0 WD M [mC B.m0 /1 M 0 : The resulting 3-manifold is a cobordism of †g;b with the obvious parameterization of its boundary. When cobordisms are considered up to homeomorphisms, the operation B is associative and the trivial cobordism †g;b Œ1; 1 WD †g;b Œ1; 1; Id is an identity element for that operation. Definition 2.1. A homology cobordism of †g;b is a cobordism .M; m/ such that both inclusions mC and m induce isomorphisms H .†g;b / ! H .M /. A homology cylinder over †g;b is a cobordism .M; m/ for which we can find an isomorphism h W H .†g;b Œ1; 1/ ! H .M / such that the following diagram commutes:

H †g;b Œ1; 1 O

h '

incl

H

@.†g;b Œ1; 1/

/ H .M / O incl

' m

/ H .@M /:

In other words, .M; m/ has the same homology type as the trivial a homology cylinder cobordism †g;b Œ1; 1; Id . The set of homology cobordisms contains the set of homology cylinders, and an application of the Mayer–Vietoris theorem shows that both sets are stable by composition. In the sequel, homology cobordisms and homology cylinders are considered

470


up to homeomorphisms. Thus, they form monoids which we denote by C †g;b C †g;b : Example 2.2 (Genus 0). For b > 1, a framed string-link in D 2 Œ1; 1 on .b 1/ strands in the sense of [31], [32] can be regarded as a cobordism of †0;b by taking its complement. More generally, the monoid C .†0;b / can be identified with the monoid of framed string-links in homology 3-balls. The monoid C .†0;b / corresponds by this identification to the monoid of framed string-links whose linking matrix is trivial. For b D 0 or b D 1, homology cobordisms of †0;b can be transformed into homology 3spheres by gluing balls along their boundary components: thus, the monoid C .†0;b / D C .†0;b / is canonically isomorphic to the monoid of homology 3-spheres with the connected sum operation.

2.2 The mapping class group and the Torelli group

We shall now identify the invertible elements of the monoid C †g;b . Definition 2.3. The mapping class group of †g;b is the group M †g;b WD HomeoC †g;b ; @†g;b = Š

of isotopy classes of homeomorphisms †g;b ! †g;b which preserve the orientation and fix the boundary pointwise. A homeomorphism f W †g;b ! †g;b (which preserves the orientation and fixes the boundary pointwise) gives rise to a cobordism †g;b Œ1; 1; .Id .1// [ .@†g;b Id/ [ .f 1/ ; called the mapping cylinder of f . This construction defines a monoid map c W M †g;b ! C †g;b : By a classical result of Baer [5], [6], any two homeomorphisms †g;b ! †g;b of the above type are homotopic relatively to the boundary if and only if they are isotopic relatively to the boundary. It follows that the homomorphism c is injective. The following result is folklore, it appears in [32] for the genus 0 case. Proposition 2.4. An element of the monoid C †g;b is left-invertible if and only if it is right-invertible, and the group of invertible elements of C †g;b coincides with the image of c W M †g;b ! C †g;b . Proof. When g D 0 and b D 0 or 1, the monoid C †g;b can be interpreted as the monoid of homology 3-spheres. (See Example 2.2.) The proposition then amounts to Alexander’s theorem [2]: the 3-manifold S 3 is irreducible. Consequently, we can assume that †g;b ¤ D 2 ; S 2 .


Let M; N 2 C †g;b be such that M B N D †g;b Œ1; 1. We are asked to show that M is a mapping cylinder. Claim. The inclusion m W †g;b ! M induces an isomorphism at the level of the fundamental group. Moreover, the manifold M is irreducible because M B N D †g;b Œ1; 1 is irreducible. Then, it follows from Theorem 10.2 in [39] that there is a homeomorphism between M and †g;b Œ1; 1 which sends m .†g;b / to †g;b .1/ and, so, mC .†g;b / to †g;b 1. The conclusion follows. To prove the claim, we denote by i W M ! M B N and j W N ! M B N the inclusions. The map i B m is equivalent to the inclusion of †g;b .1/ in †g;b Œ1; 1 and, so, induces an isomorphism at the level of the fundamental group. Thus, it is enough to prove that i W 1 .M / ! 1 .M B N / is injective. Since M B N is the trivial cylinder, the image of j nC; in 1 .M B N / contains the image of i m; . We deduce the following inclusion of subgroups of 1 .M /: m; 1 .†g;b / i 1 j .1 .N // : Besides, an application of the van Kampen theorem shows that i 1 j .1 .N // mC; 1 .†g;b / ; (2.1) so that we have m; 1 .†g;b / mC; 1 .†g;b / . Since the homomorphism i m; W 1 .†g;b / ! 1 .M B N / is surjective, we deduce that the homomorphism i mC; W 1 .†g;b / ! 1 .M B N / is surjective and, so, is an isomorphism. (Here, we use the fact that the fundamental group of a surface is Hopfian [40].) Another application of (2.1) then shows that i is injective, and we are done. Usually in the literature, the Torelli group is only defined when the surface is closed or has a single boundary component. In our situation, it is natural to define the Torelli group †g;b for any b 0 in the following way. (When b D 0 or b D 1, our definition is equivalent to the usual one recalled in §2.4.) Definition 2.5. The Torelli group of †g;b is the subgroup †g;b WD c1 C .†g;b / M †g;b : Example 2.6 (Genus 0). For b > 1, the mapping class group of †0;b can be identified with the group of framed pure braids on .b 1/ strands over the disk D 2 [11]. The Torelli group then corresponds to those pure braids with trivial linking matrix. For b D 0 or b D 1, the groups M.†0;b / and .†0;b / are trivial.

2.3 The closure construction There is a systematic way of transforming a homology cobordism into a 3-manifold without boundary: we simply “close” a homology cobordism by identifying its top and

472


bottom surfaces. This basic construction will be used several times in the following sections to define invariants of homology cylinders. So, we would like to define it carefully. Definition 2.7. The closure of an M 2 C .†g;b / is the closed connected oriented 3-manifold x WD .M= / [ S 1 D 2 [ [ S 1 D 2 : (2.2) M 1 b Here, the equivalence relation identifies mC .s/ with m .s/ for all s 2 †g;b . The resulting 3-manifold M= has a toroidal boundary which is recapped by gluing b solid tori. More precisely, we number the boundary components @1 †g;b ; : : : ; @b †g;b a base point ?i on each of them. Then, the of †g;b from 1 to b,and we choose meridian 1 @D 2 of S 1 D 2 i is glued along the circle m.?i Œ1; 1/= while its longitude S 1 1 is glued along m.@i †g;b 0/. In the special case where M D c .f / is a mapping cylinder, the closure of M is obtained from the mapping torus of f by gluing b solid tori. If b D 0, the 3-manifold c .f / is the total space of a surface bundle over S 1 and, so, one only reaches a special (although interesting) class of closed oriented 3-manifolds. If b > 0, the 3-manifold c .f / comes with an open book decomposition whose binding consists of the cores of the glued solid tori. Any closed connected oriented 3-manifold can be presented in that way [1], and one can even require the binding to be connected [27], [82], i:e: we can assume b D 1. Example 2.8. The closure of the trivial cobordism †g;b Œ1; 1 is the product †g S 1 if b D 0 and the connected sum #2gCb1 .S 2 S 1 / if b > 0. For g D 0 and b D 1, the closure operation is the same as the “recapping” operation described at the end of Example 2.2.

2.4 The closed case and the bordered case As mentioned in the introduction, we are specially interested in the closed surface †g and the bordered surface †g;1 . The monoids defined in §2.1 are denoted by Cg C g

and

Cg;1 C g;1

in the closed case and the bordered case, respectively. Observe that, for b D 0 and cylinder if and only b D 1, a homology cobordism .M; m/ of †g;b is a homology if mC and m induce the same isomorphism H †g;b ! H .M /. Similarly, the groups introduced in §2.2 are simply denoted by Mg g

Mg;1 g;1 : Observe that, for b D 0 and b D 1, an f 2 M †g;b belongs to †g;b if and only if f induces the identity in homology. This is the way the Torelli group is usually defined in the literature [47]. and


The study of Mg can be somehow “reduced” to the study of Mg;1 thanks to Birman’s exact sequence, which we now recall. For this, we need to fix a closed disk D †g . Then, we can think of the closed surface †g as the union of †g;1 with D. Theorem 2.9 (Birman’s exact sequence [9]). Give †g an arbitrary smooth structure as well as a Riemannian metric, and let U.†g / be the total space of the unit tangent bundle of †g . Then, there is an exact sequence of groups

1 U.†g /

Push

/ Mg;1

˘[ IdD

/ Mg

/1

where the “Push” map is defined below. Sketch of the proof. Let DiffeoC .†g / be the group of orientation-preserving diffeomorphisms †g ! †g . Since “diffeotopy groups” coincide with “homeotopy groups” in dimension two, we have (2.3) Mg D 0 DiffeoC .†g / : Let v be a “unit tangent vector” of D: v 2 Tp †g with kvk D 1 and p 2 D. Then, we can consider the subgroup DiffeoC .†g ; v/ consisting of diffeomorphisms whose differential fixes v. One can show that (2.4) Mg;1 ' 0 DiffeoC .†g ; v/ : The map DiffeoC .†g / ! U.†g / defined by f 7! dp f .v/ is a fiber bundle with fiber DiffeoC .†g ; v/. According to (2.3) and (2.4), the long exact sequence for homotopy groups induced by this fibration terminates with 1 DiffeoC .†g /; Id ! 1 U.†g /; v ! Mg;1 ! Mg ! 1: The map 1 U.†g /; v ! Mg;1 is called the “Push” map because it has the following description. A loop in U.†g / with base point v can be regarded as an isotopy I W D Œ0; 1 ! †g of the disk D in †g such that I.˘; 0/ D I.˘; 1/ is the inclusion D †g . This isotopy can be extended to an ambient isotopy I W †g Œ0; 1 ! †g starting with I.˘; 0/ D Id†g . Then, Push.Œ/ WD restriction of I.˘; 1/ to †g;1 D †g n int.D/ is the image of the loop in Mg;1 . For cobordisms, we have the following analogue of Birman’s exact sequence. This lemma will be used in the following sections to define homomorphisms on Cg by first defining them on Cg;1 . Lemma 2.10. Gluing a 2-handle along @†g;1 Œ1; 1 defines a surjection Cg;1 ! Cg ;

¾: M 7! M

474


¿0 if and only if M 0 can be ¾ D M Moreover, two cobordisms M and M 0 satisfy M obtained from M by surgery along a 2-component framed link K0 [ K1 in M as shown in Figure 1.

K0 K1

Figure 1. The framed link K0 [ K1 in the vicinity of the “vertical” boundary of M , i:e: the part m .@†g;1 Œ1; 1/ of @M .

¾ of †g by attaching a 2-handle along the “vertical” Proof. We define the cobordism M boundary of M : ¾ WD M [mj .D Œ1; 1/ : M @†g;1 Œ1;1 ¿0 is similarly defined from M 0 . The 2-handle D Œ1; 1 can be The cobordism M ¾ and, similarly, we have a regarded as a framed string-knot X in the 3-manifold M ¿0 . Suppose now that we are given a homeomorphism framed string-knot X 0 M ¿0 ! M ¾ . If the image Xz WD f .X 0 / happens to be isotopic to X, then f restricts fWM to a homeomorphism M 0 ! M and we are done. Otherwise, we can apply the “slam dunk” move:

K1 K0

X

Xz

(2.5)

Here, surgery is performed along the 2-component framed link K0 [ K1 ; it produces a homeomorphic manifold and the corresponding homeomorphism changes the framed z Therefore, M 0 is (up to homeomorphism) the result of the surgery string-knot X to X. in M along K0 [ K1 .


3 Johnson homomorphisms and Morita homomorphisms We review the Johnson homomorphisms and their extensions by Morita. Originally defined for the mapping class group of a surface, those homomorphisms have natural extensions to the monoid of homology cobordisms. In this section, we restrict ourselves to the bordered surface †g;1 and to the closed surface †g . We also refer to Morita’s survey [79] in Volume I of this Handbook.

3.1 The Dehn–Nielsen representation The definitions of Johnson’s and Morita’s homomorphisms are based on the action of the mapping class group on the fundamental group, which we shall first review. We start with the case of the bordered surface †g;1 . We denote WD 1 .†g;1 ; ?/ where ? 2 @†g;1 is a fixed base point. The homotopy class of boundary the oriented curve defines a special element in the group , namely WD @†g;1 . Definition 3.1. The Dehn–Nielsen representation is the group homomorphism W Mg;1 ! Aut./;

f 7! f :

A classical result by Dehn, Nielsen and Baer asserts that is injective, and that its image consists of the automorphisms that fix [98]. There is no obvious way to extend the map to the monoid Cg;1 . Nonetheless, the “nilpotent” versions of can be extended from Mg;1 to Cg;1 . For this, we consider the lower central series of , 1 2 k kC1 ; which is inductively defined by 1 WD and iC1 WD Œi ; . For each k 0, we define a monoid homomorphism k W Cg;1 ! Aut .=kC1 / ;

M 7! m; 1 B mC; :

Here the fundamental group of M is based at ?0 WD m.?; 0/, which is identified (homotopically) with mC .?/ D m.?; C1/ and to m .?/ D m.?; 1/ by the “vertical” segments m.? Œ0; 1/ and m.? Œ1; 0/. According to Stallings’ theorem [91], the map m˙; W =kC1 ! 1 .M; ?0 /=kC1 1 .M; ?0 / is a group isomorphism. (The hypothesis that m˙ is an isomorphism in homology is needed here.) Example 3.2 (Degree 1). The abelian group = 2 is H WD H1 †g;1 . The homomorphism 1 W Cg;1 ! Aut.H / gives the homology type of homology cobordisms. So, the kernel of 1 is the monoid of homology cylinders C g;1 . The image of 1 is the subgroup Sp.H / of Aut.H / consisting of the elements preserving the intersection pairing ! W H H ! Z.

476


The following filtration is introduced in [35], [25]. Definition 3.3. The Johnson filtration of Cg;1 is the sequence of submonoids Cg;1 D Cg;1 Œ0 Cg;1 Œ1 Cg;1 Œk Cg;1 Œk C 1 where Cg;1 Œk is the kernel of k for all k 0. This filtration restricts to a decreasing sequence of subgroups of the mapping class group, which was introduced by Johnson in [47] and studied by Morita in [75]. We denote it by Mg;1 D Mg;1 Œ0 Mg;1 Œ1 Mg;1 Œk Mg;1 Œk C 1 : This series of subgroups is an N -series, in the sense that the commutator subgroup of Mg;1 Œk and Mg;1 Œl is contained in Mg;1 Œk C l for all k; l 0. It is also separating, in the sense that its intersection k0 Mg;1 Œk is trivial. This fact follows directly from the injectivity of the Dehn–Nielsen representation, and from the fact that is residually nilpotent. Note that the Johnson filtration is far from being separating in the case of homology cobordisms. (For instance, every k is trivial for g D 0. This fact, according to Example 2.2, contrasts with the richness of the monoid of homology 3-spheres.) Let us now consider the case of a closed surface †g . We denote » WD 1 .†g /: We have implicitly chosen a base point ? 2 †g , but we do not include it in the notation since the definitions below will not depend on it. Definition 3.4. The Dehn–Nielsen representation is the group homomorphism /; W Mg ! Out.»

f 7! f :

In the closed case, the map is injective and its image consists of classes of / [98]. automorphisms that fix Œ†g 2 H2 .†g / ' H2 .» Similarly to the bordered case, the “nilpotent” versions of can be extended to the monoid Cg . Thus, we define for each k 0 a monoid homomorphism k W Cg ! Out .» =kC1 »/ ;

M 7! m; 1 B mC; :

To define k .M /, we need to choose a base point in M and some paths joining it to » = kC1 » mC .?/ and to m .?/ respectively, but the resulting outer automorphism of is independent of these choices. Definition 3.5. The Johnson filtration of Cg is the sequence of submonoids Cg D Cg Œ0 Cg Œ1 Cg Œk Cg Œk C 1 where Cg Œk is the kernel of k for all k 0.


This filtration restricts to a decreasing sequence of subgroups of the mapping class group Mg D Mg Œ0 Mg Œ1 Mg Œk Mg Œk C 1 which, again, is an N -series. By using the Dehn–Nielsen representation and more delicate arguments than in the bordered case [8], one can prove that the Johnson filtration is also separating in the closed case (at least for g ¤ 2).

3.2 Johnson homomorphisms We start by defining the Johnson homomorphisms in the bordered case. For this, we fix a few identifications between groups, which will be implicit in the sequel. First of all, we denote H WD H1 .†g;1 / ' = 2 : (3.1) The left-adjoint of the intersection pairing ! of the surface †g;1 defines an isomorphism between H and its dual H WD Hom.H; Z/: ' H ! H ;

h 7! !.h; /:

(In the literature, the right-adjoint of ! is also used to identify H with H . This seems, for instance, to be the case in Johnson’s and Morita’s papers.) Let L WD L.H / denote the Lie ring freely generated by H : this is a graded Lie ring whose degree is given by the lengths of brackets. Recall from [13], [65] that there is a canonical isomorphism k ' Gr D Lk D L (3.2) ! kC1 k1

k1

between the graded Lie ring associated with the lower central series of and L. Both Gr and L are generated by their degree 1 parts, and the isomorphism (3.2) is simply given in degree 1 by (3.1). Definition 3.6. For all k 1, the k-th Johnson homomorphism is the monoid map k W Cg;1 Œk ! Hom .H; kC1 =kC2 / D H ˝ kC1 = kC2 ' H ˝ LkC1 which sends an M 2 Cg;1 Œk to the map defined by fxg 7! kC1 .M /.fxg/ fxg1 for all fxg 2 = 2 . The Johnson homomorphisms are defined for homology cobordisms in [25] as a natural generalization of the homomorphisms introduced in [47], [75] for mapping class groups. Recall that @†g;1 defines an element 2 and, clearly, the automorphism kC1 .M / of = kC2 lifts to an endomorphism of = kC3 which fixes fg (namely the automorphism kC2 .M /). It can be checked that such a “boundary condition” implies that k .M / belongs to the kernel of the bracket map: DkC2 .H / WD Ker .Œ ˘; ˘ W H ˝ LkC1 .H / ! LkC2 .H // :

478


Thus, the k-th Johnson homomorphism is a monoid homomorphism k W Cg;1 Œk ! DkC2 .H / whose kernel is the submonoid Cg;1 Œk C 1. The first Johnson homomorphism 1 on the Torelli group was introduced by Johnson himself in [44] and is a key ingredient in the abelianization of the Torelli group [49] – see §6.3. Its target can be identified with the third exterior power of H by the map ' ! D3 .H /; ƒ3 H

a ^ b ^ c 7! a ˝ Œb; c C c ˝ Œa; b C b ˝ Œc; a:

(3.3)

Thus, the first Johnson homomorphism is a monoid homomorphism 1 W C g;1 ! ƒ3 H: The resulting representation of the Torelli group g;1 already appears in the background of a paper by Sullivan [92] in connection with the cohomology rings of closed oriented 3-manifolds. To understand this relation, recall from §2.3 that an x without boundary. Then, the incluM 2 C g;1 can be “closed” to a 3-manifold M x sion m˙ W †g;1 ! M M induces an isomorphism in homology so that H can be x /. identified with H1 .M Proposition 3.7 (Johnson [47]). For all M 2 C g;1 , 1 .M / coincides with the x , namely the form triple-cup product form of the closure M x / H 1 .M x / H 1 .M x / ! Z; .x; y; z/ 7! x [ y [ z; M x H 1 .M 3 1 x /; Z ' ƒ3 H1 .M x / ' ƒ3 H . which we regard as an element of Hom ƒ H .M As claimed by Johnson [47], this relationship can be generalized to the higher Johnson homomorphisms. Thus, the k-th Johnson homomorphism of an M 2 Cg;1 Œk x . This has can be computed from the length k C 1 Massey products of the closure M been proved by Kitano in the case of the mapping class group [53] and by Garoufalidis– Levine in the case of homology cobordisms [25]. Johnson proved that 1 W g;1 ! ƒ3 H is surjective (for g 2) by computing its values on some explicit elements of the Torelli group [44]. In general, the image of k W Mg;1 Œk ! DkC2 .H / is not known (except for small values of k) – see [78] for an account of this important problem. Nevertheless, we have the following result for homology cobordisms. Theorem 3.8 (Garoufalidis–Levine [25]). For every integer k 1, the monoid map k W Cg;1 Œk ! DkC2 .H / is surjective. Sketch of the proof. Let Aut .=kC1 / be the group of those automorphisms ‰ of z of = kC2 such that ‰.fg/ z = kC1 which lift to an endomorphism ‰ D fg. By using cobordism theory and surgery techniques, Garoufalidis and Levine first prove


that the monoid homomorphism k W Cg;1 ! Aut .= kC1 / is surjective ([25], Theorem 3). Similar techniques are used by Turaev in [96] to prove a realization result for nilpotent homotopy types of closed oriented 3-manifolds – see §3.3 in this connection. Next, they consider the following commutative diagram: 1

/ Cg;1 Œk k

1

/ DkC2 .H /

/ Cg;1

k

/ Aut .= kC1 /

/1

kC1

/ Aut .=kC2 /

(3.4) / Aut .= kC1 /

/ 1.

Clearly, the first row is exact. In the second row, the right-hand side map is induced by the canonical projection End.=kC2 / ! End.= kC1 /, and the surjectivity of k implies that it is surjective. The left-hand side map in the second row sends any f 2 DkC2 .H / H ˝ LkC1 ' Hom.H; kC1 = kC2 / to the automorphism ‰ of = kC2 defined by ‰.fxg/ D f .fxg/ fxg for all x 2 . It can be checked that the second row is exact ([25], Proposition 2.5) and the surjectivity of kC1 then implies that k is surjective. Habegger gives in [30] a different proof of Theorem 3.8. For this, he defines a correspondence between the set Cg;1 Œ1 D C .†g;1 / and the set C .†0;2gC1 / of framed string-links in homology 3-balls on 2g strands with trivial linking matrix. (See Example 2.2.) Although not multiplicative, this bijection makes the Johnson filtration correspond to the Milnor filtration, and the Johnson homomorphisms correspond to the Milnor invariants. Thus, Habegger calls it the “Milnor–Johnson correspondence” and the surjectivity of k then follows from the surjectivity of the Milnor invariants for string-links in homology 3-balls [32]. In the closed case, the Johnson homomorphisms are defined for the mapping class group by Morita in [76]. The definition is similar to the bordered case, but it is also more technical. Since any embedding †g;1 ! †g induces an isomorphism in homology, we use the same notation as in the bordered case: » = 2 »: H WD H1 .†g / ' The intersection pairing ! W H H ! Z defines an isomorphism H ' H and, so, is dual to a bivector ! 2 ƒ2 H . By a theorem of Labute [56], the graded Lie ring associated with the lower central series of » is canonically isomorphic to the quotient » L WD L=h!iideal , where ! is regarded as an element of L2 . Each fyg 2 k » =kC1 » induces a group homomorphism H ! kC1 » = kC2 » » = kC1 » is mapped to a subgroup defined by fxg 7! fŒy; xg. In that way, k » =kC2 » / denoted by Œk » = kC1 » ; ˘ . Besides, each element of Hom .H; kC1 2 » y 2 Lk defines by antisymmetrization of ! 2 ƒ H in H ˝ H an element ! ˝ y of »k ! L » kC1 , we get an element » k and, so, by applying the bracket H ˝ L H ˝H ˝L » kC1 . Thus, L » k is mapped to a subgroup of H ˝ L » kC1 .Id ˝Œ ˘; ˘ / .! ˝ y/ of H ˝ L

480


» k /. It is easily checked that the canonical isomorphism denoted by .Id ˝Œ ˘; ˘ / .! ˝ L » kC1 » =kC2 » / D H ˝ kC1 » = kC2 » 'H ˝L Hom .H; kC1 makes those two subgroups correspond. Definition 3.9. For all k 1, the k-th Johnson homomorphism is the monoid map k W Cg Œk !

» kC1 H ˝L Hom .H; kC1 » =kC2 »/ ' »k/ Œk » =kC1 » ; ˘ .Id ˝Œ ˘; ˘ / .! ˝ L

which sends an M 2 Cg Œk to the map defined by fxg 7! kC1 .M /.fxg/ fxg1 for ». all fxg 2 » =2 Recall that, in the closed case, kC1 .M / is an automorphism of » = kC2 » which is only defined up to some inner automorphism. This ambiguity disappears by taking » =kC1 » ; ˘ . For the same reason, the choice of the base point the quotient with Œk ? 2 †g has no incidence on the definition of k .M /. As explained in §2.4, we may think of the closed surface †g as the union of †g;1 with a disk D. We can also choose a system of meridians and parallels .˛; ˇ/ on †g;1 as shown in Figure 2. Then, for a base point ? 2 @†g;1 and some appropriate basings of those meridians and parallels, we have D @†g;1 D ˛g ; ˇg1 ˛1 ; ˇ11 2 :

ˇ1

˛1 ˇg

˛g

? Figure 2. The surface †g;1 and a system of meridians and parallels .˛; ˇ/.

Since D Œ@D is trivial in » , any representative r 2 Aut .» = kC3 » / of kC1 .M / satisfies the identity r.˛g /; r.ˇg1 / : : : r.˛1 /; r.ˇ11 / D 1 2 » = kC3 »: » kC1 actually belongs to the This implies that a representative of k .M / in H ˝ L subgroup » kC1 ! L » kC2 : » kC2 .H / WD Ker Œ ˘; ˘ W H ˝ L D


Thus, the k-th Johnson homomorphism is a monoid homomorphism » kC2 .H / D »k/ .Id ˝Œ ˘; ˘ / .! ˝ L

k W Cg Œk !

whose kernel is the submonoid Cg Œk C 1. Similarly to the bordered case, 1 was introduced by Johnson himself [44], and it plays a key role in the abelianization of g – see §6.3. Note that we have an isomorphism ' » 3 .H / ! D3 .H / ' D ƒ3 H through which the subgroup ! ^ H goes to .Id ˝Œ ˘; ˘ / .! ˝ H /. Thus, in the closed case, the first Johnson homomorphism is a monoid map ƒ3 H : !^H As in the bordered case, 1 can be interpreted in terms of the cohomology rings of x be the closure of an M 2 C g as defined closed oriented 3-manifolds. Indeed, let M in §2.3. We have a short exact sequence of abelian groups 1 W C g !

0

/H

m

/ H .M x/ 1

p

/Z

/0

(3.5)

x where m is induced by the inclusion m˙ W †g ! M M and p takes the homox /. logical intersection with m˙ .†g / 2 H2 .M Proposition 3.10. For all M 2 C g , 1 .M / is the image of the triple-cup product x under the projection form of the closure M ƒ3 s

x/ ƒ3 H1 .M

/ / ƒ3 H

/ / ƒ3 H=! ^ H

where s is a left inverse of m in (3.5). The map »˘ W Cg;1 ! Cg defined in Lemma 2.10 (when †g is decomposed into a closed disk and a copy of †g;1 ) sends the Johnson filtration to the Johnson filtration. Moreover, we have a commutative square: Cg;1 Œk »˘

k

Cg Œk

/ DkC2 .H /

k

/

»kC2 .H / D

»k / .Id ˝Œ ˘;˘ /.!˝L

(3.6) .

We deduce from Theorem 3.8 that k is surjective also in the closed case. However, computing the image of Mg Œk by k is still an unsolved and hard problem [78]. Besides, (3.6) is also useful to deduce Proposition 3.10 from Proposition 3.7.

482


3.3 Morita homomorphisms In the case of the bordered surface †g;1 , Morita introduced in [75] some refinements of the Johnson homomorphisms. From the point of view of 3-dimensional topology, Morita’s homomorphisms correspond to Turaev’s nilpotent homotopy types of closed 3-manifolds [96], which we shall now recall. Let k 1 be an integer and let G be a group of nilpotency class k (i:e: the subgroup kC1 G is trivial). Let N be a closed connected oriented 3-manifold whose k-th nilpotent quotient of the fundamental group is parameterized by the group G: ' n W G ! 1 .N /= kC1 1 .N /:

Then, Turaev defines the k-th nilpotent homotopy type of the pair .N; n/ to be k .N; n/ WD fn; .ŒN / 2 H3 .G/ n1

where fn W N ! K.G; 1/ gives 1 .N / ! 1 .N /= kC1 1 .N / ! G at the level of fundamental groups. This terminology is justified by the fact due to Thomas [94] and Swarup [93] that the oriented homotopy type of N is determined by the group 1 .N / and the image of ŒN in H3 .1 .N //. Remark 3.11. It is well known that the cobordism group SO 3 .K.G; 1// is isomorphic to H3 .G/ by the map f.L; l/g 7! l .ŒL/, for any closed connected oriented 3-manifold L and any map l W L ! K.G; 1/. Thus, an equivalent definition for the k-th nilpotent homotopy type is k .N; n/ WD f.N; fn /g 2 SO 3 .K.G; 1//. Let us now consider the closure Cx of a cobordism C 2 Cg;1 Œk defined in §2.3. Observe that the map †g;1 ! Cx , which is obtained by composing c˙ W †g;1 ! C with the inclusion C Cx , induces an isomorphism c W = kC1 ! 1 .Cx /= kC1 1 .Cx /. Definition 3.12. For all k 1, the k-th Morita homomorphism is the map Mk W Cg;1 Œk ! H3 .= kC1 / that sends a C 2 Cg;1 Œk to k Cx ; c . This map is studied by Heap in [38] at the mapping class group level, and by Sakasai in [89] for homology cobordisms. It also appears implicitly in [25]. By considering the simplicial model of K .=kC1 ; 1/, Heap shows that Mk is equivalent to Morita’s refinement of the k-th Johnson homomorphism [75]. Morita’s original definition is more algebraic: for any f 2 Mg;1 Œk, the homology class Mk .c .f // is defined in [75] from the Dehn–Nielsen representation .f / 2 Aut./ using the bar resolution of the group . Similarly, one can define Mk .C / from 2k .C / 2 Aut.= 2kC1 / using the fact (proved by Igusa and Orr [41]) that the canonical homomorphism H3 .= 2kC1 / ! H3 .=kC1 / is trivial.


However, it is not clear from the above definition that Mk is a monoid homomorphism. The additivity of Mk is a consequence of the following “variation formula” for the k-th nilpotent homotopy type. Proposition 3.13. Let N be a closed connected oriented 3-manifold. Given a cobordism C 2 Cg;1 Œk and an embedding j W †g;1 ! N , consider the 3-manifold N 0 obtained by “cutting” N along j.†g;1 / and by “inserting” C . Then, any isomorphism n W G ! 1 .N /=kC1 1 .N / induces, in a canonical way, an isomorphism n0 W G ! 1 .N 0 /=kC1 1 .N 0 / such that k .N 0 ; n0 / k .N; n/ D n1 j .Mk .C // 2 H3 .G/:

(3.7)

Similar formulas are shown in [25], Theorem 2, and [38], Theorem 5.2, by cobordism arguments – see Remark 3.11. Proof of Proposition 3.13. Denote by J the surface j.†g;1 / in N . The 3-manifold N 0 is defined by N 0 WD .N n int .J Œ1; 1// [j 0 Bc 1 C where J Œ1; 1 denotes a closed regular neighborhood of J in N and j 0 is the restriction to the boundary of the homeomorphism j Id W †g;1 Œ1; 1 ! J Œ1; 1. The van Kampen theorem shows the existence of a canonical isomorphism between 1 .N /= kC1 1 .N / and 1 .N 0 /= kC1 1 .N 0 / which is defined by the following commutative diagram: 1 .N nint.J Œ1;1// kC1 1 .N nint.J Œ1;1//

1 .N / kC1 1 .N /

QQQ n QQQ nnn n QQQ n nn QQQ( n n ( vnv n 9Š _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _/ '

1 .N 0 / : kC1 1 .N 0 /

By composing it with n, we obtain an isomorphism n0 W G ! 1 .N 0 /= kC1 1 .N 0 /. Let us now prove Formula (3.7). The closed 3-manifolds N , N 0 and Cx can be seen all together inside the following singular 3-manifold: Nz WD N n int .J Œ1; 1/ [j 0 tj 0 Bc 1 †g;1 Œ1; 1 C : Denote the corresponding inclusions by i W N ! Nz , i 0 W N 0 ! Nz and ` W Cx ! Nz . The homomorphism i W 1 .N /=kC1 1 .N / ! 1 .Nz /= kC1 1 .Nz / is an isomorphism, and we denote by nQ its pre-composition with n. Let fQ W Nz ! K.G; 1/ be 1

n Q a map which induces the composition 1 .Nz / ! 1 .Nz /= kC1 1 .Nz / ! G at the level of fundamental groups. The restrictions of fQ to N and N 0 are denoted by f and f 0 respectively, and they induce n1 and .n0 /1 at the level of fundamental groups. So, we have

k .N 0 ; n0 / k .N; n/ D f0 .ŒN 0 / f .ŒN / D fQ i 0 .ŒN 0 / i .ŒN / D fQ ` Cx :

484


But we have the following commutative square in the category of groups:

1 Cx

//

z/ 1 .N z/ kC1 1 .N

u

u

u

u

kC1

/

'

` `

1 c

x/ 1 .C x/ kC1 1 .C

u: :

o_ _ _'_ _ _ Q Q i Q QnQ 1 Q Q '

1 Nz

fQ

1 .K.= kC1 ; 1//

j

1 .N / kC1 1 .N / ' n1

Q Q (/ G

1 .K.G; 1//.

(The dashed arrows are shown here to check the commutativity.) Choose some maps a W Cx ! K.=kC1 ; 1/ and b W K.=kC1 ; 1/ ! K.G; 1/ that induce the top homomorphism and the right-hand side homomorphism, respectively, at the level of fundamental groups. Thus, b B a is homotopic to fQ B ` so that x : k .N 0 ; n0 / k .N; n/ D .fQ B `/ Cx D .b B a/ Cx D n1 j k C ; c We conclude from the definition of Mk . The k-th Morita homomorphism is a refinement of the k-th Johnson homomorphism in the following sense. Consider the (central) extension of groups 0

/ LkC1

/ = kC2

/ = kC1

/1

and the corresponding Hochschild–Serre spectral sequence: 2 D Hp =kC1 I Hq .LkC1 / H) HpCq .= kC2 / : Ep;q 2 2 2 2 W E3;0 ! E1;1 , where E3;0 ' H3 .= kC1 / We need the differential d 2 WD d3;0 2 and E1;1 ' H ˝ LkC1 . Morita uses this spectral sequence in [75] to prove that the image of d 2 is precisely the subgroup DkC2 .H /.

Theorem 3.14 (Morita [75]). For all integers k 1, we have the following commutative triangle: Cg;1 Œk

Mk

OOO OOO O k OOOO '

/ H3 .= kC1 /

d2

DkC2 .H /:

This result is stated and proved in [75] for mapping class groups, but the proof can be adapted to homology cobordisms. From a 3-dimensional viewpoint, this statement translates the fact that the k-th nilpotent homotopy type of a closed connected oriented 3-manifold determines its Massey products of length k C 1. See [38] for a topological proof of Theorem 3.14. Similar arguments appear in [25].


Example 3.15 (Degree 1). The Pontrjagin product for the homology of the abelian group H defines an isomorphism: ' ƒ3 H ! H3 .H / ' H3 .= 2 /:

Through that isomorphism, the differential d 2 W H3 .= 2 / ! D3 .H / coincides with the isomorphism (3.3). Thus, M1 is equivalent to 1 . For all integers l k 1, the following square is commutative: Cg;1 Œl Cg;1 Œk

Ml

Mk

/ H3 .= lC1 / / H3 .= kC1 / .

The right-hand side map is induced by the canonical map = lC1 ! = kC1 and, as already mentioned, Igusa and Orr proved it to be trivial for l 2k [41]. So, Mk vanishes on Cg;1 Œ2k. In fact, the computation of H3 .= kC1 / performed in [41] and the surjectivity of the Johnson homomorphisms (Theorem 3.8) can be used to prove the following stronger statement. Theorem 3.16 (Heap [38], Sakasai [89]). For all integers k 1, we have the following short exact sequence of monoids: 1

/ Cg;1 Œ2k

/ Cg;1 Œk

Mk

/ H3 .= kC1 /

/ 1:

More generally, two cobordisms C; C 0 2 Cg;1 Œk satisfy Mk .C / D Mk .C 0 / if and only if we have 2k .C / D 2k .C 0 /. (This can be deduced from Theorem 3.16 by considering group quotients of the monoid C : see §5.2 and §7.1.)

3.4 Infinitesimal versions of the Dehn–Nielsen representation We now give “infinitesimal” versions of the Dehn–Nielsen representation, which are defined on the monoid of homology cobordisms and contain all the Johnson homomorphisms. Here, the word “infinitesimal” means that we are going to replace fundamental groups by their Malcev Lie algebras. The passage from groups to Lie algebras will be useful in §4.3 to connect the Dehn–Nielsen representation to finite-type invariants. Let G be a group and let QŒG be the group algebra of G, with augmentation ideal I . The I -adic completion of QŒG

1

QŒG WD lim QŒG=I k k

1

is a complete Hopf algebra in the sense of [88]. If G is residually torsion-free nilpotent (which will always be the case in our situation), then the canonical map G ! QŒG is

486


1

injective, so that we can write G QŒG. Following Quillen [87] and Jennings [42], we define the Malcev Lie algebra of G as the primitive part of QŒG: m.G/ WD Prim QŒG :

1

1

This is a filtered Lie algebra into which G embeds by the logarithmic series: .1/nC1 .x 1/n : log W G ! m.G/; x 7! n n1

By a theorem of Quillen [88], this map induces an isomorphism ' .Gr log/ ˝ Q W Gr G ˝ Q ! Gr m.G/

between the graded Lie algebras associated with the lower central series of G and with the filtration of m.G/ respectively. This way of defining the Malcev Lie algebra is clearly functorial and, if G is residually torsion-free nilpotent, then the map m W Aut.G/ ! Aut.m.G// is injective. We now consider the case of the free group D 1 .†g;1 ; ?/. We abbreviate HQ WD H ˝ Q D .=2 / ˝ Q, and T.HQ / denotes the tensor algebra over HQ . Following our convention, the same notation is used for the degree completion of the tensor algebra. Definition 3.17. An expansion of the free group is a map W ! T.HQ / which is multiplicative and satisfies 8x 2 ;

.x/ D 1 C Œx C .deg 2/ 2 T.HQ /:

The expansion is group-like if it takes group-like values. Expansions of free groups have been considered by Lin [64] and Kawazumi [50] in connection with Milnor’s invariants and Johnson’s homomorphisms, respectively. An expansion of extends in a unique way to a filtered algebra isomorphism W QŒ ! T.HQ /, which induces the canonical isomorphism

1

1

Gr QŒ D

Ik ' ! T.HQ /; I kC1

I 3 fx 1g 7! Œx 2 HQ I2

k0

(3.8)

1

at the graded level [13], [65]. Group-like expansions correspond to Hopf algebra isomorphisms W QŒ ! T.HQ / which induce the canonical isomorphism (3.8) at the graded level. Equivalently, by restricting to primitive elements, group-like expansions of correspond to filtered Lie algebra isomorphisms W m./ ! LQ which induce the canonical isomorphism Gr m./

..Gr log/˝Q/1

(3.2)

'

'

/ Gr ˝ Q

/ LQ

at the graded level. Here LQ WD L ˝ Q denotes the Lie algebra freely generated by HQ or, depending on the context, its degree completion.


Example 3.18. If a basis of is specified – for instance, the basis .˛; ˇ/ defined by a system of meridians and parallels as in Figure 2 – then there is a unique expansion .˛;ˇ / of defined by 8i 2 f1; : : : ; gg;

.˛;ˇ / .˛i / WD exp˝ .Œ˛i / and .˛;ˇ / .ˇi / WD exp˝ .Œˇi / : (3.9)

By construction, the expansion .˛;ˇ / is group-like. Let be a group-like expansion of . The following composition can be regarded as an infinitesimal version of the Dehn–Nielsen representation, which depends on : Mg;1 /)

1

B˘B / Aut./ / m / Aut .m.// / Aut.LQ /: ' 4 U W i g Y Z \ e % d ] _ a b

(3.10)

We denote it by % . Since any automorphism of the complete free Lie algebra LQ is determined by its restriction to HQ , we can replace % by the map W Mg;1 ! Hom.HQ ; LQ /; f 7! % .f /jHQ IdHQ D B m.f / B 1 jHQ IdHQ without loss of information. This is essentially the map introduced in [50] under the name of the total Johnson map. Indeed, it can be checked from its definition that contains all the Johnson homomorphisms. Theorem 3.19 (Kawazumi [50]). The degree k part of the total Johnson map, restricted to the k-th term of the Johnson filtration, coincides with the k-th Johnson homomorphism (with rational coefficients): k D k W Mg;1 Œk ! Hom.HQ ; LQ;kC1 / ' HQ ˝ LQ;kC1 : Using Fox’s free differential calculus, Perron defines another map on Mg;1 which contains all the Johnson homomorphisms [85]. The extension of % to the monoid of homology cobordisms is straightforward. Indeed, for each integer k 1, a group-like expansion induces a filtered Lie algebra isomorphism W m.=kC1 / ! LQ =LQ;kC1 . So, we can consider the monoid homomorphism %k defined by the composition

m / B˘B 1/ Cg;1 V k / Aut.=kC1 / / .m.= Aut // Aut L =L Q kC1 Q;kC1 ' W W X 3 Y Y Z [ e f g [ \ \ ] ^ %^k _ ` ` a b b c d d e or, equivalently, we can consider the map W Cg;1 ! Hom.HQ ; LQ =LQ;kC1 /; k

f 7! %k .f /jHQ IdHQ :

Passing to the limit k ! C1, we get a map W Cg;1 ! Hom.HQ ; LQ / ' HQ ˝ LQ

488


whose restriction to Mg;1 is compatible with the preceding definition. Theorem 3.19 works with Mg;1 Œk replaced by Cg;1 Œk. Definition 3.20. The infinitesimal Dehn–Nielsen representation (induced by the grouplike expansion of ) is the monoid homomorphism % W Cg;1 ! Aut.LQ / which sends a homology cobordism C to the unique filtered automorphism of LQ whose restriction to HQ is IdHQ C .C /. The following statement can be proved from the definitions. Proposition 3.21 (See [68]). The degree Œk; 2kŒ truncation of the total Johnson map , restricted to the k-th term of the Johnson filtration, WD Œk;2kŒ

2k1

m W Cg;1 Œk !

mDk

2k1

HQ ˝ LQ;mC1

mDk

is a monoid homomorphism, and its kernel is Cg;1 Œ2k. Among group-like expansions, we prefer those which have the following property. Recall that ! 2 ƒ2 H H ˝ H is the dual of the intersection pairing. Definition 3.22. An expansion W ! T.HQ / is symplectic if it is group-like and if it sends to exp˝ .!/. It is not difficult, using the Baker–Campbell–Hausdorff formula and starting from the expansion (3.9), to construct degree-by-degree a symplectic expansion [68]. If is symplectic, the infinitesimal Dehn–Nielsen representation % has values in Aut ! .LQ / WD fa 2 Aut .LQ / W a.!/ D !g and it can be checked that Œk;2kŒ then takes values in the kernel of the bracket map:

D.HQ / WD D.H / ˝ Q D Ker .Œ ˘; ˘ W HQ ˝ LQ ! LQ / : Theorem 3.23 (See [68]). If is a symplectic expansion of , then there is a commutative diagram of the following form: H .= kC1 / /

/ H3 .= kC1 I Q/ XXXXX' XXXX+

3 jjj4 jjjj Cg;1 Œk OOO OO' Mk

Œk;2kŒ

2k1 mDk

H3 .m.= kC1 /I Q/ . jj4

DmC2 .HQ /

'

/ H3

' jjjj jjjj1 j

LQ IQ LQ;kC1


Thus, the degree Œk; 2kŒ truncation of the total Johnson map is equivalent to the k-th Morita homomorphism. The top map is injective because H3 .= kC1 / is torsion-free [41]. The righthand side top map is an application of the following result by Pickel: with rational coefficients, the homology of a torsion-free finitely-generated nilpotent group is isomorphic to the homology of its Malcev Lie algebra [86]. The bottom map is defined using diagrammatic descriptions of the spaces D.HQ / and H3 .LQ =LQ;kC1 I Q/. (See §4.3 for the diagrammatic description of D.HQ /.) The diagram is shown to be commutative using an “infinitesimal” version of Mk . We now sketch how to use symplectic expansions to deal with the case of a closed surface. If we fix a closed disk D †g (as we did in §2.4), then we have a decomposition †g D †g;1 [ D so that » D 1 .†g ; ?/ ' =hi: Thus, a symplectic expansion induces a filtered Lie algebra isomorphism ' » » W m.» / ! LQ

» Q WD LQ =h!iideal . between the Malcev Lie algebra of » and the complete Lie algebra L Then, the following composition gives an infinitesimal version of the Dehn–Nielsen representation: 1 m / » B˘B » / Out .» / Out L »Q : Mg / ) T / / Out .m .» // ' 4 U W i Y Z \ e g ]% _ a b d

(3.11)

We denote it by % . Here, for a Lie algebra g equipped with a complete filtration, Out.g/ denotes the group of filtered automorphisms of g modulo inner automorphisms, i:e: exponentials of inner derivations of g. Equivalently, we can consider the total Johnson map defined by »Q H ; L Hom Q ; W Mg ! » Q /.˘/ .˘/ expB .ad L

f 7! » B m.f / B »1 jHQ IdHQ

» Q / is quotiented by the subspace of homomorphisms of where the space Hom.HQ ; L » Q. the form .h 7! expB .ad u/.h/ h/ where u 2 L As in the bordered case, we can extend the map and the homomorphism % » Q =L » Q;kC1 and by to the monoid Cg by considering, first, the nilpotent quotients L passing, next, to the limit k ! C1. There are analogues of Theorem 3.19 and

490


Proposition 3.21 for the monoid Cg . Clearly, the following square is commutative: Cg;1 »˘

%

Cg

/ Aut ! .LQ /

%

(3.12)

/ Out L »Q .

4 The LMO homomorphism We present the LMO homomorphism, which is a diagrammatic representation of the monoid of homology cylinders. It is derived from the Le–Murakami–Ohtsuki invariant of closed oriented 3-manifolds. The LMO homomorphism dominates all the Johnson/Morita homomorphisms, and it will play a key role in Section 6. In this section, we restrict ourselves to the surfaces †g;1 and †g .

4.1 The algebra of symplectic Jacobi diagrams We start by defining the target of the LMO homomorphism. For this, we need to define some kind of Feynman diagrams which appear in the theory of finite-type invariants [7], [84]. A Jacobi diagram is a finite graph whose vertices have valence 1 (external vertices) or 3 (internal vertices). Each internal vertex is oriented, in the sense that its incident half-edges are cyclically ordered. A Jacobi diagram is colored by a set S if a map from the set of its external vertices to S is specified. A strut is a Jacobi diagram with only two external vertices and no internal vertex. Examples of Jacobi diagrams are shown in Figure 3: the custom is to draw such diagrams with dashed lines and the vertex orientations are given by the counter-clockwise orientation.

Figure 3. Some examples of Jacobi diagrams: the strut, the Y graph, the H graph, the Phi graph and the Theta graph.


As in the previous section, we start with the bordered case. We denote H WD H1 .†g;1 / and HQ WD H ˝ Q. We define the Q-vector space Jacobi diagrams without strut component Q and with external vertices colored by HQ A.HQ / WD : AS, IHX, multilinearity

The “AS” and “IHX” relations are diagrammatic analogues of the antisymmetry and Jacobi identities in Lie algebras:

D AS

C

D0

IHX

The “multilinearity” relation simply states that a Jacobi diagram D with one external vertex v colored by q1 h1 C q2 h2 (with q1 ; q2 2 Q and h1 ; h2 2 HQ ) is equivalent to the linear combination q1 D1 C q2 D2 where Di is the Jacobi diagram D with the vertex v colored by hi . The degree of a Jacobi diagram is the number of its internal vertices. Thus, A.HQ / is a graded vector space A.HQ / D

1

Ad .HQ /

d D0

where A0 .HQ / ' Q is spanned by the empty diagram ¿. Following our convention, the degree completion of A.HQ / is denoted in the same way. Example 4.1 (Degree 1). The spaces A1 .HQ / and ƒ3 HQ are isomorphic by x2

x3 7! x1 ^ x2 ^ x3 :

x1 As in the previous section, ! W HQ ˝ HQ ! Q denotes the intersection pairing of †g;1 . The group of the automorphisms of HQ that preserve !, namely the symplectic group of .HQ ; !/, is denoted by Sp.HQ /. It acts on A.HQ / in the obvious way. We shall now define on this space an Sp.HQ /-equivariant structure of Hopf algebra. If we think of HQ -colored Jacobi diagrams as a kind of tensors, the multiplication ? in A.HQ / is defined as the contraction by the pairing !=2. In more detail, let D and E be HQ -colored Jacobi diagrams, whose sets of external vertices are denoted by V and W respectively. Then, we define 1 ! color v ; color ˇ.v/ .D [ˇ E/: D ? E WD 0 2jV j 0 0 0 V V; W W '

ˇ W V 0 !W 0

v2V

492


Here, the sum is taken over all ways of identifying a subset V 0 of V with a subset W 0 of W , and D [ˇ E is obtained from D t E by gluing each vertex v 2 V 0 to ˇ.v/ 2 W 0 . The comultiplication is given on an HQ -colored Jacobi diagram D by .D/ WD D 0 ˝ D 00 ; DDD 0 tD 00

where the sum runs over all ways of dividing the connected components of D into two parts. Thus, the primitive part of A.HQ / is the subspace Ac .HQ / spanned by connected Jacobi diagrams. The counit is given by ".D/ WD ıD;; , and the antipode is the unique algebra anti-automorphism satisfying S.D/ D D if D is connected and non-empty. Definition 4.2. The Hopf algebra of symplectic Jacobi diagrams is the graded vector space A.HQ / equipped with the multiplication ? (with unit ¿), the comultiplication (with counit ") and the antipode S. The Lie algebra of symplectic Jacobi diagrams is Ac .HQ / equipped with the Lie bracket Œ ˘; ˘ ? . This Hopf algebra is introduced in [36]. Observe that all its operations respect the action of Sp.HQ /. In particular, the Lie bracket Œ ˘; ˘ ? of Ac .HQ / is Sp.HQ /equivariant, and this will play an important role in Section 6. Another Lie bracket is defined in [25], but this one is not Sp.HQ /-equivariant. We now define the algebra which will serve as a target for the LMO homomorphism in the closed case. Let I be the subspace of A.HQ / spanned by elements of the following form:

D

D

!

x1

1 !.xj ; xk / 4 1j 2/ 2 A.¿/: 2 The reader is referred to Ohtsuki’s book [84] for an introduction to this invariant. The LMO invariant can be extended to 3-manifolds with boundary in several ways [81], [17], [3]. Here, we need the LMO functor introduced in [16]. Its source is the category of “Lagrangian cobordisms” whose objects are integers g 0 and whose morphisms g ! h are cobordisms (with corners) between †g;1 and †h;1 , which are required to satisfy certain homological conditions. The target of the LMO functor is a certain category of Jacobi diagrams. We refer to [16] for the construction. It should be emphasized that the definition of the LMO functor requires two preliminary choices: (1) A Drinfeld associator must be specified. (Just like the LMO invariant, the LMO functor is constructed from the Kontsevich integral of tangles.) (2) For each g 0, a system of meridians and parallels should be fixed on †g;1 (as in Figure 2). Homology cylinders are “Lagrangian” in the sense of [16]. Thus, the LMO functor restricts to a monoid homomorphism z Y W C g;1 ! AY bgeC [ bge : Z

.M / D ¿ C .1/ˇ1 .M /

Here, the target space is the space of Jacobi diagrams without strut component and whose external vertices are colored by the finite set bgeC [ bge WD f1C ; : : : ; g C g [ f1 ; : : : ; g g: The multiplication in this space is defined by sum of all ways of gluing some of the i C -colored vertices of D D ? E WD : to some of the i -colored vertices of E, for all i D 1; : : : ; g

494


This product was discovered in [25]. (See Section 5.) Note that, in the definition of the LMO functor, the colors 1C ; : : : ; g C refer to the curves ˇ1 ; : : : ; ˇg (in the top surface of the cobordism) while the colors 1 ; : : : ; g refer to the curves ˛1 ; : : : ; ˛g (in the bottom surface of the cobordism). So, it is natural to identify AY bgeC [ bge with A.HQ / by simply changing the colors with the rules j C 7! Œˇj and j 7! Œ˛j . Unfortunately, the multiplication on A.HQ / corresponding to the multiplication ? on AY bgeC [ bge by that identification is not Sp.HQ /-equivariant. Instead of that identification, we consider the map W AY bgeC [ bge ! A.HQ / defined by .D/ WD .1/

.D/

sum of all ways of .1=2/-gluing some i -colored j C 7! ˇj vertices of D with some of its i C -colored vertices j 7! ˛j

:

Here, .D/ is the Euler characteristic of a Jacobi diagram D, and a “.1=2/-gluing” means the gluing of two vertices and the multiplication of the resulting diagram by 1=2. Itcan be provedthat is an isomorphism and that it sends the multiplication ? of AY bgeC [ bge to the multiplication ? of A.HQ /. Definition 4.4. The LMO homomorphism is the monoid map z Y W C g;1 ! A.HQ /: Z WD B Z This invariant of homology cylinders is universal among Q-valued finite-type invariants [16], [36] (see Section 6 in this connection), and it takes group-like values. Habegger defines from the LMO invariant another map C g;1 ! A.HQ / with the same properties [30], but he does not address the multiplicativity issue. We now consider the case of a closed surface. We think of †g as the union of †g;1 with a closed disk D as we did in §2.4. It can be shown from Lemma 2.10 that, if the target of the LMO functor is quotiented by some appropriate relations, then it induces a functor on the category of cobordisms between closed surfaces. Those relations in AY bgeC [ bge are sent by the isomorphism to the Hopf ideal I introduced at the end of §4.1. Definition 4.5. The LMO homomorphism is the monoid map » Q /: z Y W C g ! A.H Z WD B Z So, by construction, the following square is commutative: C g;1

Z

»˘

C g

Z

/ A.HQ / / A.H » Q /:

(4.2)


» Q / depends on the Of course, the invariant of homology cylinders Z W C g ! A.H way †g;1 is embedded on †g (as well as on the choices of a Drinfeld associator and a system of meridians and parallels on †g;1 ). Example 4.6 (Genus 0). We saw in Example 2.2 that the monoids C 0;1 and C 0 can » Q/ D be identified with the monoid of homology 3-spheres. We have A.HQ / D A.H A.¿/ for g D 0 and, by construction, the LMO homomorphism Z coincides in this case with the LMO invariant .

4.3 The tree-reduction of the LMO homomorphism A Jacobi diagram is looped if at least one of its connected components is not contractible. The subspace of A.HQ / generated by looped Jacobi diagrams is an ideal, so that we can consider the quotient Hopf algebra At .HQ / WD A.HQ /=hlooped Jacobi diagramsi: As a Q-vector space, At .HQ / can be identified with the subspace of A.HQ / generated by tree-shaped Jacobi diagrams. Thus, the composition / GLike At .HQ / C g;1 U Z / GLike A.HQ / 4 V X Z \ ] _ a b d f h Zt

is called the tree-reduction of the LMO homomorphism and is denoted by Z t . It takes values in the group-like part of the Hopf algebra At .HQ /. Besides, provided we are given a symplectic expansion of , we have the infinitesimal Dehn–Nielsen representation defined in §3.4: % W C g;1 ! IAut ! .LQ /: Here, we have restricted % to the monoid of homology cylinders, so that values are taken in the group IAut! .LQ / of automorphisms of the complete free Lie algebra LQ that induce the identity at the graded level and fix !. The logarithmic series defines a bijection ' logB W IAut! .LQ / ! Der ! .LQ ; LQ;2 / ;

a 7!

.1/nC1 n1

n

.a Id/n

between IAut! .LQ / and the Lie algebra Der ! .LQ ; LQ;2 / of derivations of LQ that vanish on ! and take values in LQ;2 . This Lie algebra appears in Kontsevich’s work [54], [55], where it is implicitly identified with the Lie algebra At;c .HQ / WD Ac .HQ /=hconnected looped Jacobi diagramsi:

496


To recall that identification, let us observe that a derivation of LQ is determined by its restriction to HQ . Hence we have an isomorphism Der .LQ ; LQ;2 / ' Hom.HQ ; LQ;2 / ' HQ ˝ LQ;2 which restricts to an isomorphism Der ! .LQ ; LQ;2 / ' D3 .HQ / D Ker .Œ ˘; ˘ W HQ ˝ LQ;2 ! LQ;3 / : The corresponding Lie algebra structure of D.HQ / appears (with integral coefficients) in Morita’s work on the Johnson homomorphisms [74], [75]. It can be proved [61] that the map .H /! D .H /; T 7 ! color.v/ ˝ comm.Tv / k W At;c Q Q kC2 k v

is an isomorphism for all k 1. Here, the sum is over all external vertices v of T and comm.Tv / is the iterated Lie bracket encoded by T “rooted” at v: we have, for instance,

h1

h2 h3 h4

D Œh1 ; ŒŒh2 ; h3 ; h4 :

comm v

It is easily checked that the isomorphism ' W At;c .HQ / ! D3 .HQ / ' Der ! .LQ ; LQ;2 /

is a Lie algebra map (which shifts the degree by 2). We can now state an algebraico-topological description of the tree-reduction of the LMO homomorphism. Theorem 4.7 (See [68]). The LMO functor defines a symplectic expansion of such that the following diagram is commutative: GLike At .HQ / j4 j j j jjjj TTTT TTTT * % IAut! .LQ /

log? '

Zt

C g;1

/ At;c .HQ / '

' logB

/ Der ! .LQ ; LQ;2 / :

Thus, the tree-reduction of the LMO homomorphism encodes the action of C g;1 on the Malcev Lie algebra m./. Since the infinitesimal Dehn–Nielsen representation (3.10) is injective on the mapping class group, we deduce the following.


Corollary 4.8 (See [16]). The LMO homomorphism is injective on g;1 . Theorem 4.7 is inspired by the work of Habegger and Masbaum [33] on the Kontsevich integral of string-links in D 2 Œ1; 1. With the following correspondence in mind, one sees that Theorem 4.7 is very close in spirit to their “global formula for Milnor’s invariants”:

monoid of string-links pure braid group Kontsevich integral Milnor’s invariants

monoid of homology cylinders Torelli group LMO homomorphism total Johnson map

We can also deduce the following from Theorem 4.7 and Theorem 3.19. A similar result was proved by Habegger in [30] for his LMO-type map C g;1 ! A.HQ /. Corollary 4.9 (See [16]). Let C 2 C g;1 . The lowest degree non-trivial term of Z t .C / 2 At .HQ / coincides, via the isomorphism , with the first non-trivial Johnson homomorphism of C . More generally, it follows from Theorem 4.7 and Theorem 3.23 that the restriction to Cg;1 Œk of the degree Œk; 2kŒ truncation of Z t corresponds (by an explicit isomorphism) to the k-th Morita homomorphism [68]. We now outline the case of a closed surface. Let I t and I t;c be the images of the ideals I and I c (introduced at the end of §4.1) in the Hopf algebra At .HQ / and in the Lie algebra At;c .HQ /, respectively. Thus, if we think of At;c .HQ / as the subspace of A.HQ / generated by connected tree-shaped Jacobi diagrams, then the subspace I t;c is generated by those diagrams having an !-colored vertex, as in (4.1). We denote »t .HQ / WD At .HQ /=I t A

and

»t;c .HQ / WD At;c .HQ /=I t;c : A

The isomorphism sends I t;c to the ideal Der.LQ ; h!iideal / C .inner derivations of LQ / \ Der ! .LQ ; LQ;2 /: » Q; L » Q;2 »t;c .HQ / to the Lie algebra ODer L So, induces an isomorphism from A »Q » Q with values in L » Q;2 , modulo inner derivations. Let IOut L of derivations of L » Q that induce the identity at the graded be the group of filtered automorphisms of L level, modulo inner automorphisms. The following is an application of Theorem 4.7 using the commutative squares (4.2) and (3.12).

498


Theorem 4.10. Let be the symplectic expansion of defined from the LMO functor in [68]. Then, the following diagram is commutative:

C g

t » .HQ / GLike A Z tjjjj5 j j j j TTTT TTTT T) % ' »Q IOut L log

B

log? '

/A »t;c .HQ / '

/ ODer L » Q; L » Q;2 .

Since the infinitesimal Dehn–Nielsen representation (3.11) is injective on the mapping class group, we have the following application of Theorem 4.10. Corollary 4.11. The LMO homomorphism is injective on g . One can also deduce from Theorem 4.10 the analogue of Corollary 4.9 for †g .

5 The Y -filtration on the monoid of homology cylinders In this section, we consider the relation of Yk -equivalence among homology cylinders, which is defined for every k 1 by surgery techniques. The Y -filtration is the decreasing sequence of submonoids of C .†g;b / obtained by considering, for all k 1, homology cylinders that are Yk -equivalent to †g;b Œ1; 1. In some respects, this filtration is similar to the lower central series of the Torelli group .†g;b /.

5.1 The Yk -equivalence relation The lower central series is a fundamental tool in the study of groups. Much of the structure of a residually nilpotent group is contained in the associated graded Lie ring of the lower central series. The family of Yk -equivalence relations of 3-manifolds (k 1) plays the same role in 3-dimensional topology as lower central series. Those equivalence relations, which we shall now recall, have been defined and studied by Goussarov [28], [29] and the first author [35]. We follow the terminology of [35]. Let M be a compact oriented 3-manifold. A graph clasper in M is a compact, connected surface C embedded in the interior of M , which is equipped with a decomposition into the following types of subsurfaces: leaves, nodes and edges. A leaf is an annulus, a node is a disc, and an edge is a band. Leaves and nodes are called constituents. Each edge connects either two distinct constituents or connects a node with itself. Any two distinct constituents are disjoint from each other. Each leaf is incident with exactly one edge by one arc in the boundary. Each node is either incident with three distinct edges or incident with two distinct edges, one of which connects the


node with itself. (Thus, in each case, the node is attached to the edges along exactly three arcs in the boundary circle.) See Figure 4 for an example of a graph clasper. a node

a leaf D an edge Figure 4. An example of graph clasper C M with 3 nodes, 3 leaves and 6 edges. (And how it is drawn with the blackboard framing convention.)

A tree clasper is a graph clasper C such that C n (the leaves of C ) is simplyconnected. The degree of a graph clasper C is defined to be the number of nodes contained in C . A graph (respectively a tree) clasper of degree k is called a Yk graph (respectively a Yk -tree). For instance, Y0 -graphs (which are also called “basic claspers” or “I-claspers” in the literature) consist of only one edge and two leaves:

Surgery along a graph clasper C M is defined as follows. We first replace each node with three leaves linking like the Borromean rings in the following way:

!

Thus, we obtain a disjoint union of Y0 -trees. Next, we replace each Y0 -tree with a 2-component framed link as follows: !

See Figure 5 for an example. Then, surgery along the graph clasper C is defined to be the surgery along the framed link thus obtained in M . The resulting 3-manifold is denoted by MC . Calculus of claspers is developed in [29], [35], [22], in the sense that some specific moves between graph claspers are shown to produce by surgery homeomorphic

500


!

!

Figure 5. The framed link associated with a Y1 -tree. (Blackboard framing convention is used here.)

3-manifolds. In fact, one can regard a node in a graph clasper as a commutator morphism for a Hopf algebra object in the category Cob of 3-dimensional cobordisms introduced in [18], [51]. A more general definition of claspers involves another kind of constituents called “boxes” which corresponds to multiplication and comultiplication for that Hopf algebra object. In this way, the calculus of claspers can be regarded as a topological commutator calculus in 3-manifolds. We refer to [35] for precise statements. Example 5.1. One of the simplest moves in the calculus of claspers consists in cutting an edge of a graph clasper and inserting a Hopf link of two leaves: edge

!

(5.1)

This (called “Move 2” in [35]) can be deduced from the “slam dunk” move (2.5). A Yk -surgery on a compact oriented 3-manifold M is defined to be the surgery along some Yk -graph in M . The Yk -equivalence is the equivalence relation generated by Yk -surgeries and orientation-preserving homeomorphisms preserving the boundary parameterization. Considering 3-manifolds up to Yk -equivalence is something like considering elements in a group up to multiplication by iterated commutators of class k. Example 5.2 (Degree 1). A Y1 -surgery is equivalent to the “Borromean surgery” introduced by Matveev in [70]. It is proved there that two closed connected oriented 3-manifolds are Borromean surgery equivalent, or Y1 -equivalent, if and only if there is an isomorphism of first homology groups for these two 3-manifolds which induces an isomorphism of torsion linking pairings. The Y1 -surgeries are used in the definition of finite-type invariants of 3-manifolds in the sense of Goussarov and the first author [28], [35]. The calculus of claspers can be used to prove general properties for the Yk -surgery and the Yk -equivalence. For instance, we deduce from (5.1) that surgeries along Yk -trees suffice to generate the Yk -equivalence. Let us give two more examples. Proposition 5.3. If 1 k l, then Yl -equivalence implies Yk -equivalence.


This fact follows from “Move 9” in [35]. Thus, the Yk -equivalence gets finer and finer as k increases. It is conjectured that two 3-manifolds are orientation-preserving homeomorphic if and only if they are Y1 -equivalent (i:e: Yk -equivalent for all k 1). As mentioned in Example 5.2, a Y1 -surgery preserves the homology (hence so do Yk -surgeries for k 1). More generally, we have the following important property. Proposition 5.4. If two 3-manifolds M and M 0 are related by a Yk -surgery, then there is a canonical isomorphism 1 .M /=kC1 1 .M / ' 1 .M 0 /= kC1 1 .M 0 / between the k-th nilpotent quotient of 1 .M / and that of 1 .M 0 /. This is another illustration of the fact that a Yk -graph can be interpreted as an iterated commutator of length k C 1.

5.2 Yk -equivalence and homology cylinders Let us now specialize the Yk -equivalence relations to the class of homology cylinders. The following statement gives a characterization of homology cylinders in terms of Y1 -equivalence. Proposition 5.5 (See [35]). A cobordism M of the surface †g;b is Y1 -equivalent to †g;b Œ1; 1 if and only if M is a homology cylinder. This surgery characterization of C .†g;b / is proved in [30], [69], and it is deduced in [67] from a result by Matveev [70]. (See Example 5.2 in this connection.) It can be used to prove results about homology cylinders by considering, first, the case of the trivial cylinder †g;b Œ1; 1 and by studying, next, how things change under Y1 -surgery. For instance, Proposition 3.7 and Proposition 3.10 can be proved in that way. For all k 1, we define a submonoid of C .†g;b / by Yk C .†g;b / WD M W M is Yk -equivalent to †g;b Œ1; 1 : Thus, by Proposition 5.3, we get a decreasing sequence of monoids C .†g;b / D Y1 C .†g;b / Y2 C .†g;b / Y3 C .†g;b / which is called the Y -filtration. For b D 0 and b D 1, it is finer than the Johnson filtration: 8k 1; Yk C g;1 Cg;1 Œk and

8k 1; Yk C g Cg Œk;

(5.2)

as follows readily from Proposition 5.4. It is easy to see that, for all k 1, the set C .†g;b /=Yk forms a monoid, with submonoid Yi C .†g;b /=Yk for all 1 i k. In fact we have the following.

502


Theorem 5.6 (See [28], [35]). For all k 1, the monoid C .†g;b /=Yk is a finitelygenerated, nilpotent group. Moreover, the submonoids Yi C .†g;b /=Yk for 1 i k are subgroups and satisfy Yi C .†g;b /=Yk ; Yj C .†g;b /=Yk Ymin.iCj;k/ C .†g;b /=Yk : In particular, Yi C .†g;b /=Yk is abelian if k=2 i k. We use claspers in this proof. The structure of the finitely-generated abelian group Yi C .†g;b /=YiC1 is discussed in Section 6 for b D 0 and b D 1. Theorem 5.6 suggests to complete the monoid C .†g;b / as follows:

b

C .†g;b / WD lim C .†g;b /=Yk : k!1

This is a group by Theorem 5.6, which is called the group of homology cylinders. For all i 1, we also set

b

Yi C .†g;b / WD lim Yi C .†g;b /=Yk ; k!1

to obtain a decreasing sequence of subgroups

b

b

b

b

C .†g;b / D Y1 C .†g;b / Y2 C .†g;b / Y3 C .†g;b / This is an N -series by Theorem 5.6, i:e: we have 8i; j 1; Yi C .†g;b /; Yj C .†g;b / YiCj C .†g;b /:

b

b

b

b

It is conjectured that the canonical map C .†g;b / ! C .†g;b / is injective, which would imply that the Y -filtration on C .†g;b / is separating. According to [67], this injectivity is equivalent to the fact that finite-type invariants (in the sense of Goussarov and the first author) distinguish homology cylinders.

5.3 Yk -equivalence and the Torelli group The Yk -equivalence can be defined also in terms of cut-and-paste operations. More precisely, we call a Torelli surgery of class k the operation which consists in cutting a 3-manifold M along a compact, connected, oriented surface † M , and in re-gluing with an element of the k-th lower central series subgroup k .†/ of the Torelli group. It is easily seen that we can restrict in that definition either to closed surfaces † which bound handlebodies, or to surfaces † with a single boundary component. Theorem 5.7 (See [35]). Let k 1. Two compact oriented 3-manifolds are Yk equivalent if and only if they are related by a Torelli surgery of class k.


We refer to [67] for a proof. Since the Torelli group of †g;b is residually nilpotent (at least for b D 0 and b D 1), Theorem 5.7 somehow supports the conjecture that the Y1 -equivalence could separate 3-manifolds. Example 5.8 (Degree 1). An element of .†g;b / is called a BP (bounding pair) map if it is represented by a pair of Dehn twists along two co-bounding simple closed curves in †g;b , where the direction of the Dehn twists are opposite to each other. Johnson proved that if g 3 then .†g / is generated by genus 1 BP maps [43]. Besides, the Torelli surgery defined by a genus 1 BP map is equivalent to a Y1 -surgery as shown in Figure 6. (See [67] for instance.) These two facts prove Theorem 5.7 for k D 1.

! right Dehn twist left Dehn twist

Figure 6. The mapping cylinder of the BP map defined by the boundary curves of †1;2 can be obtained from †1;2 Œ1; 1 by a Y1 -surgery.

By Theorem 5.7, we have the following. Corollary 5.9. For all k 1 and for all f 2 k .†g;b /, the mapping cylinder c.f / is Yk -equivalent to the trivial cylinder †g;b Œ1; 1. Therefore, the map c W .†g;b / ! C .†g;b / is a morphism of filtered monoids, i.e. we have c.k .†g;b // Yk C .†g;b / for all integers k 1. Let us now assume that b D 0 or b D 1. Then, we have three filtrations on the Torelli group: the lower central series, the restriction of the Y -filtration and the Johnson filtration. They are related in the following way: 8k 1; k .†g;b / c1 Yk C .†g;b / M.†g;b /Œk: (5.3) In the case b D 1, a “stable” version of the lower central series can also be defined. For this, we fix surface inclusions †0;1 †1;1 †2;1 , which induce a sequence of group injections 0;1 /

/ 1;1 /

/ 2;1 /

/ .

504


Thus, the Torelli group g;1 can be regarded as a subgroup of the direct limit of groups . Then, we define lim !h!1 h;1 8k 1;

kstab g;1 WD g;1 \ k lim h;1 D g;1 \ lim k h;1 : ! ! h!1

h!1

We can similarly define a “stable” version of the Y -filtration on C g;1 by 8k 1;

Ykstab C g;1 WD C g;1 \ lim Yk C h;1 ! h!1

where C g;1 is regarded as a submonoid of the monoid lim C h;1 . However, it !h!1 can be proved that Ykstab C g;1 D Yk C g;1 for all k 1 [36]. Therefore, the stable lower central series of g;1 sits between the lower central series and the restriction of the Y -filtration in the hierarchy (5.3). Conjecture 5.10. The lower central series of the Torelli group g;1 stably coincides with the restriction of the Y -filtration, i:e: we have 8k 1; kstab g;1 D c1 Yk C g;1 : (See also Problem 6.2 in this connection.)

6 The Lie ring of homology cylinders In this section, we consider the graded Lie ring induced by the Y -filtration of C .†g;b /, and we discuss its relation with the graded Lie ring induced by the lower central series of .†g;b /. When b D 0 or b D 1, this graded Lie ring can be computed with rational coefficients, using the LMO homomorphism (Section 4) and claspers (Section 5). Besides, the degree 1 part can be completely described in terms of a few invariants.

6.1 Definition of the Lie ring of homology cylinders Recall that an N -series F on a group G is a decreasing sequence G D F1 G F2 G F3 G of subgroups of G such that Fi G; Fj G FiCj G for all i; j 1. Following Lazard [57], we define the graded Lie ring induced by F as the graded abelian group Fi G Gr F G WD FiC1 G i1

with the Lie bracket defined by fgi g; fgj g WD Œgi ; gj 2 FiCj G=FiCj C1 G


on homogeneous elements fgi g 2 Fi G=FiC1 G and fgj g 2 Fj G=Fj C1 G. (One can check that the pairing is well-defined and satisfies the axioms of a Lie bracket using the Hall–Witt identities.) For instance, this construction is well known when F D is the lower central series of G, and we have already met it in the previous sections for the free group. Let us come back to the compact connected oriented surface †g;b . According to §5.2, we can apply the previous construction, for all k 1, to the group C .†g;b /=Yk equipped with the Y -filtration. We then obtain the following construction [35]. Definition 6.1. The Lie ring of homology cylinders is the graded Lie ring Yi C .†g;b /=YiC1 : Gr Y C .†g;b / WD i1

Since the mapping cylinder construction sends the lower central series to the Y filtration (Corollary 5.9), it induces a map at the graded level: Gr c W Gr .†g;b / ! Gr Y C .†g;b /: The map Gr c is at the heart of the interactions between mapping class groups and finite-type invariants of 3-manifolds. Problem 6.2. Determine whether Gr c is injective, and characterize its image. This problem is difficult to solve in general. We shall review, in the following subsections, some pieces of answer. As a preliminary remark, let us observe that the representation theory of the symplectic group may help. Indeed, we have the following lemma, which will be used in the sequel. Lemma 6.3. For b D 0 and b D 1, the conjugation action of M.†g;b / on itself and on C.†g;b / induces an action of the symplectic group Sp.H / on Gr .†g;b / and on Gr Y C .†g;b /, respectively. Moreover, the map Gr c is Sp.H /-equivariant. Proof. The conjugation action of M.†g;b / on C .†g;b / is defined by M.†g;b / C .†g;b / 3 .f; M / 7! c .f / B M B c .f 1 / 2 C .†g;b / and, obviously, it preserves the submonoid C .†g;b /. The Yi -equivalence being generated by surgeries along Yi -graphs, this action also preserves the submonoid Yi C .†g;b / of C .†g;b /. Therefore, the group M.†g;b / acts on Gr Y C .†g;b /. But we also deduce from Theorem 5.6 that 8f 2 .†g;b /; 8M 2 Yi C .†g;b /;

c .f / B M B c .f 1 / Yi C1 M:

So the action of M.†g;b / on Gr Y C.†g;b / factorizes to M.†g;b /=.†g;b / ' Sp.H /. The action of Sp.H / on Gr .†g;b / is similarly defined, and the Sp.H /-equivariance of Gr c is then obvious.

506


One way to simplify the study of the Lie ring of homology cylinders is to consider it with rational coefficients: Yi C .†g;b / ˝ Q: Gr Y C .†g;b / ˝ Q D YiC1 i 1

This Lie algebra is computed in the next subsection for b D 0 and b D 1.

6.2 The Lie algebra of homology cylinders We start with the case of a bordered surface †g;1 . The following Q-vector space was introduced in [35]. Jacobi diagrams without strut component, and with Q external vertices colored by HQ and totally ordered A< .HQ / WD : AS, IHX, multilinearity, STU-like

The AS, IHX and multilinearity relations are as defined in §4.1. The “STU-like” relation is defined as follows: D

1. The surgery map k W At;c k .H / ! Yk H g;1 =YkC1 exists with integral coefficients and is surjective, for any k 2. (See Remark 6.6.) However, the question of its injectivity seems to be open. See [61], [62] in this connection. The above results show that the Y -filtration is not very well suited to the study of the group H .†g;b /, since the associated graded Lie algebra is (for b D 0; 1 and for rational coefficients) completely determined by the Johnson homomorphisms. This inadequacy is particularly apparent in genus 0.

524


Example 7.9 (Genus 0). Recall from Example 7.3 that H 0;1 ' H 0;0 is the homology cobordism group of homology 3-spheres. We deduce from Theorem 6.14 that a homology 3-sphere M is Y2 -equivalent to S 3 if and only if its Rochlin invariant is trivial. Assume this for M . Then, it can be proved that, for all k 2, M is obtained from S 3 by surgery along a disjoint union of graph claspers which are not trees or which have degree k. We deduce from Lemma 7.6 that, for all k 2, there is a homology 3-sphere Mk which is simultaneously Yk -equivalent to S 3 and homology cobordant to M . This shows that the Y -filtration on the homology cobordism group of homology 3-spheres stabilizes starting from the degree 2. (This phenomenon is similar to the fact that the Arf invariant of knots is the only Vassiliev invariant which is preserved by concordance [83].)

7.4 Other aspects of the homology cobordism group The group H .†g;b / has been the subject of recent works in other directions. The reader is referred to Sakasai’s chapter [90] for other aspects of the homology cobordism group. To conclude, we simply mention a few developments which tend to show that H .†g;b / has quite different properties from M.†g;b /: • By using his “trace maps” [75], Morita shows that the abelianization of the group H g;1 has infinite rank [80]. In particular, the group H g;1 is not finitely generated, which contrasts with the fact that the Torelli group g;1 is finitely generated for g 3 [43]. • By using Heegaard–Floer homology, Goda and Sakasai show that the submonoid of Cg;1 consisting of irreducible homology cobordisms is not finitely generated [26]. This contrasts with the well-known fact that Mg;1 is finitely generated. • By using Reidemeister torsion, Cha, Friedl and Kim prove that the abelianization of H .†g;b / contains a direct summand isomorphic to .Z2 /1 if 2g C b > 1 and a direct summand isomorphic to Z1 ˚ .Z2 /1 if b > 1 [15]. Thus, in contrast with M.†g;b /, the group H .†g;b / is not finitely generated (and neither is it perfect). Acknowledgement. The authors would like to thank Jean–Baptiste Meilhan and Takuya Sakasai for various comments and suggestions. They are also grateful to Stefan Friedl for showing them a gap in a previous version of the proof of Proposition 7.2. The first author was partially supported by Grant-in-Aid for Scientific Research (C) 21540078. The second author was partially supported by the French ANR research project ANR-08-JCJC-0114-01.


References [1]

J. W. Alexander, Note on Riemann spaces. Bull. Amer. Math. Soc. 26 (1920), no. 8, 370–372. 472

[2]

J. W. Alexander, On the subdivision of 3-space by a polyhedron. Nat. Acad. Proc. 10 (1924), 6–8. 470

[3]

J. E. Andersen, A. J. Bene, J.-B. Meilhan, and R. C. Penner, Finite type invariants and fatgraphs. Adv. Math. 225 (2010), no. 4, 2117–2161. 493

[4]

M. F. Atiyah, Riemann surfaces and spin structures. Ann. Sci. Ecole Norm. Sup. (4) 4 (1971), 47–62. 513

[5]

R. Baer, Kurventypen auf Flächen. J. Reine Angew. Math. 156 (1927), 231–246. 470

[6]

R. Baer, Isotopie von Kurven auf orientierbaren, geschlossenen Flächen und ihr Zusammenhang mit der topologischen Deformation der Flächen. J. Reine Angew. Math. 159 (1928), 101–116. 470

[7]

D. Bar-Natan, On the Vassiliev knot invariants. Topology 34 (1995), no. 2, 423–472. 490

[8]

H. Bass and A. Lubotzky, Linear-central filtrations on groups. In The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math. 169, Amer. Math. Soc., Providence, R.I., 1994, 45–98. 477

[9]

J. S. Birman, Mapping class groups and their relationship to braid groups. Comm. Pure Appl. Math. 22 (1969), 213–238. 473

[10] J. S. Birman, On Siegel’s modular group. Math. Ann. 191 (1971), 59–68. 466 [11] J. S. Birman, Braids, links, and mapping class groups. Ann. of Math. Stud. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo 1974. 471 [12] J. S. Birman and R. Craggs, The -invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold. Trans. Amer. Math. Soc. 237 (1978), 283–309. 466, 512, 513 [13] N. Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres, Chapitre III: Groupes de Lie, Actualités Scientifiques et Industrielles 1349, Hermann, Paris 1972. 477, 486 [14] G. W. Brumfiel and J. W. Morgan, Quadratic functions, the index modulo 8, and a Z=4Hirzebruch formula. Topology 12 (1973), 105–122. 512 [15] J. C. Cha, S. Friedl, and T. Kim, The cobordism group of homology cylinders. Compos. Math. 147 (2011), no. 3, 914–942. 520, 524 [16] D. Cheptea, K. Habiro, and G. Massuyeau, A functorial LMO invariant for Lagrangian cobordisms. Geom. Topol. 12 (2008), no. 2, 1091–1170. 493, 494, 497, 508 [17] D. Cheptea and T. T. Q. Le, A TQFT associated to the LMO invariant of three-dimensional manifolds. Comm. Math. Phys. 272 (2007), no. 3, 601–634. 493 [18] L. Crane and D. Yetter, On algebraic structures implicit in topological quantum field theories. J. Knot Theory Ramifications 8 (1999), no. 2, 125–163. 500 [19] R. Fintushel and R. J. Stern, Instanton homology of Seifert fibred homology three spheres. Proc. London Math. Soc. (3) 61 (1990), no. 1, 109–137. 521 [20] M. Furuta, Homology cobordism group of homology 3-spheres. Invent. Math. 100 (1990), no. 2, 339–355. 521

526


[21] S. Garoufalidis, The mystery of the brane relation. J. Knot Theory Ramifications 11 (2002), no. 5, 725–737. 507 [22] S. Garoufalidis, M. N. Goussarov, and M. Polyak, Calculus of clovers and finite type invariants of 3-manifolds. Geom. Topol. 5 (2001), 75–108. 466, 499, 507 [23] S. Garoufalidis and J. Levine, Finite type 3-manifold invariants, the mapping class group and blinks. J. Differential Geom. 47 (1997), no. 2, 257–320. 466 [24] S. Garoufalidis and J. Levine, Finite type 3-manifold invariants and the structure of the Torelli group. I. Invent. Math. 131 (1998), no. 3, 541–594. 466 [25] S. Garoufalidis and J. Levine, Tree-level invariants of 3-manifolds, Massey products and the Johnson homomorphism. In Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc., Providence, R.I., 2005, 173–203. 476, 477, 478, 479, 482, 483, 484, 492, 494, 507, 520 [26] H. Goda and T. Sakasai, Abelian quotients of monoids of homology cylinders. Geom. Dedicata 151 (2011), 387–396. 524 [27] F. González-Acuña, 3-dimensional open books. Lectures, University of Iowa Topology Seminar, 1974/75. 472 [28] M. N. Goussarov, Finite type invariants and n-equivalence of 3-manifolds. Compt. Rend. Acad. Sc. Paris Série I Math. 329 (1999), no. 6, 517–522. 466, 498, 500, 502 [29] M. N. Goussarov, Variations of knotted graphs. The geometric technique of n-equivalence. Algebra i Analiz 12 (2000), no. 4, 79–125; English transl. St. Petersburg Math. J. 12 (2001), no. 4, 569–604. 466, 498, 499, 507 [30] N. Habegger, Milnor, Johnson and tree-level perturbative invariants. Preprint 2000; http://www.math.sciences.univ-nantes.fr/~habegger/ 479, 494, 497, 501, 507 [31] N. Habegger and X.-S. Lin, The classification of links up to link-homotopy. J. Amer. Math. Soc. 3 (1990), no. 2, 389–419. 470, 521 [32] N. Habegger and X.-S. Lin, On link concordance and Milnor’s N invariants. Bull. London Math. Soc. 30 (1998), no. 4, 419–428. 470, 479, 521 [33] N. Habegger and G. Masbaum, The Kontsevich integral and Milnor’s invariants. Topology 39 (2000), no. 6, 1253–1289. 497 [34] N. Habegger and C. Sorger, An infinitesimal presentation of the Torelli group of a surface with boundary. Preprint 2000;http://www.math.sciences.univ-nantes.fr/~habegger/ 510 [35] K. Habiro, Claspers and finite type invariants of links. Geom. Topol. 4 (2000), 1–83. 466, 476, 498, 499, 500, 501, 502, 505, 506, 507, 511, 512, 516, 519 [36] K. Habiro and G. Massuyeau, Symplectic Jacobi diagrams and the Lie algebra of homology cylinders. J. Topology 2 (2009), no. 3, 527–569. 492, 493, 494, 504, 506, 508, 510, 511, 512 [37] R. M. Hain, Infinitesimal presentations of the Torelli groups. J. Amer. Math. Soc. 10 (1997), no. 3, 597–651. 509, 510, 511 [38] A. Heap, Bordism invariants of the mapping class group. Topology 45 (2006), no. 5, 851–886. 482, 483, 484, 485 [39] J. Hempel, 3-Manifolds. Ann. of Math. Stud. 86, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo 1976. 471

Chapter 8. From mapping class groups to monoids of homology cobordisms: a survey 527 [40] H. Hopf, Beiträge zur Klassifizierung der Flächenabbildungen. J. Reine Angew. Math. 165 (1931), 225–236. 471 [41] K. Igusa and K. E. Orr, Links, pictures and the homology of nilpotent groups. Topology 40 (2001), no. 6, 1125–1166. 482, 485, 489 [42] S. A. Jennings, The group ring of a class of infinite nilpotent groups. Canad. J. Math. 7 (1955), 169–187. 486 [43] D. L. Johnson, Homeomorphisms of a surface which act trivially on homology. Proc. Amer. Math. Soc. 75 (1979), no. 1, 119–125. 503, 524 [44] D. L. Johnson, An abelian quotient of the mapping class group g . Math. Ann. 249 (1980), no. 3, 225–242. 478, 481 [45] D. L. Johnson, Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22 (1980), no. 2, 365–373. 513 [46] D. L. Johnson, Quadratic forms and the Birman–Craggs homomorphisms. Trans. Amer. Math. Soc. 261 (1980), no. 1, 235–254. 513, 518 [47] D. L. Johnson, A survey of the Torelli group. In Low-dimensional topology (San Francisco, Calif., 1981), Contemp. Math. 20, Amer. Math. Soc., Providence, R.I., 1983, 165–179. 466, 472, 476, 477, 478 [48] D. L. Johnson, The structure of the Torelli group. I. A finite set of generators for . Ann. of Math. (2) 118 (1983), no. 3, 423–442. 466 [49] D. L. Johnson, The structure of the Torelli group. III. The abelianization of . Topology 24 (1985), no. 2, 127–144. 466, 478, 509, 512, 517, 519 [50] N. Kawazumi, Cohomological aspects of Magnus expansions. Preprint 2005; arXiv:math/0505497v3 [math.GT]. 486, 487 [51] T. Kerler, Bridged links and tangle presentations of cobordism categories. Adv. Math. 141 (1999), no. 2, 207–281. 500 [52] R. C. Kirby, The topology of 4-manifolds. Lecture Notes in Math. 1374, Springer-Verlag, Berlin 1989. 512, 518 [53] T. Kitano, Johnson’s homomorphisms of subgroups of the mapping class group, the Magnus expansion and Massey higher products of mapping tori. Topology Appl. 69 (1996), no. 2, 165–172. 478 [54] M. Kontsevich, Formal (non)commutative symplectic geometry. In The Gel’fand Mathematical Seminars, 1990–1992, Birkhäuser, Boston, Mass., 1993, 173–187. 495 [55] M. Kontsevich, Feynman diagrams and low-dimensional topology. In First European Congress of Mathematics (Paris, 1992), Vol. II, Progr. Math. 120, Birkhäuser, Basel 1994, 97–121. 495 [56] J. P. Labute, On the descending central series of groups with a single defining relation. J. Algebra 14 (1970), 16–23. 479 [57] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 101–190. 504 [58] T. T. Q. Le, J. Murakami, and T. Ohtsuki, On a universal perturbative invariant of 3-manifolds. Topology 37 (1998), no. 3, 539–574. 493 [59] C. Lescop, Global surgery formula for the Casson-Walker invariant. Ann. of Math. Stud. 140, Princeton University Press, Princeton, N.J., 1996. 493

528


[60] J. Levine, Homology cylinders: an enlargement of the mapping class group. Algebr. Geom. Topol. 1 (2001), 243–270. 520, 521, 523 [61] J. Levine, Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group”. Algebr. Geom. Topol. 2 (2002), 1197–1204. 496, 523 [62] J. Levine, Labeled binary planar trees and quasi-Lie algebras. Algebr. Geom. Topol. 6 (2006), 935–948. 523 [63] W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds. Ann. of Math. (2) 76 (1962), 531–540. 466 [64] X.-S. Lin, Power series expansions and invariants of links. Geometric topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math. 2.1, Amer. Math. Soc., Providence, R.I., 1997, 184–202. 486 [65] W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations. Interscience Publishers, New York 1966. 477, 486 [66] G. Masbaum, Quantum representations of mapping class groups. In Groupes et géométrie, SMF Journ. Annu. 2003, Soc. Math. France, Paris 2003, 19–36. 466 [67] G. Massuyeau, Finite-type invariants of 3-manifolds and the dimension subgroup problem. J. London Math. Soc. (2) 75 (2007), no. 3, 791–811. 501, 502, 503 [68] G. Massuyeau, Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant. Bull. Soc. Math. France 140 (2012), no. 1, 101–161. 488, 496, 497, 498 [69] G. Massuyeau and J.-B. Meilhan, Characterization of Y2 -equivalence for homology cylinders. J. Knot Theory Ramifications 12 (2003), 493–522. 501, 512, 515, 516, 517, 519 [70] S. V. Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres. Mat. Zametki 42 (1987), no. 2, 268–278, 345; English transl. Math. Notes 42 (1987), no. 1–2, 651–656. 500, 501 [71] J. Milnor, Spin structures on manifolds. Enseignement Math. 9 (1963), no. 2, 198–203. 512 [72] S. Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles. I. Topology 28 (1989), no. 3, 305–323. 466, 510 [73] S. Morita, On the structure of the Torelli group and the Casson invariant. Topology 30 (1991), no. 4, 603–621. 466, 510 [74] S. Morita, Mapping class groups of surfaces and three-dimensional manifolds. Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Vol. I, Math. Soc. Japan, Tokyo 1991, 665–674. 466, 496 [75] S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces. Duke Math. J. 70 (1993), 699–726. 466, 476, 477, 482, 484, 496, 524 [76] S. Morita, A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles. In Topology and Teichmüller spaces (Katinkulta, 1995), World Scientific, River Edge, N.J., 1996, 159–186. 466, 479 [77] S. Morita, Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles. J. Differential Geom. 47 (1997), no. 3, 560–599. 466 [78] S. Morita, Structure of the mapping class groups of surfaces: a survey and a prospect. Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr. 2, Geometry & Topology Publications, Coventry 1999, 349–406. 466, 478, 481

Chapter 8. From mapping class groups to monoids of homology cobordisms: a survey 529 [79] S. Morita, Introduction to mapping class groups of surfaces and related groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, EMS Publishing House, Zurich 2007, 353–386. 475 [80] S. Morita, Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces. In Groups of diffeomorphisms, Adv. Stud. Pure Math. 52, Math. Soc. Japan, Tokyo 2008, 443–468. 524 [81] J. Murakami and T. Ohtsuki, Topological quantum field theory for the universal quantum invariant. Comm. Math. Phys. 188 (1997), no. 3, 501–520. 493 [82] R. Myers, Open book decompositions of 3-manifolds. Proc. Amer. Math. Soc. 72 (1978), no. 2, 397–402. 472 [83] K. Y. Ng, Groups of ribbon knots. Topology 37 (1998), no. 2, 441–458. 524 [84] T. Ohtsuki, Quantum invariants. A study of knots, 3-manifolds, and their sets. Ser. Knots Everything 29, World Scientific Publishing, River Edge, N.J., 2002. 490, 493 [85] B. Perron, Homomorphic extensions of Johnson homomorphisms via Fox calculus. Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 1073–1106. 487 [86] P. F. Pickel, Rational cohomology of nilpotent groups and Lie algebras. Comm. Algebra 6 (1978), no. 4, 409–419. 489 [87] D. G. Quillen, Rational homotopy theory. Ann. of Math. 90 (1969), no. 2, 205–295. 486 [88] D. G. Quillen, On the associated graded ring of a group ring. J. Algebra 10 (1968), 411–418. 485, 486 [89] T. Sakasai, Homology cylinders and the acyclic closure of a free group. Algebr. Geom. Topol. 6 (2006), 603–631. 482, 485 [90] T. Sakasai, A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2012, 531–594. 468, 524 [91] J. Stallings, Homology and central series of groups. J. Algebra 2 (1965), 170–181. 475, 520 [92] D. Sullivan, On the intersection ring of compact three manifolds. Topology 14 (1975), no. 3, 275–277. 478 [93] G. A. Swarup, On a theorem of C. B. Thomas. J. London Math. Soc. (2) 8 (1974), 13–21. 482 [94] C. B. Thomas, The oriented homotopy type of compact 3-manifolds. Proc. London Math. Soc. (3) 19 (1969), 31–44. 482 [95] V. G. Turaev, Cohomology rings, linking coefficient forms and invariants of spin structures in three-dimensional manifolds. In Russian: Mat. Sb. (N.S.) 120 (162) (1983), no. 1, 68–83, 143; English transl. Math. USSR Sb. 48 (1984), 65–79. 512, 513 [96] V. G. Turaev, Nilpotent homotopy types of closed 3-manifolds. Topology (Leningrad, 1982), Lecture Notes in Math. 1060, Springer-Verlag, Berlin 1984, 355–366. 479, 482 [97] C. T. C. Wall, Non-additivity of the signature. Invent. Math. 7 (1969), 269–274. 514 [98] H. Zieschang, E. Vogt, and H.-D. Coldewey, Surfaces and planar discontinuous groups. Translated from the German by John Stillwell, Lecture Notes in Math. 835, SpringerVerlag, Berlin 1980. 475, 476

Chapter 9

A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces Takuya Sakasai

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fox calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fox derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Magnus representation for a free group . . . . . . . . . . . . 2.3 Homology and cohomology of groups . . . . . . . . . . . . . . . 2.4 Fox derivatives in low-dimensional topology . . . . . . . . . . . . 3 Magnus representations for automorphism groups of free groups . . . . 4 Magnus representations for mapping class groups . . . . . . . . . . . . 4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Symplecticity and its topological interpretation . . . . . . . . . . 4.3 Non-faithfulness and decompositions . . . . . . . . . . . . . . . 4.4 Determinant of the Magnus representation . . . . . . . . . . . . . 4.5 The Johnson filtration and Magnus representations . . . . . . . . 4.6 Applications to three-dimensional topology . . . . . . . . . . . . 5 Homology cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . 5.2 Stallings’ theorem and the Johnson filtration . . . . . . . . . . . . 6 Magnus representations for homology cylinders I . . . . . . . . . . . . 6.1 Observation on fundamental groups of homology cylinders . . . . 6.2 The acyclic closure of a group . . . . . . . . . . . . . . . . . . . 6.3 Dehn–Nielsen type theorem . . . . . . . . . . . . . . . . . . . . 6.4 Extension of the Magnus representation . . . . . . . . . . . . . . 7 Magnus representations for homology cylinders II . . . . . . . . . . . 8 Applications of Magnus representations to homology cylinders . . . . . 8.1 Higher-order Alexander invariants and homologically fibered knots 8.2 Bordism invariants and signature invariants . . . . . . . . . . . . 8.3 Abelian quotients of groups of homology cylinders . . . . . . . . 8.4 Generalization to higher-dimensional cases . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction Let †g;n be a compact connected oriented smooth surface of genus g with n boundary components (n may be 0). We take a base point p in @†g;n when n 1 and arbitrarily when n D 0. The mapping class group Mg;n of †g;n is defined as the group of all isotopy classes of orientation-preserving diffeomorphisms of †g;n which fix the boundary pointwise. The area of research covered by the mapping class group in contemporary mathematics is very broad: algebraic geometry, differential geometry, hyperbolic geometry, complex analysis, topology, combinatorial group theory, mathematical physics etc. The group Mg;n serves as the modular group of the Teichmüller space, which is the main theme of this Handbook, and this yields a strong connection among the above subjects. Now let us consider Mg;n from a topological point of view, which is our approach for studying Mg;n in this chapter. While Mg;n does not act on the surface †g;n itself, it works as transformation groups of many discretized objects related to †g;n : the set of isotopy classes of curves, the fundamental group 1 .†g;n ; p/ (when n 1), the homology groups etc. In many cases, these discretized (simplified in a sense) data lose no topological information on the objects involved, for the classical theory of surface topology says that homotopical information of a surface governs its topology. For example, we use Mg;n more often than the diffeomorphism group of †g;n in the construction of three-dimensional manifolds by Heegaard splittings or Dehn surgery, and also that of surface bundles over a manifold. When n D 1, a theorem of Dehn and Nielsen says that an element in Mg;1 is completely characterized by its action on 1 .†g;1 ; p/, which is a free group of rank 2g. This suggests a method for studying Mg;n through the theory of the automorphism group of a free group. Magnus representations are matrix representations for free groups and their automorphism groups. The definition is usually given in terms of the Fox derivative and it looks like a Jacobian matrix of a differentiable map. With their relationship to homology and cohomology of groups, Magnus representations have been used as a fundamental tool for studying various groups by researchers in combinatorial group theory for many years. En route, applications to Mg;n have also been given. Compared with the history of mapping class groups and automorphism groups of free groups, the study of monoids and groups of homology cobordisms over a surface is a quite new theme of research. This research was independently initiated by Goussarov [43] and Habiro [48] in their investigations of three-dimensional manifolds via socalled clover or clasper surgery, which are known to be essentially the same. These surgery techniques are said to be a “topological commutator calculus”, which invokes a connection to Mg;n as automorphisms of 1 .†g;n ; p/. In fact, Garoufalidis–Levine [35] established a connection between the above surgery techniques and classical algebraic topology related to Mg;n in terms of Massey products on the first cohomology of three-dimensional manifolds. In this chapter, we use the word “homology cylinders” for homology cobordisms over a surface with fixed markings of their boundaries. These markings are necessary

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to define a product operation on the set of homology cylinders as a generalization of the mapping class group. We can consider, for example, homology 3-spheres and pure string links with n strings to be homology cylinders over †0;1 and †0;nC1 . For a given homology cylinder over †g;n , we can construct another one by changing its markings by using Mg;n . Therefore homology cylinders enable us to treat important objects in three-dimensional topology simultaneously. This motivates the study of homology cylinders with particular stress on their algebraic structure. The purpose of this chapter is to survey research on structures of mapping class groups and monoids (groups) of homology cylinders through their Magnus representations as a common tool for study. Note that homology cylinders are also discussed in the chapter of Habiro and Massuyeau [49]. However, our approach here is distinct from theirs and more group-theoretical. The author hopes that their chapter and the present one could complement each other and offer the readers an introduction to this fruitful subject. Here we briefly mention the content of this chapter. Basically, it is divided into two parts: The first part is intended for serving as a survey of Magnus representations for automorphism groups of free groups and mapping class groups of surfaces. In Section 2, we first recall the Fox calculus as a tool for many computations in this chapter. We put stress on its relationship to homology and cohomology of groups. Section 3 is devoted to give a machinery of Magnus representations with relations to automorphisms of the derived quotients of a free group. Finally, in Section 4, we overview applications of Magnus representations to mapping class groups of surfaces following works of Morita and Suzuki. The second part of this chapter begins in Section 5, where the monoid and related groups of homology cylinders over a surface are introduced. Sections 6 and 7 are devoted to discussing methods for extending Magnus representations for mapping class groups to homology cylinders. In Section 8, we present several topics on homology cylinders where Magnus representations play important roles. Convention. All maps act on elements from the left. We often use the same notation to write a map and induced maps on quotients of its source or target. Homology and cohomology groups are assumed to be with coefficients in the ring of integers Z and all manifolds are assumed to be smooth unless otherwise indicated. The author is deeply grateful to Athanase Papadopoulos for giving him a chance to write this chapter and providing many valuable suggestions. The author also would like to thank Kazuo Habiro, Nariya Kawazumi, Teruaki Kitano, Gwénaël Massuyeau, Takayuki Morifuji, Takao Satoh and Masaaki Suzuki for their careful reading and helpful comments on the manuscript, and Hiroshi Goda for permitting the author to use the pictures in Example 8.3. This work was partially supported by Grant-in-Aid for Scientific Research, (No. 21740044), Ministry of Education, Science, Sports and Technology, Japan.

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2 Fox calculus We begin by recalling the Fox calculus, which was defined by Fox in the 1950s and which has been known as an important tool in the study of free groups and their automorphisms. Magnus representations are defined as an application of this machinery. Historically, Magnus representations were defined without the Fox calculus. However, we use the Fox calculus since it is now widely accepted to be standard and offers a clear connection to low-dimensional topology. Good references for this topic have been the original paper of Fox [32] and Birman’s book [14]. Our discussion is almost parallel to the latter with particular stress on the relationship of the Fox calculus to homology and cohomology of groups. The relation becomes a key to generalizing the machinery so that it can be applied to objects in more broad range than free groups and their automorphisms in the latter half of this chapter. Since we shall work in non-commutative rings almost everywhere in this chapter, we first need to fix our notation in detail. Notation. For a group G and two of its subgroups G1 and G2 , we denote by ŒG1 ; G2 the commutator subgroup of G1 and G2 . We set Œx; y D xyx 1 y 1 for x, y 2 G. The integral (or rational) group ring of G is denoted by ZŒG (or QŒG). For a matrix A with entries in a ring R and a ring homomorphism ' W R ! R0 , we denote by ' A the matrix obtained from A by applying ' to each entry. A> denotes the transpose of A. When R is ZŒG or its right field of fractions (if it exists), we denote by Ax the matrix obtained from A by applying the involution induced from .x 7! x 1 ; x 2 G/ to each entry. For a module M , we write M n the module of column vectors with n entries in M .

2.1 Fox derivatives Let G be a group and let M be a left ZŒG-module. A crossed homomorphism (or derivation) from G to M is a map f W G ! M satisfying f .xy/ D f .x/ C xf .y/ for all x; y 2 G. In other words, it is a homomorphism .f; idG / W G ! M Ì G, where M Ì G is the semi-direct product with the group structure given by .m1 ; g1 / .m2 ; g2 / D .m1 C g1 m2 ; g1 g2 / for m1 ; m2 2 M and g1 ; g2 2 G. When G is Fn , a free group of rank n, the latter description shows that a crossed homomorphism f W Fn ! M is determined by its values on any generating set of Fn . In general, if G has a (not necessarily finite) presentation hx1 ; x2 ; : : : j r1 ; r2 ; : : :i, crossed homomorphisms f W G ! M correspond to crossed homomorphisms f W hx1 ; x2 ; : : :i ! M satisfying f .ri / D 0 for all i D 1; 2; : : :, where hx1 ; x2 ; : : :i is the free group generated by fx1 ; x2 ; : : :g.


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Let us define Fox derivatives for Fn . We take a basis f1 ; 2 ; : : : ; n g of Fn . The group Fn acts on ZŒFn by left multiplication, so that ZŒFn is a left ZŒFn -module. Definition 2.1. The Fox derivative (or free derivative) with respect to j in the basis f1 ; 2 ; : : : ; n g is the crossed homomorphism @ W Fn ! ZŒFn @j defined by

@i @j

D ıij (Kronecker’s delta). We use the same notation for its extension @ W ZŒFn ! ZŒFn @j

to ZŒFn as an additive map. Fundamental properties of Fox derivatives are as follows. P Z be the PLet t W ZŒFn ! trivializer (or augmentation homomorphism) defined by t. v2Fn av v/ D v2Fn av . Proposition 2.2. (1) The equality (2) For g; h 2 ZŒFn , we have

@i1 @j

D ıij i1 holds.

@g @h @.gh/ D t.h/ C g : @j @j @j (3) (Chain rule) Let ' W Fn ! Fn be an endomorphism of Fn . For any w 2 Fn , we have n X @w @'.k / @'.w/ D ' : @j @k @j kD1

(4) (“Fundamental formula” of Fox calculus) For g 2 ZŒFn , we have n X @g .j 1/: g t.g/ D @j j D1

(5) Let W Fn ! be a homomorphism. Then v 2 Fn satisfies @v D 0 for j D 1; 2; : : : ; n @j if and only if v 2 ŒKer ; Ker . (1) and (2) are easily proved. As for (3), (4) and (5), we prove them in the next subsections by relating them to homology and cohomology of groups.

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2.2 The Magnus representation for a free group Now we define the Magnus representation for Fn by using Fox derivatives. Let S D fs1 ; s2 ; : : : ; sn g be a set of formal parameters. We denote by .ZŒFn /ŒS the polynomial ring over ZŒFn with variables S, where the elements of S are supposed to commute with one another and with the elements of ZŒFn . Definition 2.3. For w 2 Fn , we define a matrix Pn @w ! w j D1 @j sj : .w/ WD 0 1 Then the map w 7! .w/ gives a homomorphism Fn ! GL.2; .ZFn /ŒS / called the Magnus representation for Fn . This representation was first given by Magnus [77] without using Fox derivatives (see Remark 2.4). It is clearly injective. On the other hand, it follows from Proposition 2.2 (5) that for a homomorphism W Fn ! , the kernel of the homomorphism w 7! .w/ is ŒKer ; Ker . In particular, if we take the abelianization map a W Fn ! H1 WD H1 .Fn / Š Zn as , we obtain an injection of the metabelian quotient Fn =ŒŒFn ; Fn ; ŒFn ; Fn into GL.2; .ZŒH1 /ŒS /, where the definition of .ZŒH1 /ŒS is given by replacing Fn with H1 in the definition of .ZŒFn /ŒS . Remark 2.4. The Magnus representation of Fn can be described by using crossed homomorphisms. Indeed, we can unify the Fox derivatives @@j .j D 1; 2; : : : ; n/ into a homomorphism

@ @ @ ; ;:::; @1 @2 @n

>

; idFn W Fn ! .ZŒFn /n Ì Fn :

This map is equivalent to the Magnus representation. On the other hand, if we consider a homomorphism Fn ! .ZŒFn /n Ì Fn given by j 7!

j

.0; : : : ; 0; 1 ; 0; : : : ; 0/> ; j ;

we can define the Fox derivatives as its first projection. This corresponds to Magnus’ original description.

2.3 Homology and cohomology of groups In this subsection, we interpret the definition and fundamental properties of Fox derivatives and the Magnus representation for Fn in terms of homology and cohomology. This interpretation makes it easier to give their topological applications. As space is limited, however, we refer to Brown’s book [16] for the general theory. Instead,


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here we give explicit chain and cochain complexes to calculate the homology and cohomology of a given group. Let G be a group and M be a left ZŒG-module. We denote by Ci the free ZŒG-module generated by the symbols Œg1 jg2 j : : : jgi corresponding to i -tuples of elements g1 ; g2 ; : : : ; gi of G (C0 Š ZŒG generated by Œ). We define the chain complex C .GI M / and cochain complex C .GI M / by Ci .GI M / D Ci ˝ZŒG M; C i .GI M / D HomZŒG .Ci ; M /: Here the tensor product is taken for the right ZŒG-module Ci and the left ZŒG-module M , and HomZŒG .Ci ; M / consists of ZŒG-“equivariant” homomorphisms f in the sense that f satisfies f .cg/ D g 1 f .c/ for c 2 Ci and g 2 G. (These conventions are slightly different from those in [16].) The boundary operator @i W Ci .GI M / ! Ci 1 .GI M / is given by @i .Œg1 jg2 j : : : jgi ˝ m/ D Œg2 jg3 j : : : jgi ˝ g11 m Œg1 g2 jg3 j : : : jgi ˝ m C Œg1 jg2 g3 jg4 j : : : jgi C .1/i1 Œg1 j : : : jgi2 jgi1 gi ˝ m C .1/i Œg1 j : : : jgi2 jgi1 ˝ m and the coboundary operator ıi W C i .GI M / ! C iC1 .GI M / is given by .ıi f /.Œg1 jg2 j : : : jgiC1 / D g1 f .Œg2 jg3 j : : : jgiC1 / f .Œg1 g2 jg3 j : : : jgiC1 / C f .Œg1 jg2 g3 jg4 j : : : jgiC1 / C .1/i f .Œg1 j : : : jgi1 jgi giC1 / C .1/iC1 f .Œg1 j : : : jgi1 jgi / for f 2 C i .GI M / regarded as a function from the set of symbols Œg1 jg2 j : : : jgi to M . We denote the corresponding homology and cohomology groups by H .GI M / and H .GI M /. We obtain the following explicit description of homology and cohomology in degree 0 by observing the complexes. H0 .GI M / D M=hm gm j m 2 M; g 2 Gi; H 0 .GI M / D fm 2 M j gm D m for any g 2 Gg: Next we take a close look at H 1 .GI M /. The condition for f 2 C 1 .GI M / to be a cocycle is that 0 D .ı1 f /.Œg1 jg2 / D g1 f .Œg2 / f .Œg1 g2 / C f .Œg1 / holds for any g1 ; g2 2 G. If we naturally identify a cochain in C 1 .GI M / with a map from G to M , this cocycle condition is nothing more than the definition of crossed homomorphisms from G to M mentioned in Section 2.1. Therefore the module of 1-cocycles is written as the module Cross.GI M / of crossed homomorphisms from

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G to M . A 1-coboundary is obtained from each element m 2 M Š C 0 .GI M / corresponding to the function Œ 7! m, and we have .ı0 m/.Œg/ D gm m for m 2 M and g 2 G. We call such a crossed homomorphism (after the above identification) a principal crossed homomorphism and denote the module of principal crossed homomorphisms by Prin.GI M /. Consequently we have H 1 .GI M / Š Cross.GI M /=Prin.GI M /: Example 2.5. Let M D Z with the trivial G-action. We have H0 .GI Z/ Š H 0 .GI Z/ Š Z; H1 .GI Z/ Š G=ŒG; G; the abelianization of G; H 1 .GI Z/ Š Hom.G; Z/ D Hom.H1 .GI Z/; Z/: Remark 2.6. For any ZŒG-bimodule M (for example, M D ZŒG), C .GI M / and C .GI M / have natural right actions of G, with our convention. Moreover, the operators @i and ıi are equivariant with respect to this right action, so that H .GI M / and H .GI M / become right ZŒG-modules. Let us return to our concern. The Fox derivative 1

@ @j

can be seen as a 1-cocycle

in Cross.Fn I ZŒFn / C .Fn I ZŒFn / sending j to 1 and i .i ¤ j / to 0. Since a crossed homomorphism Fn ! ZŒF ˚ n is determined by its values on any basis of Fn , we see that @@1 ; @@2 ; : : : ; @@n forms a basis of Cross.Fn I ZŒFn / as a right ZŒFn -module. Proof of Proposition 2.2 (3). Let ' ZŒFn be the left ZŒFn -module whose underlying abelian group is ZŒFn but on which Fn acts through '. We easily see that Cross.Fn I ' ZŒFn / is a free right ZŒFn -module of rank n generated by ³ ² @ @ @ ; ' ; :::; ' ; ' @1 @2 @n where Fn acts from the right by the usual, not through ', multiplication. Now @the map Pn @'.w/ @'./ w 7! @j is in Cross.Fn I ' ZŒFn /, so that we can put @j D kD1 ' @k gk . If we substitute k in this equality, we have gk D

@'.k / @j

and our claim follows.

Proof of Proposition 2.2 (4). It suffices to show our claim when g D v 2 Fn , a monomial. We can easily check that the equality n X @ .j 1/ 2 C 1 .Fn I ZŒFn / ı0 1 D @j j D1

0

holds for 1 2 C .Fn I ZŒFn / Š ZŒFn . Applying this 1-cochain to v, we obtain the desired equality.


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Remark 2.7. For groups H G and a left ZŒG-module M , the relative homology H .G; H I M / (resp. cohomology H .G; H I M /) can be defined by considering C .GI M /=C .H I M / (resp. Ker.C .GI M / ! C .H I M //) as usual.

2.4 Fox derivatives in low-dimensional topology Fox derivatives and low-dimensional topology are connected by using the topological definition of (co)homology of groups. Convention. For a connected CW-complex X , we denote by Xz its universal covering. z The group We take a base point p of X and a lift pQ of p as a base point of X. G WD 1 .X; p/ acts on Xz from the right through its deck transformation group, namely, the lift of 2 G starting from pQ reaches p Q 1 . When X is a finite complex, we regard the cellular chain complex C .Xz / of Xz , on which G acts from the right, as a collection of free right ZŒG-modules consisting of column vectors together with boundary operators given by left multiplication of matrices. For a left ZŒG-module M , the twisted chain complex C .X I M / is given by the tensor product of the right z and the left ZŒG-module M . This complex gives the twisted ZŒG-module C .X/ homology group H .X I M /. The twisted cochain complex C .X I M / and the twisted cohomology group H .X I M / are defined similarly. In topology, the homology H .GI M / and cohomology H .GI M / of a group G with coefficients in a left ZŒG-module M are defined as twisted (co)homology groups H .GI M / D H .K.G; 1/I M /; H .GI M / D H .K.G; 1/I M /; where K.G; 1/ is the Eilenberg–MacLane space of G. The space K.G; 1/ is characterized uniquely up to homotopy equivalence as a connected CW-complex satisfying 1 .K.G; 1// D G;

i .K.G; 1// D 0 .i 2/:

Remark 2.8. There are several methods for checking that this definition coincides with that in the previous section. We refer to Brown’s book [16] again. One method is to see that the complex in the previous section is derived from the fat realization of a (semi-)simplicial structure of K.G; 1/. Note that for any CW-complex X with 1 X D G, we have H0 .X I M / Š H0 .GI M /;

H1 .X I M / Š H1 .GI M /;

H 0 .X I M / Š H 0 .GI M /;

H 1 .X I M / Š H 1 .GI M /

since K.G; 1/ can be obtained from X by attaching 3-cells, 4-cells,: : :, so that all higher homotopy groups are eliminated. We also see that there exists an epimorphism H2 .X/

/ / H2 .G/:

(2.1)

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The kernel is exactly the image of the Hurewicz homomorphism 2 X ! H2 .X /. For a group G with a presentation hx1 ; x2 ; : : : j r1 ; r2 ; : : :i, we construct a 2complex X consisting of one 0-cell, say p, one 1-cell for each generator and one 2-cell for each relation with an attaching map according to the word. Then 1 X D G. Using this construction, we now look for a practical method for calculating H1 .GI M / in case G has a finite presentation hx1 ; x2 ; : : : ; xk j r1 ; r2 ; : : : ; rl i. Consider the chain complex @2

@1

C2 .X I M / ! C1 .X I M / ! C0 .X I M / ! 0: This complex can be rewritten as D2

D1

M l ! M k ! M ! 0 with matrices D1 , D2 . Observing the lifts of the loops x1 ; : : : ; xk and r1 ; : : : rl starting z we see that from the base point pQ of X, D1 D x11 1 x21 1 xk1 1 ;

D2 D

@rj @xi

1i k 1j l

over ZŒG. Here and hereafter “over ZŒG” means that we are considering all entries to be in ZŒG as images of the natural homomorphism ZŒhx1 ; : : : ; xk i ! ZŒG. The relation D1 D2 D O follows from Proposition 2.2 (4). From this expression of D2 , we actually see that Fox derivatives contribute to low-dimensional topology in the calculation of H1 .GI M / D H1 .X I M /. Example 2.9. Let W G ! be an epimorphism. Take a CW-complex X with 1 X D G. Then C .X I ZŒ/ just corresponds to the cellular complex of the covering X of X with respect to . Hence we have H1 .GI ZŒ/ Š H1 .X I ZŒ/ Š H1 .X / Š H1 .Ker /: Proof of Proposition 2.2 (5). First, we may suppose that W Fn ! is an epimorphism. We now consider H1 .Fn I ZŒ/. The space K.Fn ; 1/ is given by a bouquet X of n circles corresponding to the generating system f1 ; 2 ; : : : ; n g of Fn . By an observation similar to the one for the matrix D2 above, we see that the abelianization map Ker Š 1 X ! H1 .X / D f1-cycles on X g C1 .X / Š .ZŒ/n is given by

!>

v 7!

@v @1

@v @2

@v @n

:

The kernel of this map is ŒKer ; Ker and our claim immediately follows.


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Example 2.10. Let X be the bouquet in the above proof of Proposition 2.2 (5) with base point p. Consider the homology exact sequence H1 .X I ZŒFn / ! H1 .X; fpgI ZŒFn / ! H0 .fpgI ZŒFn / ! H0 .X I ZŒFn / ! H0 .X; fpgI ZŒFn /; which can be regarded as that for the pair .Fn ; f1g/ of groups. Clearly H1 .X I ZŒFn / D H1 .Xz / D 0 and H0 .X I ZŒFn / D H0 .Xz / D Z (see Example 2.9). From the cell structures, we immediately see that H1 .X; fpgI ZŒFn / D C1 .X; fpgI ZŒFn / Š .ZŒFn /n , H0 .X; fpgI ZŒFn / Š 0 and H0 .fpgI ZŒFn / Š ZŒFn . Hence the above exact sequence is rewritten as

0 ! .ZŒFn /n ! ZŒFn ! Z ! 0: The third map is given by the trivializer t with the kernel I.Fn / called the augmentation ideal of ZŒFn . Consequently, the map induces an isomorphism Š

W .ZŒFn /n ! I.Fn /

(2.2)

as right ZŒFn -modules. We can easily check that ..a1 ; a2 ; : : : ; an /> / D

n X .i1 1/ai iD1

for .a1 ; a2 ; : : : ; an / 2 .ZŒFn / . That is, f11 1; 21 1; : : : ; n1 1g forms a right free basis of I.Fn /. Note that the map Fn ! .ZFn /n sending v 2 Fn to 1 .v 1 1/ recovers the Fox derivatives (cf. Proposition 2.2 (4)): @v @v > @v 1 1 .v 1/ D : @1 @2 @n >

n

We close this section by the following application of the Fox calculus to knot theory: Example 2.11 (The Alexander polynomial of a knot). Let K be a knot in the 3-sphere S 3 . That is, K is a smoothly embedded circle in S 3 . The fundamental group of the knot exterior E.K/ WD S 3 N.K/ of K, where N.K/ is an open tubular neighborhood of K, is called the knot group G.K/ of K. We have H1 .E.K// Š H1 .G.K// Š Z generated by the meridian t of K with a fixed orientation. The Alexander module AZ .K/ of K is defined as AZ .K/ WD H1 .E.K/I ZŒhti/ D H1 .G.K/I ZŒht i/: Then the Alexander polynomial K .t/ of K is defined as the determinant of a square (say k k) matrix D representing AZ .K/, namely D fits into an exact sequence D

ZŒht ik ! ZŒht ik ! AZ .K/ ! 0:

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It can be checked that det D up to multiplication by ˙t m (m 2 Z) does not depend on the choice of D. To obtain such a matrix D, take a Wirtinger presentation of G.K/, which gives a presentation of the form hx1 ; x2 ; : : : ; xkC1 j r1 ; r2 ; : : : ; rk i. Using

the arguments in this subsection, we can see that the square matrix represents AZ .K/.

@rj @xi

1i;j k

3 Magnus representations for automorphism groups of free groups In this section, we overview generalities of Magnus representations for the automorphism group Aut Fn of a free group Fn D h1 ; 2 ; : : : ; n i. Applications to the mapping class group of a surface are discussed in the next section. Definition 3.1. The (universal ) Magnus representation for Aut Fn is the map r W Aut Fn ! M.n; ZŒFn / assigning to ' 2 Aut Fn the matrix r.'/ WD

@'.j / @i

; i;j

r W Aut Fn ! M.n; ZŒFn / which we call the Magnus matrix for '. While we call the map r the Magnus “representation”, it is actually a crossed homomorphism in the following sense: Proposition 3.2. For ',

2 Aut Fn , the equality r.' / D r.'/ ' r. /

holds. In particular, the image of r is included in the set GL.n; ZŒFn / of invertible matrices. Proof. Although we need to be careful about the noncommutativity of ZŒFn , the proof is an easy application of Proposition 2.2 (3) together with the fact that r.idFn / D In . Remark 3.3. The second assertion of Proposition 3.2 is part of Birman’s inverse function theorem [13] stating that an endomorphism W Fn ! Fn is an automorphism if and only if r. /, which makes sense, belongs to GL.n; ZŒFn /.


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The map r is injective since '.i / is recovered from r.'/ by applying Proposition 2.2 (4) to the i-th column for each ' 2 Aut Fn , namely there is no lack of information. To obtain a genuine representation, a homomorphism, we need to reduce information of the map r as follows. Let W Fn ! be an epimorphism whose kernel is characteristic, namely Ker is invariant under all automorphisms of Fn . Then induces a natural homomorphism Aut Fn ! Aut . We consider the matrix r .'/ obtained from the Magnus matrix r.'/ by applying the map W ZŒFn ! ZŒ to each entry. Proposition 3.4. Let W Fn ! be an epimorphism whose kernel is characteristic. Then the restriction of the map r W Aut Fn ! GL.n; ZŒ/ to Ker.Aut Fn ! Aut / is a homomorphism. Moreover, the kernel of r coincides with that of the natural homomorphism Aut Fn ! Aut .Fn =ŒKer ; Ker /: Proof. The first half of our assertion follows from Proposition 3.2. To show the second, we first note that the map Aut Fn ! Aut is decomposed as Aut Fn ! Aut .Fn =ŒKer ; Ker / ! Aut .Fn = Ker / D Aut ; so that Ker.Aut Fn ! Aut .Fn =ŒKer ; Ker // Ker.Aut Fn ! Aut /. Suppose ' 2 Ker.Aut Fn ! Aut .Fn =ŒKer ; Ker //, then there exists vi 2 ŒKer ; Ker such that '.i / D i vi for each i . Using Proposition 2.2 (2) and (5), we have @'.i / @.i vi / @i @vi C i D D D ıij : @j @j @j @j Therefore r .'/ D In and Ker.Aut Fn ! Aut .Fn =ŒKer ; Ker // Ker r . On the other hand, if ' 2 Ker r , we have 1 @.i1 '.i // @i @'.i / C i1 D @j @j @j 1 1 @'.i / D .ıij i / C i @j D ıij .i1 / C .i1 /ıij D 0: By Proposition 2.2 (5), we see that i1 '.i / 2 ŒKer ; Ker . This means that ' 2 Ker.Aut Fn ! Aut .Fn =ŒKer ; Ker // and hence Ker r Ker.Aut Fn ! Aut .Fn =ŒKer ; Ker // follows. This completes the proof.

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Remark 3.5. If we use the description of Fox derivatives in Example 2.10, the Magnus representation r associated with as above can be regarded as a transformation of I.Fn / ˝ZŒFn ZŒ by the diagonal action of Aut .Fn /. Example 3.6. The trivializer t W ZŒFn ! Z is induced from the trivial homomorphism Fn ! f1g and Aut Fn acts trivially on f1g. Hence we have a homomorphism rt W Aut Fn ! GL.n; Z/: Note that rt coincides with the action of Aut Fn on H1 WD H1 .Fn / Š Zn . Nielsen studied the map rt and showed that it is surjective [94]. The kernel IAn WD Ker rt is called the IA-automorphism group. A finite generating set of IAn was first given by Magnus [76]. We can see a summary of these works in Morita’s chapter in the first volume of this Handbook ([89], Chapter 7). Example 3.7. The abelianization homomorphism a W Fn ! H1 is most frequently used as a reduction homomorphism . In this case, the restriction of ra to IAn yields a homomorphism ra W IAn ! GL.n; ZŒH1 /: This representation was first introduced by Bachmuth [10], who defined the representation as a transformation of the .1; 2/-entry of the Magnus representation Fn ! GL.2; .ZŒH1 /ŒS/ in Section 2.2 and studied the automorphism group of the metabelian quotient of Fn . In relation to topology, we consider the braid group Bn and the pure braid group Pn Bn of n strings. For the definitions, we refer to Paris’ chapter [98] in the second volume of this Handbook as well as to Birman’s book [14]. We now focus on Artin’s theorem [5], [6] saying that there exists a natural embedding of Bn into Aut Fn , which embeds Pn into IAn . By postcomposing ra with this embedding, we obtain a homomorphism ra W Pn ! GL.n; ZŒH1 / called the Gassner representation [36]. To obtain a representation for Bn , we further reduce ZŒH1 to ZŒht i by the homomorphism b W H1 ! ht i sending each j to t. Then we obtain a homomorphism rbBa W Bn ! GL.n; ZŒht i/ called the Burau representation [18]. Note that in their papers, Burau and Gassner did not use the Fox calculus, but gave explicit formulas to construct their representations. Continuing rt and ra , we now consider the following system consisting of a filtration of Aut Fn and a sequence of Magnus representations. Let Fn.0/ WD Fn Fn.1/ Fn.2/ Fn.3/


545

be the derived series of Fn defined by Fn.kC1/ D ŒFn.k/ ; Fn.k/ for k 0. Fn.k/ is a characteristic subgroup of Fn and we denote by pk W Fn ! Fn =Fn.k/ the natural projection. Note that p0 D t and p1 D a. Define a filtration I 0 An WD Aut Fn I 1 An I 2 An I 3 An of Aut Fn by I k An WD Ker.Aut Fn ! Aut .Fn =Fn.k/ // for k 1. Note that I 1 An D IAn . By Proposition 3.4, the map rpk W I k An ! GL.n; ZŒFn =Fn.k/ / is a homomorphism with Ker rpk D Ker.Aut Fn ! Aut .Fn =ŒFn.k/ ; Fn.k/ // D Ker.Aut Fn ! Aut .Fn =Fn.kC1/ // D I kC1 An : That is, each of the Magnus representations rpk plays the role of an obstruction for an automorphism in I k An to be in the next filter. It seems difficult to determine the image of rpk in general. However, if we put ³ ² Pn 1 1/aij D j1 1 i D1 .i ; Gn.k/ WD .aij /i;j 2 GL.n; ZŒFn =Fn.k/ / .k/ in ZŒFn =Fn for all j .

then we can check that Gn.k/ is a subgroup of GL.n; ZŒFn =Fn.k/ / and it includes the image of rpk since we can apply Proposition 2.2 (4) under the condition that '.j / D j 2 Fn =Fn.k/ for all j . Proposition 3.8. There exists a canonical injective homomorphism ˆ W Gn.k/ ! Aut .Fn =Fn.kC1/ / and we have an exact sequence ˆ

1 ! Gn.k/ ! Aut .Fn =Fn.kC1/ / ! Aut .Fn =Fn.k/ /: Proof. We first define a homomorphism ˆ W Gn.k/ ! Aut .Fn =Fn.kC1/ /. Let A D P .aij /i;j 2 Gn.k/ . Then niD1 .i1 1/aij D j1 1 holds for j D 1; 2; : : : ; n. By

Theorem 3.7 of [14], there exists vj 2 Fn satisfying pk .vj / D j 2 Fn =Fn.k/ and pk

@vj @i

D aij 2 ZŒFn =Fn.k/ . Moreover such a vk is unique up to Fn.kC1/ . Define

an endomorphism 'A W Fn ! Fn by 'A .j / D vj . By construction, it induces the identity map on Fn =Fn.k/ . Then Proposition 2.2 (3) shows that A 7! 'A defines a homomorphism ˆ from Gn.k/ to the monoid of endomorphisms of Fn =Fn.kC1/ . Since Gn.k/ is a group, the image of ˆ should be included in Aut .Fn =Fn.kC1/ /. Consequently we obtain a homomorphism ˆ W Gn.k/ ! Aut .Fn =Fn.kC1/ /. The compo-

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sition Gn.k/ ! Aut .Fn =Fn.kC1/ / ! Aut .Fn =Fn.k/ / is trivial by construction. On the other hand, the homomorphism rpk W I k An ! Gn.k/ induces a homomorphism Ker.Aut .Fn =Fn.kC1/ / ! Aut .Fn =Fn.k/ // ! Gn.k/ , which gives the inverse of ˆ. This shows the exactness of the sequence. Remark 3.9. The above system can be regarded as a refinement of the Andreadakis filtration of Aut Fn , which is defined by using the lower central series of Fn (see Section 4.5 and Morita’s chapter [89] in Volume I of this Handbook). For a characteristic subgroup G Fn , an automorphism of Fn =G is said to be tame if it is induced from one of Fn . The study of tame automorphisms has been of great interest among researchers in combinatorial group theory. We refer to the surveys by Gupta [44] and Gupta–Shpilrain [45] for details. As for Aut .Fn =Fn.k/ /, the following results are known. Bachmuth [10] and Bachmuth–Mochizuki [11], [12] showed that all the automorphisms of Fn =Fn.2/ are tame if n D 2 or n 4 while there exists a non-tame automorphism when n D 3. The latter fact was first shown by Chein [22]. Moreover it was shown by Shpilrain [109] that non-tame automorphisms of Fn =Fn.k/ do exist and the map Aut .Fn =Fn.kC1/ / ! Aut .Fn =Fn.k/ / is not surjective for n 4 and k 3. Recently, Satoh [108] showed that I 2 An is not finitely generated and moreover H1 .I 2 An / has infinite rank for n 2 (see also Church–Farb [23] mentioned in the next section). We here pose the following problems: Problems 3.10. (1) Determine the image of rpk . (2) Find a structure on the filtration fI k An g. For example, the associated graded module, namely the direct sum of the successive quotients of the Andreadakis filtration has a natural graded Lie algebra structure.

4 Magnus representations for mapping class groups Now we start our discussion on applications of Magnus representations to mapping class groups of surfaces.

4.1 Definition Let †g;n be a compact oriented surface of genus g with n boundary components. We take a base point p in @†g;n when n 1 and arbitrarily when n D 0. The fundamental group 1 †g;n of †g;n with respect to the base point p is a free group of rank 2g Cn1 except the cases n D 0, when 1 †g;0 is given as a quotient of a free group of rank 2g by one relation.


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The mapping class group of †g;n is the group of all isotopy classes of orientationpreserving diffeomorphisms of †g;n , where all diffeomorphisms and isotopies are assumed to fix @†g;n pointwise when n > 0. For a general theory of mapping class groups, we refer to Birman [14], Ivanov [59], Farb–Margalit [31] as well as Morita’s chapter [89]. The case where n D 1 is now of particular interest and we assume it hereafter. In our context, the following theorem, which is often called the Dehn–Nielsen theorem, is crucial for applying the techniques mentioned in the previous sections to Mg;1 . We put WD 1 †g;1 for simplicity. Theorem 4.1 (The Dehn–Nielsen theorem). The natural action of Mg;1 on induces an isomorphism Mg;1 Š f' 2 Aut j '. / D g; where 2 corresponds the boundary loop @†g;1 . The corresponding injection W Mg;1 ,! Aut is called the Dehn–Nielsen embedding. Remark 4.2. When n D 0, Mg;0 acts on 1 †g;0 up to conjugation and the corresponding Dehn–Nielsen theorem is stated in terms of the outer automorphism group Out .1 †g;0 / of 1 †g;0 . When n 2, the homomorphism Mg;n ! Aut .1 †g;n / is not injective since the Dehn twist along a loop parallel to one of the boundaries not containing p acts trivially on 1 †g;n . Therefore we need a special care in treating such boundaries. Filling these n 1 boundaries by n 1 copies of †1;1 , we have an embedding †g;n ,! †gCn1;1 . A diffeomorphism of †g;n naturally extends to one of †gCn1;1 such that it restricts to the identity map on †gCn1;1 †g;n . This induces a homomorphism Mg;n ! MgCn1;1 , which is known to be injective. Hence we have Mg;n MgCn1;1 Aut .1 †gCn1;1 /. We now take 2g oriented loops 1 ; 2 ; : : : ; 2g as in Figure 1. They form a basis of and we often identify with the free group F2g D h1 ; 2 ; : : : ; 2g i of rank 2g. We have D Œ1 ; gC1 Œ2 ; gC2 : : : Œg ; 2g . We put H WD H1 .†g;1 / D H1 .†g;1 ; @†g;1 /. The group H can be identified with Z2g by choosing f1 ; 2 ; : : : ; 2g g as a basis of H , where we write j again for j as an element of H D =Œ; . Poincaré duality endows H with a non-degenerate anti-symmetric bilinear form W H ˝ H ! Z called the intersection pairing. The above basis of H is a symplectic basis with respect to , namely we have .i ; j / D .gCi ; gCj / D 0;

.i ; gCj / D .gCj ; i / D ıij

for i; j D 1; 2; : : : ; g. The action of Mg;1 on H Š Z2g defines a homomorphism

W Mg;1 ! GL.2g; Z/. Since this action preserves the intersection pairing, the image

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Takuya Sakasai g

2

1

gC1

gC2

2g

p Figure 1. Our basis of 1 †g;1 .

of is included in the symplectic group (or the Siegel modular group) Sp.2g; Z/ D fX 2 GL.2g; Z/ j X > JX D J g; O I where J D Ig Og . Note that Sp.2g; Z/ SL.2g; Z/. It is classically known that

W Mg;1 ! Sp.2g; Z/ is surjective. Consequently we have an exact sequence

1 ! g;1 ! Mg;1 ! Sp.2g; Z/ ! 1; where g;1 WD Ker is called the Torelli group. Using the Dehn–Nielsen theorem, we define the (universal) Magnus representation r W Mg;1 ! GL.2g; ZŒ/ for Mg;1 by assigning to ' 2 Mg;1 the matrix r.'/ WD

@'.j / @i

i;j

. The map r is

an injective crossed homomorphism by Proposition 3.2 and the paragraph subsequent to it. By definition, D rt holds and we have g;1 D Mg;1 \ IA2g . Also we have a crossed homomorphism ra W Mg;1 ! GL.2g; ZŒH /; which restricts to a homomorphism ra jg;1 W g;1 ! GL.2g; ZŒH /; called the Magnus representation for the Torelli group. Applications of these Magnus representations to Mg;1 were first given by Morita in [86], [87]. After that, the study of the Magnus representation for the Torelli group has been intensively pursued by Suzuki [112], [113], [114], [115]. Remark 4.3. In [3], Andersen–Bene–Penner constructed groupoid lifts of the Dehn– Nielsen embedding W Mg;1 ,! Aut and the Magnus representation r to the Ptolemy groupoid Pt.†g;1 / of †g;1 . This groupoid may be regarded as a discrete model of paths in the (decorated) Teichmüller space [100] and Mg;1 can be embedded

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as the oriented paths starting from a fixed vertex v and reaching vertices in the same Mg;1 -orbit as v.

4.2 Symplecticity and its topological interpretation Now we overview known properties of the Magnus representations r and ra . The first one is called the (twisted) symplecticity. Theorem 4.4 (Morita [87], Suzuki [114], Perron [101]). For any ' 2 Mg;1 , the Magnus matrix r.'/ satisfies the equality

where Jz D 0 B B B J1 D B B @ 0 B B B J2 D B B B @

J1 J2 J3 J4

r.'/> Jz r.'/ D ' Jz; 2 GL.2g; ZŒ/ is defined by

1 1 .1 2 /.1 11 / .1 3 /.1 11 / :: : .1 g /.1 11 /

1 1 2 .1 3 /.1 21 / :: : .1 g /.1 21 /

1 1 gC1 1 .1 2 /.1 gC1 / 1 .1 3 /.1 gC1 / :: : 1 .1 g /.1 gC1 /

C C C C; C A

0 1 3 ::

:

1 g 1

1 2 gC2 1 .1 3 /.1 gC2 / :: : 1 .1 g /.1 gC2 /

C C C C; C C A

0 1 3 gC3

::

: 1 g 2g

0

1 11 gC1 B .1 gC2 /.1 11 / B B 1 J3 D B B .1 gC3 /.1 1 / B :: @ : .1 2g /.1 11 / 0 B B B J4 D B B B @

1 1 gC1 1 .1 gC2 /.1 gC1 / 1 .1 gC3 /.1 gC1 / :: : 1 .1 2g /.1 gC1 /

1 1

21

gC2

.1 gC3 /.1 21 / :: : .1 2g /.1 21 /

C C C C; C C A

0 ::

:

1 g1 2g 1

1 1 gC2 1 .1 gC3 /.1 gC2 / :: : 1 .1 2g /.1 gC2 /

0 1

1 gC3

::

: 1 1 2g

C C C C: C C A

Note that the matrix Jz first appeared in Papakyriakopoulos’ paper [97] and it is mapped to the matrix J by the trivializer t W ZŒ ! Z. Morita used a finite generating system of Mg;1 to show that the equality holds for each element of the system. On

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the other hand, Suzuki gave a topological description of the Magnus representation and showed that the equality holds for any element of Mg;1 . Perron’s proof is similar to Suzuki’s. Suzuki’s description is as follows. First, consider the twisted homology H1 .†g;1 ; fpgI ZŒ/; which coincides with the usual homology

e

H1 .†g;1 ; f 1 .p//

e e

of the universal covering f W †g;1 ! †g;1 . This module is isomorphic to .ZŒ/2g and the set of lifts 1 ; 2 ; : : : ; 2g (see Convention in Section 2.4) of 1 ; 2 ; : : : ; 2g forms a basis as a right ZŒ-module. The action of ' 2 Mg;1 on †g;1 is uniquely lifted on †g;1 so that pQ is fixed. It induces a right ZŒ-equivariant isomorphism of H1 .†g;1 ; fpgI ZŒ/. Suzuki showed that the matrix representation of this equivariant isomorphism under the above basis coincides with r.'/. Next Suzuki considered an intersection pairing

e

h ; i W H1 .†g;1 ; fpgI ZŒ/ H1 .†g;1 ; fpgI ZŒ/ ! ZŒ on H1 .†g;1 ; fpgI ZŒ/ called the higher intersection number in [114] by using Papakyriakopoulos’ idea of biderivations [97] in ZŒ. Note that Turaev [116] also gave a construction similar to biderivations. Let c1 ; c2 be paths on †g;1 connecting two points of f 1 .p/. We take another base point q 2 @†g;1 and decompose @†g;1 into two segments A and B as in Figure 2.

e

1 2 2 2g

1 @†g;1 B

p

A

q

B

Figure 2. Decomposition of @†g;1 .

We slide c2 along the lifts of A so that the resulting path connects two points of f 1 .q/. Then we set X .c Q 1 ; c2 /; hc1 ; c2 i D 2

Q 1 ; c2 / is the where c1 is the path obtained from c1 by the right action of and .c usual intersection number of c1 and c2 on †g;1 . We can naturally extend this pairing of paths to the desired pairing of H1 .†g;1 ; fpgI ZŒ/ so that

e

huf; vi D fNhu; vi;

hu; vf i D hu; vif


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holds for any f 2 ZŒ and u; v 2 H1 .†g;1 ; fpgI ZŒ/. This pairing is clearly preserved by the action of Mg;1 . Then the twisted symplecticity is obtained by writing this invariance under our basis of H1 .†g;1 ; fpgI ZŒ/. Remark 4.5. Sliding the path c2 in the above procedure has the following homological meaning, which was pointed out in Turaev [116]. In the source of the pairing h ; i, we identify the left H1 .†g;1 ; fpgI ZŒ/ with H1 .†g;1 ; AI ZŒ/ by using the inclusion .†g;1 ; fpg/ ,! .†g;1 ; A/ and similarly the right with H1 .†g;1 ; BI ZŒ/ by .†g;1 ; fpg/ ,! .†g;1 ; A/

- .†g;1 ; fqg/ ,! .†g;1 ; B/;

which corresponds to the slide. Then we can take a homological intersection between the pairs .†g;1 ; A/ and .†g;1 ; B/ arising from Poincaré–Lefschetz duality.

4.3 Non-faithfulness and decompositions Here we focus on the Magnus representation ra W g;1 ! GL.2g; ZŒH / for the Torelli group. We first mention the following fact first found by Suzuki: Theorem 4.6 (Suzuki [112]). The Magnus representation for the Torelli group g;1 is not faithful, namely ker ra ¤ f1g, for g 2. Suzuki’s proof exhibits an example, which looks not so complicated but needs a long computation. After that he gave an improvement [115] based on the topological interpretation of ra . See also Perron [101]. Along this line, the following remarkable result was recently shown by Church–Farb: Theorem 4.7 (Church–Farb [23]). The kernel Ker ra is not finitely generated. Moreover, H1 .Ker ra / has infinite rank for g 2. Note that their argument can be also applied to I 2 A. On the other hand, whether the Gassner representation, the corresponding representation for braids, is faithful or not is unknown for n 4. When n D 3, it was shown to be faithful by Magnus–Peluso [78] (see also [14], Theorem 3.15). A decisive difference between the Magnus representation for g;1 and the Gassner representation for Pn appears in their irreducible decompositions. Here, the word “irreducible” means that there exist no invariant direct summands of .ZŒH /2g (or .ZŒH1 /n ), which is a slight abuse of terminology. It is easily checked that the Gassner representation has a 1-dimensional trivial subrepresentation (see [14], Lemma 3.11.1). Moreover, Abdulrahim [1] showed by using a technique of complex specializations that the Gassner representation is the direct sum of the trivial representation and an .n 1/-dimensional irreducible representation. As for the Magnus representation for g;1 , Suzuki gave the following decomposition of ra after extending the target:

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Theorem 4.8 (Suzuki [113]). Let ˙1 R D ZŒ1˙1 ; : : : ; 2g ; 1=.1 gC1 /; : : : ; 1=.1 2g /:

Then the Magnus representation ra W g;1 ! GL.2g; R/ for the Torelli group with an extension of its target has a 1-dimensional subrepresentation which is not a direct summand. Moreover the quotient .2g 1/-dimensional representation has a .2g 2/-dimensional subrepresentation which is not a direct summand and whose quotient is a 1-dimensional trivial representation. In the proof, Suzuki gave a matrix P 2 GL.2g; R/ such that 1 0 1 C B 0 C B C B :: 1 0 P ra .'/P D B : ra .'/ C C B A @ 0 0 0 0 1

(4.1)

holds for any ' 2 g;1 with an irreducible representation ra0 W g;1 ! GL.2g 2; R/. Remark 4.9. Here we comment on the topological meaning of the above decomposition. For simplicity, we use the quotient field KH WD ZŒH .ZŒH f0g/1 of ZŒH and consider ra W g;1 ! GL.2g; KH /. The homology exact sequence shows that 0 ! H1 .†g;1 I KH / ! H1 .†g;1 ; fpgI KH / ! H0 .fpgI KH / ! 0 is exact and it can be written as 2g1 2g 0 ! KH ! KH ! KH ! 0: 2g ! KH coincides with @1 W C1 .†g;1 ; fpgI KH / ! C0 .†g;1 ; fpgI KH /. The map KH 2g Now the representation ra works as a transformation of H1 .†g;1 ; fpgI KH / Š KH 2g1 and we can check that it preserves H1 .†g;1 I KH / Š KH , namely we have a 2g subrepresentation as a transformation of H1 .†g;1 I KH /. Taking a basis of KH from 2g1 ones of KH and KH , we obtain the first decomposition corresponding to the upper left .2g 1/-matrix of (4.1). One more step is obtained by finding the vector !> @ @ @ @1 @2 @2g > 1 1 1 2g 11 1 g1 1 D 1 gC1 2g1 to be an invariant vector belonging to Ker @1 D H1 .†g;1 I KH / Š KH . A similar observation can be applied to the Gassner representation for Pn . In this case, however, the invariant vector corresponding to the trivial subrepresentation does not belong to


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the subspace Ker @1 , so that we cannot obtain an .n2/-dimensional subrepresentation from this. The following observation might be useful for further comparison of the two representations. Let L be a pure braid with g strings. Consider a closed tubular neighborhood of the union of the loops gC1 ; gC2 ; : : : ; 2g in †g;1 (see Figure 1) to be the image of an embedding W †0;gC1 ,! †g;1 of a g holed disk †0;gC1 as in Figure 3. †0;gC1

ι

Figure 3. The embedding W †0;gC1 ,! †g;1 .

Since Pg can be regarded as a subgroup of M0;gC1 , we have an injective homomorphism I W Pg ,! Mg;1 by a method similar to that mentioned in Remark 4.2. The construction of the map I is due to Oda [95] and Levine [74] (see also Gervais–Habegger [38]). As in the following way, we can compare the restriction of the universal Magnus representation r for Mg;1 to Pg with that for Aut Fg D Aut .1 †0;gC1 / denoted here by rG W Pg ! GL.g; ZŒ1 †0;gC1 /. Note that we are now identifying 1 †0;gC1 with the subgroup of generated by gC1 ; : : : ; 2g . By construction, we obtain the following: I 0g . Proposition 4.10. For any pure braid L 2 Pg , r.I.L// D g rG .L/ Here we must remark that the embedding Pg ,! Mg;1 has an ambiguity due to framings, which count how many times one applies Dehn twists along each of the loops parallel to the inner boundary of †0;gC1 . However we can check that the lower right part of r.I.L// is independent of the framings. While the entire image I.Pg / is not included in g;1 , we can easily check that I.ŒPg ; Pg / g;1 . Suppose L 2 Pg is in the kernel of the Gassner representation. Then L 2 ŒPg ; Pg (see [14], Theorem 3.14), so that I.L/ 2 g;1 . The symplecticity of r shows that the lower left part of ra .I.L// is O. Consequently, we have observed that for L 2 Pg , L is in the kernel of the Gassner representation if and only if I.L/ is in the kernel of the Magnus representation ra for g;1 .

4.4 Determinant of the Magnus representation Now we focus on the Magnus representation ra W Mg;1 ! GL.2g; ZŒH / as a crossed homomorphism, whose importance was first pointed out by Morita. We put k WD

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Takuya Sakasai

det B ra W Mg;1 ! .ZŒH / D ˙H , where ˙H is regarded as the multiplicative group of monomials in ZŒH . The image of k is included in H since t.k.'// D det. .'// D 1. We here turn the group H as a multiplicative group into the additive one as usual. Theorem 4.11 (Morita [84], [86]). H 1 .Mg;1 I H / Š Z for g 2 and it is generated by k. This cohomology class, which has many natural representatives as crossed homomorphisms arising from various contexts, is referred as to the Earle class in Kawazumi’s chapter [64] of the second volume of this Handbook. Consider the composition [

H 1 .Mg;1 I H / ˝ H 1 .Mg;1 I H / ! H 2 .Mg;1 I H ˝ H / ! H 2 .Mg;1 / and apply it to k ˝ k, where [ denotes the cup product and denotes the map which contracts the coefficient H ˝ H to Z by the intersection pairing on H . At the cocycle level, .k [ k/ is given by .k [ k/.Œ'j / D .k.'/; '.k. /// for ',

2 Mg;1 . Then the following was shown by Morita:

Theorem 4.12 (Morita [85]). For g 2, we have .k [ k/ D e1 2 H 2 .Mg;1 /; where e1 is the first Miller–Morita–Mumford class. With Meyer’s results [80], Harer [50] (see also Korkmaz–Stipsicz [67]) showed that H 2 .Mg;1 / Š Z for g 3 and it is known that e1 is twelve times the generator up to sign. We refer to Morita’s paper [83] and Kawazumi’s chapter [64] for the definition and generalities on the Miller–Morita–Mumford classes. Remark 4.13. Satoh [107] proved that H 1 .Aut Fn I H1 .Fn // Š Z for n 3 and we can check that the generator is represented by the crossed homomorphism j det ra j W Aut Fn ! H1 .Fn / sending ' 2 Aut Fn to h 2 H1 .Fn / with det.ra .'// D ˙h 2 ˙H1 .Fn /. In particular, the pullback map H 1 .Aut F2g I H / ! H 1 .Mg;1 I H / is an isomorphism under an identification H1 .F2g / Š H . However, there exist no corresponding statement to Theorem 4.12 since we do not have a natural intersection pairing on H1 .Fn /. In fact, Gersten [37] showed that H 2 .Aut Fn / Š Z=2Z for n 5. See Kawazumi [63], Theorem 7.2, for more details.


555

4.5 The Johnson filtration and Magnus representations A filtration of Mg;1 is obtained by taking intersections with the filtration fI k A2g g1 kD0 of Aut D Aut F2g . However, as far as the author knows, nothing is known about I k A2g \ Mg;1 for k 3. This reflects the difficulty in treating the derived series of F2g . In the study of Mg;1 , instead, the filtration arising from the lower central series of is frequently used. Recall that the lower central series 1 .G/ WD G 2 .G/ 3 .G/ of a group G is defined by kC1 .G/ D ŒG; k .G/ for k 1. The group k .G/ is a characteristic subgroup of G. We denote the k-th nilpotent quotient G=. k .G// of G by Nk .G/. Here N2 .G/ D H1 .G/. Let qk W ! Nk ./ be the natural projection. Consider the composition

k W Mg;1 ,! Aut ! Aut .Nk .// of the Dehn–Nielsen embedding and the map induced from qk . This defines a filtration Mg;1 Œ1 WD Mg;1 Mg;1 Œ2 Mg;1 Œ3 Mg;1 Œ4 called the Johnson filtration of Mg;1 by setting Mg;1 Œk WD Ker k . Note that Mg;1 Œ2 D g;1 . The corresponding filtration for Aut Fn was studied earlier by Andreadakis [4] and we here call filtration T . Since is known to Tit1the Andreadakis k be residually nilpotent, namely k1 ./ D f1g, we have 1 k1 Mg;1 Œk D f1g. The Andreadakis filtration of Aut Fn has a similar property. In the above cited paper, Andreadakis constructed an exact sequence 1 ! Hom.H; k ./= kC1 .// ! Aut .NkC1 .// ! Aut .Nk .// ! 1; from which we obtain a homomorphism

k WD kC2 jMg;1 ŒkC1 W Mg;1 Œk C 1 ! Hom.H; kC1 ./= kC2 .// with Ker k D Mg;1 Œk C 2, called the k-th Johnson homomorphism for k 1. That is, the successive quotients of the Johnson filtration are described by the Johnson homomorphisms. We refer to Johnson’s survey [60] for his original description and to the chapters of Morita [89] and Habiro–Massuyeau [49] for the details of these homomorphisms. The theory of the Johnson homomorphisms has been studied intensively by many researchers and is now highly developed (see Morita [88] for example). Remark 4.14. It is known that there exist non-tame automorphisms of Nk ./. In fact, Bryant–Gupta [17] showed that if n k 2, Aut .Nk .Fn // is generated by the tame automorphisms and one non-tame automorphism written explicitly. It follows from Andreadakis’ exact sequence that we may use Coker k to detect the non-tameness. Morita [87] studied Coker k by using his trace maps and showed they are non-trivial for general k.

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Corresponding to the Johnson filtration, we have a crossed homomorphism rqk W Mg;1 ! GL.2g; ZŒNk .//; whose restriction to Mg;1 Œk is a homomorphism rqk W Mg;1 Œk ! GL.2g; ZŒNk .// for each k 2. Note that rq2 D ra , the Magnus representation for g;1 . Problem 4.15. Determine whether rqk is faithful or not for k 3. Also, determine the image of rqk for k 2. As for the relationship between the Johnson filtration and Magnus representations, Morita [86] gave a method for computing k1 .'/ from rqk .'/ for ' 2 Mg;1 . For example, we can easily calculate 1 .'/ from det.rq2 .'//. Note also that Morita’s trace maps mentioned above are highly related to det rq2 . Here we pose the converse as a problem. Problem 4.16. Describe explicitly how we can reproduce rqk from the “totality” of the Johnson homomorphisms. Suzuki [114] showed that Mg;1 Œk 6 Ker rq2 for every k 2 by using the topological description of rq2 . Another approach to the Johnson homomorphisms using the Magnus expansion is studied by Kawazumi [63] (see also the chapters by Kawazumi [64] and Habiro– Massuyeau [49] in this Handbook). It would be interesting to compare his construction with Magnus representations.

4.6 Applications to three-dimensional topology We close the first part of this survey by briefly mentioning some relationships between the Magnus representation rq2 and three-dimensional topology. It also serves as an original model for the results discussed in the second part. There exist several methods for making a three-dimensional manifold from an element of Mg;1 such as Heegaard splittings, mapping tori and open book decompositions. We here recall the last two. For a diffeomorphism ' of †g;1 fixing @†g;1 pointwise, the mapping torus T'@ of ' is defined as T'@ WD †g;1 Œ0; 1=..x; 1/ D .'.x/; 0//;

x 2 †g;1 :

The manifold T'@ is a †g;1 -bundle over S 1 . We fill the boundary of T'@ by a solid torus S 1 D 2 , so that each disk fxg D 2 caps a fiber †g;1 ft g. Then we obtain a closed 3-manifold T' also called the mapping torus of '. If we change the attaching


557

of S 1 D 2 so that each disk fxg D 2 caps fqg S 1 .@†g;1 / S 1 D @T'@ , then we have another closed 3-manifold C' called the closure of '. We also say that C' has an open book decomposition. The core S 1 f.0; 0/g of the glued solid torus in C' is called the binding and ' is called the monodromy. Note that the above constructions of T'@ , T' and C' depend only on the isotopy class of ', so that they are well-defined for each element of Mg;1 . More precisely, they depend on the conjugacy class in Mg;1 . From the presentation D h1 ; 2 ; : : : 2g i of , we can easily obtain 1 T'@ D h1 ; 2 ; : : : 2g ; j i '.i /1 1 .1 i 2g/i; Q 1 T' D h1 ; 2 ; : : : 2g ; j jgD1 Œj ; gCj ; i '.i /1 1 ; .1 i 2g/i; 1 C' D h1 ; 2 ; : : : 2g j i '.i /1 .1 i 2g/i; where corresponds to the loop fpg S 1 in T'@ and T' . The (multi-variable) Alexander polynomial G is an invariant of finitely presentable groups. It can be regarded as an invariant of compact manifolds by considering their fundamental groups. For a finitely presentable group G, the polynomial G is computed from the Alexander module AZ .G/ WD H1 .GI ZŒH1 .G// by a purely algebraic procedure. We here omit the details and refer to Turaev’s book [118] for the definition and its relationship to torsions. For a knot group G.K/, the polynomial G.K/ with replaced by t coincides with the Alexander polynomial K .t / of K mentioned in Example 2.11. When ' 2 g;1 , H D H1 .†g;1 / is naturally embedded in H1 .T'@ /, H1 .T' / and H1 .CM /. Then we can easily check that the Magnus representation rq2 .'/ can be used to describe the multi-variable Alexander polynomials of T'@ , T' and C' . For example, we have : det.I2g rq2 .'// 2 ZŒH1 .T' / D ZŒH hi; 1 T' D .1 /2 : where D means that the equality holds up to multiplication by monomials. A generalization of this formula was given by Kitano–Morifuji–Takasawa [65] in their study of L2 -torsion invariants of mapping tori. Another application is given when C' D S 3 . In this case, we focus on the binding, which gives a knot K in S 3 , of the open book decomposition. Such a K is called a fibered knot. We can check that the Alexander polynomial K .t / is given by : K .t/ D det.I2g t 2 .'// D det.I2g t rt .'//:

(4.2)

Since H collapses to the trivial group in H1 .E.K// Š Z, we cannot readily have a formula which generalizes (4.2) by using rq2 . In Section 8.1, we discuss the details about this in a more general situation.

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Takuya Sakasai

5 Homology cylinders Now we start the second half of our survey. In this section, we introduce homology cylinders over a surface and give a number of examples. We also describe how Johnson homomorphisms are extended to the monoid and group of homology cylinders.

5.1 Definition and examples The definition of homology cylinders goes back to Goussarov [43], Habiro [48], Garoufalidis–Levine [35] and Levine [75] in their study of finite type invariants of 3-manifolds. Strictly speaking, the definition below is closer to that in [35] and [75]. Note that homology cylinders are called “homology cobordisms” in the chapter of Habiro–Massuyeau [49], where the terminology “homology cylinders” is used for a more restricted class of 3-manifolds. Definition 5.1. A homology cylinder over †g;n consists of a compact oriented 3manifold M with two embeddings iC ; i W †g;n ,! @M , called the markings, such that: (i) iC is orientation-preserving and i is orientation-reversing; (ii) @M D iC .†g;n /[i .†g;n / and iC .†g;1 /\i .†g;1 / D iC .@†g;n / D i .@†g;n /; (iii) iC j@†g;n D i j@†g;n ; (iv) iC ; i W H .†g;n / ! H .M / are isomorphisms, namely M is a homology product over †g;n . We denote a homology cylinder by .M; iC ; i / or simply M . Two homology cylinders .M; iC ; i / and .N; jC ; j / over †g;n are said to be Š

! N isomorphic if there exists an orientation-preserving diffeomorphism f W M satisfying jC D f BiC and j D f Bi . We denote by Cg;n the set of all isomorphism classes of homology cylinders over †g;n . We define a product operation on Cg;n by .M; iC ; i / .N; jC ; j / WD .M [i B.jC /1 N; iC ; j / for .M; iC ; i /, .N; jC ; j / 2 Cg;n , which endows Cg;n with a monoid structure. Here the unit is .†g;n Œ0; 1; id 1; id 0/, where collars of iC .†g;n / D .id 1/.†g;n / and i .†g;n / D .id 0/.†g;n / are stretched half-way along .@†g;n / Œ0; 1 so that iC .@†g;n / D i .@†g;n /. Example 5.2. For each diffeomorphism ' of †g;n which fixes @†g;n pointwise, we can construct a homology cylinder by setting .†g;n Œ0; 1; id 1; ' 0/ with the same treatment of the boundary as above. It is easily checked that the isomorphism class of .†g;n Œ0; 1; id1; ' 0/ depends only on the (boundary fixing)


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isotopy class of ' and that this construction gives a monoid homomorphism from the mapping class group Mg;n to Cg;n . In fact, it is an injective homomorphism (see Garoufalidis–Levine [35], Section 2.4, Levine [75], Section 2.1, Habiro–Massuyeau’s chapter in this volume ([49], Section 2.2) and Proposition 1 in [40]). By this example, we may regard Cg;n as an enlargement of Mg;n , where the usage of the word “enlargement” comes from the title of Levine’s paper [75]. In fact, we will see that the Johnson homomorphisms and Magnus representations for Mg;n are naturally extended. In [35], Garoufalidis–Levine further introduced homology cobordisms of homology cylinders, which give an equivalence relation among homology cylinders. Definition 5.3. Two homology cylinders .M; iC ; i / and .N; iC ; i / over †g;n are said to be homology cobordant if there exists a compact oriented smooth 4-manifold W such that: (1) @W D M [ .N /=.iC .x/ D jC .x/; i .x/ D j .x//; x 2 †g;n ; (2) the inclusions M ,! W , N ,! W induce isomorphisms on the integral homology. We denote by Hg;n the quotient set of Cg;n with respect to the equivalence relation of homology cobordism. The monoid structure of Cg;n induces a group structure of Hg;n . It is known that Mg;n can be embedded in Hg;n (see Cha–Friedl–Kim [21], Section 2.4). We call Hg;n the homology cobordism group of homology cylinders. Example 5.4. Homology cylinders were originally introduced in the theory of clasper (clover) surgery and finite type invariants of 3-manifolds due to Goussarov [43] and Habiro [48] independently. Since clasper surgeries do not change the homology of 3-manifolds, the theory fits well to the setting of homology cylinders. It is known that every homology cylinder is obtained from the trivial one by doing some clasper surgery and then changing the markings by the mapping class group (see Massuyeau– Meilhan [79]). While clasper surgery brings a quite rich structure to Cg;n , here we do not take it up in detail. See the chapter of Habiro–Massuyeau [49] and references therein. Another approach from the theory of finite type invariants to homology cylinders was obtained by Andersen–Bene–Meilhan–Penner [2]. The following constructions give us direct methods for obtaining homology cylinders whose underlying 3-manifolds are not product manifolds. Example 5.5. For each homology 3-sphere X, the connected sum ..†g;n Œ0; 1/ # X; id 1; id 0/ gives a homology cylinder. It can be checked that this correspondence is an injective monoid homomorphism from the monoid Z3 of all (integral) homology 3-spheres

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whose product is given by connected sum to Cg;n . In fact, it induces isomorphisms Z3 Š C0;1 Š C0;0 . Moreover, this construction is compatible with homology cobordisms, so that we have a homomorphism from the homology cobordism group ‚3Z to Cg;n , which is also shown to be injective (see Cha–Friedl–Kim [21], Proof of Theorem 1.1). It is a challenging problem to extract new information on ‚3Z from the theory of homology cylinders. At present, no result has been obtained. Example 5.6 (Levine [74]). A string link is a generalization of a braid defined by Habegger–Lin [47]. While we omit here the definition, the difference between the two notions is clear from Figure 4.

braid

string link Figure 4. Braid and string link.

From a pure string link L D 2 Œ0; 1 with g strings, we can construct a homology cylinder as follows. Recall the embedding W †0;gC1 ,! †g;1 of a g holed disk †0;gC1 D 2 in Section 4.2 and Figure 3. Let C be the complement of an open tubular neighborhood of L in D 2 Œ0; 1. For any choice of a framing of L, a homeomorphism Š

! @. .†0;gC1 / Œ0; 1/ is fixed. (Note that a string link with a framing h W @C itself can be regarded as a homology cylinder over †0;gC1 .) Then the manifold ML obtained from †g;1 Œ0; 1 by removing .†0;gC1 / Œ0; 1 and regluing C by h becomes a homology cylinder with the same marking as the trivial homology cylinder. This construction can be seen as a generalization of the embedding Pg ,! Mg;1 in Section 4.2 and it gives an injective monoid homomorphism from the monoid of pure string links to Cg;1 . Moreover it induces an injective group homomorphism from the concordance group of pure string links with g strings to Hg;1 . Habegger [46] gave another construction of homology cylinders from pure string links. Example 5.7 ([40], [42]). Let K be a knot in S 3 with a Seifert surface R of genus g. By cutting open the knot exterior E.K/ along R, we obtain a manifold MR . The boundary @MR is the union of two copies of R glued along their boundary circles, which are just the knot K. The pair .MR ; K/ is called the complementary sutured manifold of R. We can check that the following properties are equivalent to each other. (a) The Alexander polynomial K .t/ is monic and its degree is equal to twice the genus of g D g.K/ of K.


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(b) The Seifert matrix S of any minimal genus Seifert surface R of K is invertible over Z. (c) The complementary sutured manifold .MR ; K/ for any minimal genus Seifert surface R is a homology product over R. We call a knot having the above properties a homologically fibered knot, where the name comes from the fact that fibered knots satisfy them. Thus if we fix an identification of †g;1 with R for a homologically fibered knot, we obtain a homology cylinder over †g;1 . Note that aside from the name, the equivalence of the above conditions (a), (b), (c) was mentioned in Crowell–Trotter [30]. There exists a similar discussion for links. We close this subsection by two observations on connections between homology cylinders and the theory of 3-manifolds. First, the constructions of closed 3-manifolds mentioned in Section 4.6 have their analogue for homology cylinders. For example, the closure CM of a homology cylinder .M; iC ; i / 2 Cg;1 is defined as CM WD M=.iC .x/ D i .x//

x 2 †g;1 :

By a topological consideration, we see that this construction is the same as gluing †g;1 Œ0; 1 to M along their boundaries and also as the description of Habiro– Massuyeau [49], Definition 2.7. The closure construction is compatible with the homology cobordism relation, denoted by H-cob, namely we have the following commutative diagram: G closing Cg;1 / / fclosed 3-manifoldsg g0

G Hg;1 g0

closing

/ / fclosed 3-manifoldsg=(H-cob.)

Therefore, roughly speaking, Hg;1 might be regarded as a group structure on the set of homology cobordism classes of closed 3-manifolds. We have a similar discussion for clasper surgery equivalence. Second, irreducibility of 3-manifolds often plays an important role in the theory of 3-manifolds (see Hempel’s book [56] for generalities). Correspondingly, we define: Definition 5.8. A homology cylinder .M; iC ; i / 2 Cg;1 is said to be irreducible if irr the subset of Cg;1 the underlying 3-manifold M is irreducible. We denote by Cg;1 consisting of all irreducible homology cylinders. irr By a standard argument using irreducibility, we can show that Cg;1 is a submonoid irr irr of Cg;1 . In particular, C0;0 Š C0;1 Š f1g. Irreducible homology cylinders are all

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Haken manifolds since jH1 .M /j D 1 for any M 2 Cg;n unless .g; n/ D .0; 0/ or .0; 1/ For every .M; iC ; i / 2 Cg;1 , the underlying 3-manifold M has a prime decomposition of the form M Š M0 # X1 # X2 # # Xn ; where M0 is the unique prime factor having @M and X1 ; X2 ; : : : ; Xn are homology 3irr spheres. Note that .M0 ; iC ; i / 2 Cg;1 . Using Myers’ theorem (Theorem 3.2 in [91]), we have the following description on the homology cobordism group of irreducible homology cylinders: Proposition 5.9. Every homology cylinder in Cg;1 with g 1 is homology cobordant to an irreducible one. That is, irr =.H-cob./ D Hg;1 : Cg;1

5.2 Stallings’ theorem and the Johnson filtration From now on, we limit our discussion to the case where n D 1 as in the first part of this chapter. In this subsection, we briefly recall how to extend the (reduced versions of) Dehn–Nielsen embedding and Johnson homomorphisms to homology cylinders. Convention. We use the point p 2 @†g;1 as the common base point of †g;1 , iC .†g;1 /, i .†g;1 /, a homology cylinder M , etc. For a given .M; iC ; i / 2 Cg;1 , two homomorphisms iC ; i W 1 †g;1 ! 1 M are not generally isomorphisms. However, the following holds: Theorem 5.10 (Stallings [110]). Let A and B be groups and let f W A ! B be a 2-connected homomorphism. Then the induced map f W Nk .A/ ! Nk .B/ is an isomorphism for each k 2. Here, a homomorphism f W A ! B is said to be 2-connected if f induces an isomorphism on the first homology, and an epimorphism on the second homology. In this chapter, the words “Stallings’ theorem” always means Theorem 5.10. Using the epimorphism (2.1), we can see that two homomorphisms iC ; i W D 1 †g;1 ! 1 M are both 2-connected for any .M; iC ; i / 2 Cg;1 . Therefore, they induce isomorphisms on the nilpotent quotients of and 1 M . For each k 2, we can define a map k W Cg;1 ! Aut .Nk .// by

k .M; iC ; i / WD .iC /1 B i ; which gives a monoid homomorphism. It can be checked that k .M; iC ; i / depends only on the homology cobordism class of .M; iC ; i /, so that we have a group homomorphism k W Hg;1 ! Aut .Nk .//. The restriction of k to the subgroup


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Mg;1 Cg;1 coincides with the homomorphism k mentioned in Section 4.5. The homomorphisms k .k D 2; 3; : : :/ define filtrations Cg;1 Œ1 WD Cg;1 Cg;1 Œ2 Cg;1 Œ3 Cg;1 Œ4 ; Hg;1 Œ1 WD Hg;1 Hg;1 Œ2 Hg;1 Œ3 Hg;1 Œ4 called the Johnson filtration of Cg;1 and Hg;1 by setting Cg;1 Œk WD Ker k and Hg;1 Œk WD Ker k . By definition, the image of the homomorphism k is included in ˇ ³ ² ˇ There exists a lift 'Q 2 End of ' : Aut 0 .Nk .// WD ' 2 Aut .Nk .// ˇˇ satisfying '. / Q

mod kC1 ./. On the other hand, Garoufalidis–Levine and Habegger independently showed the following: Theorem 5.11 (Garoufalidis–Levine [35], Habegger [46]). For k 2, the image of

k coincides with Aut 0 .Nk .//. As seen in Section 4.5, the k-th Johnson homomorphism is obtained by restricting

kC2 to Cg;1 Œk C 1 and Hg;1 Œk C 1. Garoufalidis–Levine [35], Proposition 2.5, showed that Andreadakis’ exact sequence in Section 4.5 (with k shifted) restricts to the exact sequence 1 ! hg;1 .k/ ! Aut0 .NkC2 .// ! Aut0 .NkC1 .// ! 1: Here hg;1 .k/ Hom.H; kC1 ./= kC2 .// is defined as the kernel of the composition Hom.H; kC1 ./= kC2 .// Š H ˝ . kC1 ./= kC2 .// Š H ˝ . kC1 ./= kC2 .// D . 1 ./= 2 .// ˝ . kC1 ./= kC2 .// ! kC2 ./= kC3 ./; where we used the Poincaré duality H D H in the second row and the last map is obtained by taking commutators. Corollary 5.12. The k-th Johnson homomorphisms k W Cg;1 Œk C 1 ! hg;1 .k/ and

k W Hg;1 Œk C 1 ! hg;1 .k/ are surjective for any k 1. Recall that in the case of the mapping class group, all the k W Mg;1 ! Aut .Nk .// for k 2 are induced from a single homomorphism W Mg;1 ! Aut . Then we pose the following question: Does there exist a homomorphism Hg;1 ! Aut G for some group G which induces k W Hg;1 ! Aut .Nk .// for all k 2? One of the answers is to use the map nil W Hg;1 ! Aut . nil / obtained by combining the homomorphisms

k for all k 2, where nil WD lim Nk ./ is the nilpotent completion of . In fact, it k

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was shown by Bousfield [15] that Nk .G/ Š Nk .G nil / holds for any finitely generated group G and hence we have a natural homomorphism Aut .G nil / ! Aut .Nk .G//. However, G nil is in general enormous and difficult to treat. In Section 6, we introduce the acyclic closure (or HE-closure) G acy of a group G as a reasonable extension and apply it to our situation.

6 Magnus representations for homology cylinders I In this section, we extend the (universal) Magnus representation r to Cg;1 and Hg;1 . The construction is based on Le Dimet’s argument [70] for the extension of the Gassner representation for string links. However what we present here is its generalized version: We construct our extended representations as crossed homomorphisms and use more general (not necessarily commutative) rings. In the construction, there are two key ingredients: the acyclic closure of a group G and the Cohn localization ƒG of ZŒG. The former is used to give a generalization of the Dehn–Nielsen theorem with no reduction and then we construct the (universal) Magnus representation r for Cg;1 and Hg;1 with the aid of the latter.

6.1 Observation on fundamental groups of homology cylinders The definition of the acyclic closure of a group is given purely in terms of group theory, whose relationship to topology seems to be unclear at first glance. Here we digress and give an observation to see the background. For a homology cylinder .M; iC ; i / 2 Cg;1 , if we could have a natural assignment of an automorphism of , there would be no problem. However, it seems in general difficult (maybe impossible) to do so because 1 M can be “bigger” than 1 †g;1 : iC

1 †g;1 o

'

7 1 M

i

The observation we now start is intended to give an “estimation” of how big 1 M can be. The usual handle decomposition theory and Morse theory say that M , a homology cobordism over †g;1 , is obtained from †g;1 Œ0; 1 by attaching a number of 1-handles h11 ; h12 ; : : : ; h1m and the same number of 2-handles h21 ; h22 ; : : : ; h2m to †g;1 f1g. Then 1 M can be written as 1 M Š

hx1 ; x2 ; : : : ; xm i ; hr1 ; r2 ; : : : ; rm i

where xi corresponds to attaching h1i and rj to hj2 , and hx1 ; x2 ; : : : ; xm i denotes the free product of and hx1 ; x2 ; : : : ; xm i. Put Fm D hx1 ; x2 ; : : : ; xm i. The condition


565

H .M; i .†g;1 // D 0 implies that the image of fr1 ; r2 ; : : : ; rm g under the map proj:

Fm ! Fm ! H1 .Fm / forms a basis of H1 .Fm / Š Zm . Repeating Tietze transformations, we can rewrite the above presentation into one of the form 1 M Š

Fm 1 1 i hx1 v1 ; x2 v21 ; : : : ; xm vm

proj:

with vj 2 Ker. Fm ! Fm ! H1 .Fm // for j D 1; 2; : : : ; m. On the other hand, given a group of the form GD

Fm 1 1 i hx1 w1 ; x2 w21 ; : : : ; xm wm

(6.1)

proj:

with wj 2 Ker. Fm ! Fm ! H1 .Fm // for j D 1; 2; : : : ; m, we can construct a cobordism W over †g;1 Œ0; 1 with 1 W Š G by attaching (4-dimensional) 1-handles h11 ; h12 ; : : : ; h1m and 2-handles h21 ; h22 ; : : : ; h2m to .†g;1 Œ0; 1/ f1g .†g;1 Œ0; 1/ Œ0; 1 according to the words xi wi1 . We denote by M the opposite side of .†g;1 Œ0; 1/ f0g in @W , namely @W D .†g;1 Œ0; 1/ [ .M /. By construction, the manifold M with the same markings as .†g;1 Œ0; 1; id 1; id 0/ defines a homology cylinder in Cg;1 and W is a homology cobordism between M and †g;1 Œ0; 1. By the duality of handle decompositions, the cobordism W is also obtained from M Œ0; 1 by attaching 2-handles and 3-handles. Therefore we have a surjective homomorphism 1 M 1 W Š G. That is, roughly speaking, 1 M is “bigger” than G. (In higher-dimensional cases discussed in Section 8.4, we have an isomorphism 1 M Š G.) Consequently, we have: Proposition 6.1. The fundamental group 1 M of .M; iC ; i / 2 Cg;1 can be written proj:

in the form (6.1) for some m with wj ’s in Ker. Fm ! Fm ! H1 .Fm //. Conversely, for any group G having such a form, there exists a homology cylinder .M; iC ; i / 2 Cg;1 such that 1 M surjects onto G.

6.2 The acyclic closure of a group The concept of the acyclic closure (or HE-closure in [72]) of a group was defined as a variation of the algebraic closure of a group by Levine [71], [72]. Topologically, the algebraic (acyclic) closure of a group G can be obtained as the fundamental group of (a variation of) the Vogel localization of any CW-complex X with 1 X Š G (see Le Dimet’s book [69]). We summarize here the definition and fundamental properties. We also refer to Hillman’s book [57] and Cha’s paper [20]. The proofs of the propositions in this subsection are almost the same as those for the algebraic closures in [71].

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Definition 6.2. Let G be a group, and let Fm D hx1 ; x2 ; : : : ; xm i be a free group of rank m. (i) w D w.x1 ; x2 ; : : : ; xm / 2 G Fm , a word in x1 ; x2 ; : : : ; xm and elements of G, is said to be acyclic if

proj

w 2 Ker G Fm ! Fm ! H1 .Fm / : (ii) Consider the following “equation” with variables x1 ; x2 ; : : : ; xm : 8 x1 D w1 .x1 ; x2 ; : : : ; xm /; ˆ ˆ ˆ ˆ <x D w .x ; x ; : : : ; x /; 2 2 1 2 m : ˆ :: ˆ ˆ ˆ : xm D wm .x1 ; x2 ; : : : ; xm /: When all words w1 ; w2 ; : : : ; wm 2 G Fm are acyclic, we call such an equation an acyclic system over G. (iii) A group G is said to be acyclically closed (AC, for short) if every acyclic system over G with m variables has a unique “solution” in G for any m 0, where a “solution” means a homomorphism ' that makes the following diagram commutative: G RRRR RRR RRR id RRR RRR RRR RR( G Fm / G. 1 1 ' hx1 w1 ; : : : ; xm wm i Example 6.3. Let G be an abelian group. For g1 ; g2 ; g3 2 G, consider the equation ´ x1 D g1 x1 g2 x2 x11 x21 ; x2 D x1 g3 x11 ; which is an acyclic system. Then we have a unique solution x1 D g1 g2 ; x2 D g3 . As we can expect from this example, all abelian groups are AC. Moreover, all nilpotent groups and the nilpotent completion of a group are AC, which can be deduced from the following fundamental properties of AC-groups: Proposition 6.4 ([71], T Proposition 1). (a) Let fG˛ g˛ be a family of AC-subgroups of an AC-group G. Then ˛ G˛ is also an AC-subgroup Q of G. (b) Let fG˛ g˛ be a family of AC-groups. Then ˛ G˛ is also an AC-group. (c) When G is a central extension of H , then G is an AC-group if and only if H is an AC-group. (d) For any direct system (resp. inverse system) of AC-groups, the direct limit (resp. inverse limit) is also an AC-group.


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Let us define the acyclic closure of a group. Proposition 6.5 ([71], Proposition 3). For any group G, there exists a pair of a group G acy and a homomorphism G W G ! G acy satisfying the following properties: (1) G acy is an AC-group. (2) Let f W G ! A be a homomorphism and suppose that A is an AC-group. Then there exists a unique homomorphism f acy W G acy ! A which satisfies f acy B G D f . Moreover such a pair is unique up to isomorphism. Definition 6.6. We call G (or G acy ) obtained above the acyclic closure of G. Taking the acyclic closure of a group is functorial, namely, for each group homoacy acy morphism f W G1 ! G2 , we have the induced homomorphism f acy W G1 ! G2 acy by applying the universal property of G1 to the homomorphism G2 B f , and the composition of homomorphisms induces that of the corresponding homomorphisms on acyclic closures. The most important properties of the acyclic closure are the following: Proposition 6.7 ([71], Proposition 4). For every group G, the acyclic closure G W G ! G acy is 2-connected. Proposition 6.8 ([71], Proposition 5). Let G be a finitely generated group and H a finitely presentable group. For each 2-connected homomorphism f W G ! H , the induced homomorphism f acy W G acy ! H acy on acyclic closures is an isomorphism. From Proposition 6.7 and Stallings’ theorem, the nilpotent quotients of a group and those of its acyclic closure are isomorphic. Note that the homomorphism G is not necessarily injective: consider a perfect group G and the 2-connected homomorphism G ! f1g. As for a free group Fm , its residual nilpotency shows that Fm is injective. Proposition 6.9 ([71], Proposition 6). For any finitely presentable group G, there exists a sequence of finitely presentable groups and homomorphisms G D P0 ! P1 ! P2 ! ! Pk ! PkC1 ! satisfying the following properties: (1) G acy D lim Pk , and G W G ! G acy coincides with the limit map of the above !k sequence. (2) G ! Pk is a 2-connected homomorphism for any k 1. From this proposition, we see, in particular, that the acyclic closure of a finitely presentable group is a countable set.

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6.3 Dehn–Nielsen type theorem Now we return to our discussion on homology cylinders. For each homology cylinder .M; iC ; i / 2 Cg;1 , we have a commutative diagram

i

iC

1 M

/ 1 M o

acy

acy i

Š

/ .1 M /acy o

acy

iC

Š

acy

by Proposition 6.8. From this, we obtain a monoid homomorphism defined by

acy W Cg;1 ! Aut . acy /;

.M; iC ; i / 7! .iC /1 B iacy ; acy

and it induces a group homomorphism acy W Hg;1 ! Aut . acy /. Here we describe a generalization of the Dehn–Nielsen theorem. Recall that 2 acy is a word corresponding to the boundary loop of †g;1 . Theorem 6.10 ([103], Theorem 6.1). The image of acy W Hg;1 ! Aut . acy / is Aut 0 . acy / WD f' 2 Aut . acy / j '. / D 2 acy g: In the proof, we immediately see that the image of acy is included in Aut0 . acy / since iC . / D i . / 2 1 M for every .M; iC ; i / 2 Cg;1 . Conversely, given an element ' 2 Aut 0 . acy /, we need to construct a homology cylinder M D .M; iC ; i / satisfying acy .M / D '. The construction is based on that of Theorem 5.11 due to Garoufalidis–Levine ([35], Theorem 3). In our context, however, we must pay extra attention because there is a difference between our situation and theirs: Although the composition ! acy ! Nk . acy / Š Nk ./ is surjective, W ! acy is not. Note that Hg;1 É1Ê WD Ker. acy / is non-trivial in contrast with the case of the mapping class group. Indeed the homology cobordism group ‚3Z of homology 3spheres is included in it. See also Section 8.2 for more about Hg;1 É1Ê. As for the mysterious group Hg;1 É1Ê, we have the following problem: Problem 6.11. Determine whether Hg;1 É1Ê coincides with the group \ Hg;1 Œ1 WD Ker. k W Hg;1 ! Aut .Nk ./// k2

D Ker. nil W Hg;1 ! Aut . nil //; in which Hg;1 É1Ê is included. This is closely related to the question of whether the natural map acy ! nil is injective or not.

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6.4 Extension of the Magnus representation As in the original case, the (universal) Magnus representation for homology cylinders acy is obtained from that for Aut .Fn / through the Dehn–Nielsen type theorem. The resulting representation is also a crossed homomorphism. For the construction, we need another tool called (a special case of) the Cohn localization or the universal localization. We refer to Section 7 of [29] for details. Proposition 6.12 (Cohn [29]). Let G be a group with the trivializer t W ZŒG ! Z. Then there exists a pair of a ring ƒG and a ring homomorphism lG W ZŒG ! ƒG satisfying the following properties: (1) For every matrix m with coefficients in ZŒG, if t m is invertible then invertible.

lG

m is also

(2) The pair .ƒG ; lG / is universal among all pairs having the property .1/. Furthermore the pair .ƒG ; lG / is unique up to isomorphism. Note that any automorphism of a group G induces an automorphism of ZŒG and moreover of ƒG by the universal property of ƒG . Example 6.13. When G D H1 .Fn /, we have ˚ ˇ ƒH1 .Fn / Š fg ˇ f; g 2 ZŒH1 .Fn /; t.g/ D ˙1 : acy

We write i again for the image of i by Fn W Fn D h1 ; 2 ; : : : ; n i ,! Fn . Now we can check the following facts on ƒFnacy :

Fn

acy

lF acy n

Lemma 6.14. (1) The composition ZŒFn ! ZŒFn ! ƒFnacy is injective.

(2) Let G be a finitely presentable group and let f W Fn ! G be a 2-connected homomorphism. Then Hi .G; f .Fn /I ƒG / D 0 holds for i D 0; 1; 2. Moreover, we acy may take Fn as G. acy

Sketch of proof. Consider the composition of the ring homomorphism ZŒFn ! ZŒFnnil with the Magnus expansion, which can be extended to ZŒFnnil . It is known that the Magnus expansion is injective on ZŒFn . We can check that this composition satisfies Property .1/ of Proposition 6.12, so that the Magnus expansion can be extended to ƒFnacy . Hence .1/ follows. For the proof of the first assertion of .2/, see [103], Lemma 5.11. We may put acy G D Fn by Proposition 6.9 and commutativity of homology and direct limits. Lemma 6.14 .2/ leads us to show the following, which can be regarded as a generalization of the isomorphism (2.2). The proof is almost the same as that of Proposition 1.1 in [70].

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Proposition 6.15. The homomorphism

sending .a1 ; : : : ; an />

W ƒnF acy ! I.Fnacy / ˝ZŒFnacy ƒFnacy n P n 2 ƒF acy to niD1 .i1 1/ ˝ ai is an isomorphism of right n

acy

acy

ƒFnacy -modules, where I.Fn / WD Ker.t W ZŒFn ! Z/. Definition 6.16. For 1 i n, we define the extended Fox derivative @ W Fnacy ! ƒFnacy @i by the formula @ @ > acy @ ; ;:::; W Fn ! ƒnF acy ; n @1 @2 @n

v 7! 1 ..v 1 1/ ˝ 1/:

The extended Fox derivatives coincide with the original ones if we restrict them to Fn (cf. Example 2.10). They share many properties as mentioned in [70], Proposition 1.3. In particular, we have the equality n X @v .i1 1/ ˝ 2 I.Fnacy / ˝ZŒFnacy ƒFnacy .v 1 1/ ˝ 1 D @i iD1

for any v 2

acy Fn . acy

Definition 6.17. The (universal ) Magnus representation for Aut .Fn / is the map r W Aut .Fnacy / ! M.n; ƒFnacy / acy

assigning to ' 2 Aut .Fn / the matrix r.'/ WD

@'.j / @i

! : i;j

We can easily check that the Magnus representation r is a crossed homomorphism and the image of r is included in the set GL.n; ƒFnacy / of invertible matrices. By Lemma 6.14 .1/, we see that the Magnus representation defined here gives a generalization of the original. Example 6.18. Consider the monoid End2 .Fn / of all 2-connected endomorphisms of acy Fn . We have a natural homomorphism End2 .Fn / ! Aut .Fn / by Proposition 6.8. For any f 2 End2 .Fn /, the Magnus matrix r.f / can be obtained by using the original Fox derivatives. In particular, r is injective on End2 .Fn /. Therefore, we see acy that End2 .Fn / is a submonoid of Aut .Fn /. Every automorphism of Aut .Nk .Fn // can be lifted to a 2-connected endomorphism of Fn . Hence the homomorphisms acy acy Aut .Fn / ! Aut .Nk .Fn // Š Aut .Nk .Fn // are surjective for all k 2.


571

Finally, by using the Dehn–Nielsen type theorem for n D 2g, we obtain the (universal) Magnus representation r W Cg;1 ! GL.2g; ƒ acy / for homology cylinders, which induces r W Hg;1 ! GL.2g; ƒ acy /.

7 Magnus representations for homology cylinders II In this section, we discuss another method for extending Magnus representations by using twisted homology of homology cylinders. This time, we follow the Kirk– Livingston–Wang’s construction [66]. In connection with it, we also mention another invariant of homology cylinders arising from torsion. For our purpose, we first recall the setting of higher-order Alexander invariants originating in Cochran–Orr–Teichner [27], Cochran [24] and Harvey [51], [52], where PTFA groups play an important role. A group is said to be poly-torsion-free abelian (PTFA) if it has a sequence D 1 F 2 F F n D f1g whose successive quotients i =iC1 .i 1/ are all torsion-free abelian. An advantage of using PTFA groups is that the group ring ZŒ of is known to be an Ore domain so that it can be embedded into the field (skew field in general) K WD ZŒ.ZŒ f0g/1 called the right field of fractions. We refer to the books of Cohn [29] and Passman [99] for generalities of localizations of non-commutative rings. A typical example of a PTFA group is Zn , where KZn is isomorphic to the field of rational functions with n variables. More generally, free nilpotent quotients Nk .Fn / and Nk ./ are known to be PTFA. Let M D .M; iC ; i / 2 Cg;1 . We fix a homomorphism W 1 M ! into a PTFA group . The following lemma is crucial in our construction of Magnus matrices (cf. Lemma 6.14). For the direct proof, see Proposition 2.1 of [66]. See also Cochran–Orr–Teichner [27], Section 2, for a more general treatment. Lemma 7.1. For ˙ 2 fC; g, we have H .M; i˙ .†g;1 /I K / D 0. Equivalently, K / ! H .M; fpgI K / i˙ W H .†g;1 ; fpgI i˙

is an isomorphism of right K -vector spaces. Remark 7.2. The same conclusion as in the above lemma holds for the homology with coefficients in any ZŒ1 M -algebra R satisfying: Every matrix with entries in ZŒ1 M sent to an invertible one by the trivializer t W ZŒ1 M ! Z is also invertible

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in R (cf. Proposition 6.12). By a theorem of Strebel [111], we see that K satisfies this property for any homomorphism 1 M ! into a PTFA group . S Since S WD 2g iD1 i †g;1 (see Figure 1) is a deformation retract of †g;1 relative to p, we have Š 1 S and H1 .†g;1 ; fpgI i˙ K / Š H1 .S; fpgI i˙ K / D C1 .Sz/ ˝ZŒ i˙ K Š K2g

e

z ˝ZŒ i K as a right K -vector space with basis f 1 ˝ 1; : : : ; 2g ˝ 1g C1 .S/ ˙ (see Section 4.2). Definition 7.3. For M D .M; iC ; i / 2 Cg;1 and a homomorphism 1 M ! into a PTFA group , the Magnus matrix r .M / 2 GL.2g; K / associated with is defined as the representation matrix of the right K -isomorphism i

K2g Š H1 .†g;1 ; fpgI i K / ! H1 .M; fpgI K / Š

1 iC

! H1 .†g;1 ; fpgI iC K / Š K2g ; Š

where the first and the last isomorphisms use the basis mentioned above. A method for computing r .M / is given in Section 4 of [40], which is based on one of Kirk–Livingston–Wang [66]. An admissible presentation of 1 M is defined to be one of the form hi .1 /; : : : ; i .2g /; z1 ; : : : ; zl ; iC .1 /; : : : ; iC .2g / j r1 ; : : : ; r2gCl i

(7.1)

for some integer l 0. That is, it is a finite presentation with deficiency 2g whose generating set contains i .1 /; : : : ; i .2g /; iC .1 /; : : : ; iC .2g / and is ordered as above. Such a presentation always exists. For any admissible presentation, we define 2g .2g C l/, l .2g C l/ and 2g .2g C l/ matrices A, B, C by @rj @rj @rj ; BD ; C D AD 1il @i .i / 1i 2g @zi @iC .i / 1i2g 1j 2gCl

1j 2gCl

1j 2gCl

over ZŒ K . A is invertible over K Proposition 7.4 ([40], Proposition 3). The square matrix B and we have 1 I2g A r .M / D C 2 GL.2g; K /: (7.2) 0.l;2g/ B Remark 7.5. We shall meet the same formula (7.2) when we compute the Magnus matrix following the definition in the previous section. From this, we can conclude that the definitions in this and the previous sections are the same. Formula (7.2) gives the following properties of Magnus matrices:

573


Proposition 7.6. Let be a PTFA group. (1) For ' 2 Mg;1 ,! Aut and a homomorphism W 1 .†g;1 Œ0; 1/ D ! , we have ! @'.j / r ..†g;1 Œ0; 1; id 1; ' 0// D : @i i;j

(2) (Functoriality) For M; N 2 Cg;1 and a homomorphism W 1 .M N / ! , we have r .M N / D rBi .M / rBj .N /; where i W 1 M ! 1 .M N / and j W 1 N ! 1 .M N / are the induced maps from the inclusions M ,! M N and N ,! M N . (3) (Homology cobordism invariance ) Suppose M; N 2 Cg;1 are homology cobordant by a homology cobordism W . For any homomorphism W 1 W ! , we have rBi .M / D rBj .N /; where i W 1 M ! 1 W and j W 1 N ! 1 W are the induced maps from the inclusions M ,! W and N ,! W . Hence Magnus matrices are invariants of pairs of a homology cylinder and a homomorphism . To obtain a map from a submonoid of Cg;1 solely, we need a natural choice of for all homology cylinders involved that have some compatibility with respect to the product operation in Cg;1 . For that purpose, we here use the nilpotent quotient Nk ./ with fixed k 2. Using Stallings’ theorem, we can consider the composition 1 iC

qk W 1 M ! Nk .1 M / ! Nk ./ Š

for every .M; iC ; i / 2 Cg;1 . Then we have a map rqk W Cg;1 ! GL.2g; KNk ./ / and Proposition 7.6 can be rewritten as follows: Proposition 7.7 ([105]). Let k 2 be an integer. (1) The map rqk extends the corresponding Magnus representation for Mg;1 . (2) The map rqk is a crossed homomorphism, namely, the equality rqk .M1 M2 / D rqk .M1 / k .M1 / .rqk .M2 // holds for any M1 ; M2 2 Cg;1 by using k W Cg;1 ! Aut .Nk .//. (3) rqk induces a crossed homomorphism rqk W Hg;1 ! GL.2g; KNk ./ /.

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As in the case of Mg;1 , the restrictions of rqk to Cg;1 Œk and Hg;1 Œk give genuine homomorphisms. We can naturally generalize the arguments in Sections 4.2 and 4.3. For example, the (twisted) symplecticity rqk .M /> qkJz rqk .M / D k .M / .qkJz/ 2 GL.2g; KNk ./ /

(7.3)

holds. Note that the proof in [104] is also applicable to the universal Magnus representation r. Remark 7.8. The author does not know whether we can define (crossed) homomorphisms from Cg;1 and Hg;1 by using derived quotients of 1 M . This is because there are no results for derived quotients of groups corresponding completely to Stallings’ theorem except that Cochran–Harvey [25] gave a partial result, which was used to define homology cobordism invariants of 3-manifolds arising from L2 -signature invariants (see Harvey [53] for example). Example 7.9 ([105], Example 4.4). Let L be the pure string link of Figure 5 with 2 strings.

iC .3 /

iC .4 / z

i .3 /

i .4 /

Figure 5. String link L.

By Levine’s construction given in Example 5.6, L yields a homology cylinder .ML ; iC ; i / 2 Cg;1 with 1 ML having an admissible presentation: *

i .1 /; : : : ; i .4 / z iC .1 /; : : : ; iC .4 /

iC .1 /i .3 /1 iC .4 /i .1 /1 ; ŒiC .1 /; iC .3 /iC .2 /zi .2 /1 Œi .3 /; i .1 /; iC .4 /i .3 /iC .4 /1 z 1 ; i .3 /iC .3 /1 i .3 /1 z; i .4 /z 1 iC .4 /1 z

+ :

Let us compute the Magnus matrix rq2 .ML /. We identify H D N2 ./ and N2 .1 ML / D H1 .ML / by using iC . From the presentation, we have z D i .3 / D

575


3 , i .4 / D 4 , i .2 / D 2 3 and i .1 / D 1 31 4 in H . Then 0 1 1 31 1 0 0 0 C B 0 1 0 0 0 B 1 C A 1 1 1 C; 1 1 0 DB 3 3 4 3 1 1 4 B C B @ 0 A 0 0 0 1 1 1 1 3 3 3 4 0 2 0 1 1 1 1 3 0 0 0 B 0 C 1 0 0 0 C; C DB 1 @ 0 0 1 0 A 1 1 0 1 31 0 3 11 3 A 1 I4 over ZŒH . The Magnus matrix rq2 .ML / D C B 0.1;4/ is given by 0 B B B B B B B B B B @

1

0

0

0

0

1

0

0

11 1 3 C41 1

21 31 41 41 C1

31 1 3 C41 1

41 .41 1/

31 C41 1

11 3 41

.131 /.21 31 21 1/

31 1

31 41 C31 C241 1

31 C41 1

31 C41 1

31 C41 1

31 C41 1

Note that det.rq2 .ML // D 31 41

1 C C C C C C: C C C C A

31 C41 1

3 C 4 1 : 31 C 41 1

Since rq2 .ML / has an entry not belonging to ZŒH , we see that ML is not in Mg;1 . In other words, L is not a braid. We close this section by introducing another invariant of homology cylinders called the -torsion. We refer to Milnor [82], Turaev [116] and Rosenberg [102] for generalities of torsions and basics of K1 -group. Here we only recall that for a ring R, the abelian group K1 .R/ is defined as the abelianization of the group GL.R/ D lim GL.n; R/ of invertible matrices with entries in R. By Lemma 7.1, !n the relative complex C .M; iC .†g;1 /I K / obtained from any cell decomposition of .M; iC .†g;1 // is acyclic, so that the torsion .C .M; iC .†g;1 /I K // can be defined. Definition 7.10. Let M D .M; iC ; i / 2 Cg;1 with a homomorphism W 1 M ! into a PTFA group . The -torsion C .M / of M is defined by

C .M / WD .C .M; iC .†g;1 /I K // 2 K1 .K /= ˙ .1 M /: Note that for any field K, the Dieudonné determinant gives an isomorphism K1 .K/ Š H1 .K /, where K D K f0g denotes the unit group. The -torsion

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Takuya Sakasai

is trivial when .M; iC ; i / 2 Cg;1 is contained in Mg;1 since M D †g;1 Œ0; 1 collapses to iC .†g;1 /. The -torsion satisfies the following properties: Proposition 7.11. Suppose is a PTFA group and M; N 2 Cg;1 . .1/ For a homomorphism W 1 M ! , the -torsion C .M / can be computed from A any admissible presentation of 1 M and is given by B 2 K1 .K /=˙.1 M /. .2/ (Functoriality) For a homomorphism W 1 .M N / ! , we have C C

C .M N / D Bi .M / Bj .N /;

where i W 1 M ! 1 .M N / and j W 1 N ! 1 .M N / are the induced maps from the inclusions M ,! M N and N ,! M N . By an argument similar to rqk , we can obtain a crossed homomorphism

qCk W Cg;1 ! K1 .KNk ./ /=.˙Nk .// for D Nk ./. Example 7.12. For the homology cylinder ML in Example 7.9, we have det. qC2 .ML // D 1 C 3 3 41 : Since it is non-trivial, we see again that ML … Mg;1 .

8 Applications of Magnus representations to homology cylinders In this final section we present a number of applications of Magnus representations to homology cylinders. The following subsections are independent of each other.

8.1 Higher-order Alexander invariants and homologically fibered knots Let G be a group and let W G ! be a homomorphism into a PTFA group . For a pair .G; /, the higher-order Alexander module A .G/ is defined by A .G/ WD H1 .GI ZŒ/; where ZŒ is regarded as a ZŒG-module through . Higher-order Alexander invariants generally indicate invariants derived from A .G/. After having been defined and developed by Cochran–Orr–Teichner [27], Cochran [24] and Harvey [51], [52], many applications to the theory of knots and 3-manifolds were obtained. In the theory of higher-order Alexander invariants, one of the important problems is to find methods for computing the invariants explicitly and extract topological information from


577

them. This problem arises from the difficulty in non-commutative rings involved in the definition. Let K be a knot in S 3 . We fix a homomorphism W G.K/ D 1 .E.K// ! into a PTFA group . It was shown in Cochran–Orr–Teichner [27], Section 2, and Cochran [24], Section 3, that H .E.K/I K / D 0 if is non-trivial. Then we can define the torsion

.E.K// WD .C .E.K/I K // 2 K1 .K /= ˙ .G.K//: Friedl observed in [33] that this torsion .E.K// can be regarded as a higherorder Alexander invariant for G.K/. In the case where is the abelianization map K .t / 1 W G.K/ ! ht i, Milnor’s formula [81] 1 .E.K// D 1t is recovered. We now try to understand the higher-order invariant .E.K// for a homologically fibered knot K by factorizing it into the invariants we have seen in the previous section. The formula is given as follows: Theorem 8.1 ([42], Theorem 3.8). Let K be a homologically fibered knot with a minimal genus Seifert surface R of genus g and let .MR ; iC ; i / 2 Cg;1 be the corresponding homology cylinder. For any non-trivial homomorphism W G.K/ ! into a PTFA group , a loop representing the meridian of K satisfies ./ ¤ 1 2 and we have a factorization

.E.K// D

C .MR / .I2g ./r .MR // 1 ./

2 K1 .K /= ˙ .G.K//

(8.1)

of the torsion .E.K//. When K is a fibered knot and D 1 , the abelianization map, we recover the formula (4.2) by using Milnor’s formula mentioned above. The explicit computation of .E.K// is still difficult after the factorization (8.1) in general. However, when we consider the projection 2 W G.K/ ! G.K/=G.K/.2/ to the metabelian quotient, which is known to be PTFA (see Strebel [111]), then the situation gets interesting as follows. In the group extension 1 ! G.K/.1/ =G.K/.2/ ! G.K/=G.K/.2/ ! G.K/=G.K/.1/ Š Z ! 1; we have G.K/.1/ =G.K/.2/ Š H1 .R/ Š H1 .MR / since it coincides with the first homology of the infinite cyclic covering of E.K/, which can be seen as the product of infinitely many copies of MR . In particular, we may regard H Š H1 .MR / as a natural, independent of choices of minimal genus Seifert surfaces, subgroup of G.K/=G.K/.2/ . We can easily observe that C2 .MR / D qC2 .MR / and r2 .MR / D rq2 .MR /, namely they can be determined by computations on a commutative subfield KH Š KH1 .MR / in KG.K/=G.K/.2/ .

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Takuya Sakasai

Remark 8.2. From the formula (8.1) with the above observation, it seems reasonable to say that after applying the Dieudonné determinant, C2 .MR / D qC2 .MR / is the “bottom coefficient” of 2 .E.K// with respect to ./. Note that qC2 .MR / may be regarded as a special case of a decategorification of the sutured Floer homology as shown by Friedl–Juhász–Rasmussen [34]. Example 8.3 ([42], Example 6.8). Let K and K 0 be the knots obtained as the boundaries of the Seifert surfaces R and R0 in Figure 6. Here the side with the darker color in R and R0 means the C-side.

2 2

1

1

R0

R

p

p z2

z3

z4 z6

z5

z1

z3

z2

z1

z7 z8

z9

Figure 6. Homologically fibered knots K and K 0 (Pictures are taken from [42].)

K 0 is the trefoil knot, which is a fibered knot with fiber R0 . We can easily check that K is a homologically fibered knot with a minimal genus Seifert surface R. It is known that .MR ; iC ; i /, .MR0 ; jC ; j / give homology cobordant homology cylinders in C1;1 . An admissible presentation of 1 MR is given by *

i .1 /; i .2 / z1 ; : : : ; z9 iC .1 /; iC .2 /

z1 z2 z3 ; z1 z9 z8 ; z4 z5 z41 z21 ; z41 z5 z31 z51 ; z3 z6 z31 z4 ; z7 z5 z8 z51 ; z71 z9 z7 z51 ; i .1 /z1 z7 z41 z2 z51 z3 z81 z5 ; i .2 /z81 z7 z41 z11 ; iC .1 /z7 z41 z2 z51 z3 z81 z5 ; iC .2 /z7 z41 z11

+ :

579


From this, we have 2 1 1 2 2 1 C 1 C 2 ; det. qC2 .MR // D 3 1 2 2 1 1 1 1 2 rq2 .MR / D ; 1 1 2 1 11 where the value of det. qC2 .MR // shows that K is not fibered. On the other hand, an admissible presentation of 1 MR0 is given by + * i .1 /; i .2 / z1 ; z2 ; z3 iC .1 /; iC .2 /

z1 z2 z3 ; i .1 /z31 ; i .2 /z31 z11 ; iC .1 /z2 ; iC .2 /z11

and we have det. qC2 .MR // D rq2 .MR / D

1 ; 2

1

11 2

21 : 1 11

From this example, we see that the torsion C is not preserved under homology cobordism relation in general. See also the formula (8.3) mentioned later. More examples are exhibited in [42] with particular interest in non-fiberedness of homologically fibered knots.

8.2 Bordism invariants and signature invariants In this subsection, we introduce two kinds of invariants of homology cylinders of topological nature: bordism invariants and signature invariants. Then we discuss how Magnus matrices behave in their interrelationship. Let us first introduce bordism invariants, which naturally generalize those for Mg;1 given by Heap [55]. Since (the infinitesimal version of) those homomorphisms are fully discussed in the chapter of Habiro–Massuyeau ([49], Section 3.3), we here recall it only briefly. Š

Let .M; iC ; i / 2 Cg;1 Œk. Then we have iC D i W Nk ./ ! Nk .1 M /. Consider the composition .iC /1 D .i /1

fM W M ! K.1 M; 1/ ! K.Nk .1 M /; 1/ ! K.Nk ./; 1/ of continuous maps. We can assume fM B iC D fM B i W †g;1 ! K.Nk ./; 1/ after adjusting by homotopy, if necessary. fM induces a continuous map fQM W CM ! K.Nk ; 1/ from the closure CM of M . Define a map k W Cg;1 Œk ! 3 .Nk .// by k .M; iC ; i / WD .CM ; fQM /; where 3 .Nk .// denotes the third bordism group of K.Nk ./; 1/.

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Then we have the following. Theorem 8.4 ([103], Theorem 7.1). For k 2, k is a homomorphism and factors through Hg;1 Œk. Moreover, the induced homomorphism k W Hg;1 Œk ! 3 .Nk .// gives an exact sequence k

1 ! Hg;1 Œ2k 1 ! Hg;1 Œk ! 3 .Nk .// ! 1: Sketch of proof. The proof is divided into the following steps: (1) k factors through Hg;1 Œk; (2) k is actually a homomorphism; (3) k is onto; (4) Ker k D Hg;1 Œ2k 1. (1) and (2) follow from standard topological constructions. We use arguments in Orr [96] and Levine [71] to reduce the proof of (3) to that of Theorem 6.10. The proof of (4) proceeds as follows. We have a natural isomorphism 3 .Nk .// Š H3 .Nk .// by assigning f .ŒX/ 2 H3 .Nk .// to .X; f / 2 3 .Nk .//, where ŒX 2 H3 .X / is the fundamental class of a closed oriented 3-manifold X . Igusa–Orr [58] showed that the homomorphism H3 .N2k1 .// ! H3 .Nk .// induced by the natural projection N2k1 ./ ! Nk ./ is trivial. From this, we see that Hg;1 Œ2k 1 Ker k . On the other hand, the induced homomorphism k W Hg;1 Œk=Hg;1 Œ2k 1 ! 3 .Nk .// turns out to be an epimorphism between free abelian groups of the same rank, which shows that it is an isomorphism. In particular, the identity Hg;1 Œ2k 1 D Ker k follows. Next, we briefly review Atiyah–Patodi–Singer’s -invariant in [7], [8]. Let .M; g/ be a .2l 1/-dimensional compact oriented Riemannian manifold, and let ˛ W 1 M ! U.m/ be a unitary representation. Consider the self-adjoint operator B˛ W even .M I V˛ / ! even .M I V˛ / on the space of all differential forms of even degree on M with values in the flat bundle V˛ associated with ˛ defined by p B˛ ' WD . 1/l .1/pC1 .d˛ d˛ /' for ' 2 2p .M I V˛ /. Here, is the Hodge star operator. Then we define the spectral function ˛ .s/ of B˛ by X .sign/jjs ; ˛ .s/ WD

¤0

where runs over all non-zero eigenvalues of B˛ with multiplicities. This function converges to an analytic function for s 2 C having sufficiently large real part, and is continued analytically as a meromorphic function on the complex plane so that it


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takes a finite value at s D 0. The value ˛ .0/ is called the -invariant of .M; g/ associated with ˛. We simply write .0/ for the -invariant associated with the trivial representation 1 M ! U.1/. Theorem 8.5 (Atiyah–Patodi–Singer [8]). The value ˛ .M / WD ˛ .0/ m .0/ does not depend on a metric of M , so that it defines a diffeomorphism invariant of M called the -invariant associated with ˛. Moreover, if there exists a compact smooth manifold N such that M D @N and if ˛ can be extended to a unitary representation ˛Q W 1 N ! U.m/ of 1 N , then ˛ .M / D m sign.N / sign˛Q .N / holds, where sign.N / and sign˛Q .N / denote the signature and the twisted signature of N . Levine [73] applied the theory of -invariants to the following situation and obtained some invariants of links. Let Rm .G/ be the space of all unitary representations G ! U.m/ of a group G. If G is generated by l elements, Rm .G/ can be realized as a real algebraic subvariety of the direct product U.m/l of l-tuples of U.m/. We endow Rm .G/ with the usual (Hausdorff) topology as a subspace of U.m/l . For a pair .M; ˛/ consisting of an odd-dimensional closed manifold M and a group homomorphism ˛ W 1 M ! G, we define a function

.M; ˛/ W Rm .G/ ! R by .M; ˛/./ WD B˛ .M /. This function has the following properties. Theorem 8.6 (Levine [73]). (1) For each pairˇ .M; ˛/, there exists a proper algebraic subvariety † of Rm .G/ such that .M; ˛/ˇRm .G/† is a continuous real valued function. (2) If .M; ˛/ and .M 0 ; ˛ 0 / are homology G-bordant, there exists an algebraic subvariety †0 of Rm .G/ such that ˇ ˇ

.M; ˛/ˇRm .G/†0 D .M 0 ; ˛ 0 /ˇRm .G/†0 : Here, two pairs .M; ˛/; .M 0 ; ˛ 0 / are said to be homology G-bordant if there exists a pair .N; ˛/ Q such that @N D M 0 [ M , H .N; M / D H .N; M 0 / D 0, and the pullback of ˛Q on 1 M (resp. 1 M 0 ) coincides with ˛ (resp. ˛ 0 ) up to conjugation in G. Note that by an argument in [9], .M; ˛/ mod Z is continuous on Rm .G/. From this, we can show that .M; ˛/ is a bounded function on Rm .G/. Now we return to our situation. We now consider R1 .N2 .// D R1 .H / to construct an invariant of Hg;1 Œ2. Fix a diffeomorphism R1 .H / Š T 2g , where T 2g denotes the 2g-dimensional torus, by using a basis of H . We give a standard measure

582 d normalized by

Takuya Sakasai

R T 2g

d D 1 to T 2g . Then we define H;1 W Hg;1 Œ2 ! R

by

Z H;1 .M; iC ; i / WD

T 2g

.CM ; fQM /. / d:

Note that for each element of Hg;1 Œ2, .CM ; fQM / is uniquely determined up to homology H -bordism. Since .CM ; fQM / is bounded, continuous and takes the same value for two homology H -bordant manifolds almost everywhere in T 2g , the map H;1 is well-defined. Theorem 8.7. The map H;1 W Hg;1 Œ2 ! R has the following properties: (1) The restriction of H;1 to Ker rq2 is a homomorphism; (2) H;1 .Hg;1 É1Ê/ is an infinitely generated (over Z / subgroup of R. Proof. For k D 2, the bordism invariant 2 gives an exact sequence 2

1 ! Hg;1 Œ3 ! Hg;1 Œ2 ! 3 .H / ! 1: From this, we see that if .M; iC ; i / 2 Hg;1 Œ3, then the pair consisting of the closure CM of M and the homomorphism fQM W 1 CM ! H induced from the continuous map fQM W CM ! K.H; 1/ is the boundary of a pair .WM ; fWM /. Then the function

.CM ; fQM / has an interpretation as a signature defect and Z

.CM ; fQM /. / d H;1 .M; iC ; i / D T 2g Z sign.WM / sign BfW .WM / d D M T 2g Z sign BfW .WM / d D sign.WM / T 2g

M

follows, where sign BfW .WM / is the signature of the intersection form induced on M H2 .WM I C BfWM / with coefficients in the left 1 WM -module C on which 1 WM acts through B fWM W 1 WM ! U.1/. To show (1), it suffices to show that both sign.WM / and sign BfW .WM / are additive. M Let M1 D .M1 ; iC ; i /, M2 D .M2 ; jC ; j / 2 Ker rq2 . Note that Ker rq2 Hg;1 Œ3. We take a pair .WMi ; fWMi / satisfying .CMi ; fQMi / D @.WMi ; fWMi /. By performing surgeries on WMi preserving the H -bordism class, if necessary, we can assume that 1 WMi Š H1 .WMi / Š H . Then the manifold W WD WM1 [†g;1 Œ0;1 WM2 obtained from WM1 and WM2 by gluing along †g;1 Œ0; 1 CMi together with the homomorphism fW WD fWM1 [ fWM2 satisfy @.W; fW / D .M1 M2 ; fQM1 M2 /. See Figure 7.

583

Chapter 9. A survey of Magnus representations M1 WM1 †g;1 [ .†g;1 /

†g;1 Œ0; 1

†g;1 [ .†g;1 /

WM2 M2

Figure 7. The manifold W .

If we apply Wall’s non-additivity theorem [119] of signatures to W , WM1 , WM2 , we see that the correction term is zero when M1 ; M2 2 Hg;1 Œ2 by an argument associated with the Meyer cocycle [80], and therefore the additivity of signatures follows. To see the additivity of the integration of sign BfW .WM /, we need to use a loM cal coefficient system version of Wall’s theorem in [80]. Indeed, almost everywhere in R1 .H /, the local coefficient C BfWM becomes a KH -vector space and the veci tor spaces used in the calculation of the correction term coincide with each other if rq1 .M1 / D rq2 .M2 / D I2g . This implies that the correction term is equal to zero almost everywhere in R1 .H / and hence its integration over R1 .H / is also equal to zero. The additivity of the integration of sign BfW .WM / follows from this. M (2) is shown by the following explicit examples. Note that these examples are based on those in Cochran–Orr–Teichner [27], [28] and Harvey [53] to show the infinite generation of some subgroups of the knot (or string link) concordance group. For .†g;1 Œ0; 1; id 1; id 0/ 2 Cg;1 , we take a loop l in the interior of †g;1 Œ0; 1 representing 1 2 H Š H1 .†g;1 Œ0; 1/. We remove an open tubular neighborhood N.l/ of l from †g;1 Œ0; 1 and then glue the exterior E.K/ of a knot K S 3 so that the canonical longitude (resp. the meridian) of E.K/ corresponds to the meridian (resp. the inverse of the longitude) of N.l/. We can check that the resulting manifold MK becomes a homology cylinder. Moreover it belongs to Hg;1 É1Ê since 1 .†g;1 Œ0; 1 N.l// ! 1 .†g;1 Œ0; 1/ Š ! acy extends to 1 MK . Then we can show that Z H;1 .MK / D

.K/ d; 2S 1

where .K/ is the Levine–Tristram signature of the knot K associated with 2 S 1 . It was shown in [28], Section 5, that the above values move around an infinitely generated subgroup of R when K runs over all knots. Therefore (2) follows. Corollary 8.8. The groups Ker rq2 , Hg;1 É1Ê and their abelianizations are all infinitely generated.

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We can consider results of Cochran–Harvey–Horn [26] to be a further generalization of the invariant H;1 . They constructed von Neumann -invariants for homology cylinders by using the theory of L2 -signature invariants. Note that Magnus matrices also appear in their context as obstructions to the additivity of invariants. In fact, we can see that the correction term vanishes under the triviality of the corresponding Magnus matrix by rewriting Wall’s argument [119] word-by-word in terms of L-groups. We close this subsection by posing the following problem: acy

Problem 8.9. Determine H3 .Fn /. This problem is an analogue of a similar problem for the algebraic closure of a free group. It was shown in [103] that the bordism invariant similar to k gives an epimorphism W Hg;1 É1Ê H3 . acy /. At present, however, we cannot extract any acy information on Hg;1 É1Ê from since it is not known even whether H3 .Fn / is trivial or not.

8.3 Abelian quotients of groups of homology cylinders Abelian quotients of a monoid or group are helpful not only to know how big the monoid or group is, but to extract information on its structure. In this subsection, we focus on abelian quotients of monoids and homology cobordism groups of homology cylinders and we compare them to the corresponding results for mapping class groups. We assume that g 1. As we have seen in Sections 4.5 and 5.2, the Johnson homomorphisms give finite rank abelian quotients of Mg;1 Œk, Cg;1 Œk and Hg;1 Œk for each k 2. Indeed the image of Cg;1 Œk and Hg;1 Œk is generally bigger than that of Mg;1 Œk. Before discussing further, as commented in [41], we point out that Cg;1 has the monoid Z3 of homology 3-spheres as an abelian quotient. In fact, we have a forgetful homomorphism F W Cg;1 ! Z3 defined by F .M; iC ; i / D S 3 # X1 # X2 # # Xn irr for the prime decomposition M D M0 # X1 # X2 # # Xn of M with M0 2 Cg;1 3 and Xi 2 Z .1 i n/ (recall Section 5.1). The map F owes its well-definedness to the uniqueness of the prime decomposition of 3-manifolds. The map F gives a splitting of the construction of Example 5.5 and is surjective. The underlying 3-manifolds of homology cylinders obtained from Mg;1 are all †g;1 Œ0; 1 and, in particular, irreducible. Therefore it seems more reasonable to irr . Until now, many infinitely generated abelian quotients for compare Mg;1 with Cg;1 monoids and homology cobordism groups of irreducible homology cylinders have been given, which are completely different from the corresponding cases for mapping class groups. We present them in order. irr \ Cg;1 Œk for k 2 and Theorem 8.10 ([105], Corollary 6.16). The submonoids Cg;1 irr Ker.Cg;1 ! Hg;1 / have abelian quotients isomorphic to .Z0 /1 .


585

The proof uses homomorphisms constructed from the torsions qCk . Precisely speaking, irreducibility was not discussed in [105]. However, we can modify the argument. Theorem 8.11 (Morita [90], Corollary 5.2). Hg;1 Œ2 has an abelian quotient isomorphic to Z1 . For the construction, Morita used the trace maps mentioned in Remark 4.14 with a deep observation of the Johnson filtration of Hg;1 . It was shown by Harer [50] that Mg;1 is a perfect group for g 3 (see also Farb– irr Margalit [31]). By taking into account the similarity between Mg;1 , Cg;1 and Hg;1 irr as we have seen, it had been conjectured that Cg;1 and Hg;1 do not have non-trivial abelian quotients. However, Goda and the author showed the following: irr has an abelian quotient isoTheorem 8.12 ([41], Theorem 2.6). The monoid Cg;1 1 morphic to .Z0 / .

Sketch of proof. The proof uses some results of sutured Floer homology (a variant of Heegaard Floer homology) developed by Ni [92], [93] and Juhász [61], [62]. For each homology cylinder .M; iC ; i / 2 Cg;1 , we have a natural decomposition @M D iC .†g;1 / [iC .@†g;1 /Di .@†g;1 / i .†g;1 / of @M . Such a decomposition defines a sutured manifold .M; / with the suture D iC .@†g;1 / D i .@†g;1 /. Since the sutured manifold obtained from a homology cylinder is balanced in the sense of Juhász ([61], Definition 2.2), the sutured Floer homology SFH.M; / is defined. By taking irr ! Z0 defined by R.M; iC ; i / D the rank of SFH, we obtain a map R W Cg;1 rank Z .SFH.M; //. Deep results of Ghiggini [39], Ni [92], [93] and Juhász [61], [62] irr to the monoid Z show that the map R is a monoid homomorphism from Cg;1 >0 of positive integers whose product is given by multiplication. By the uniqueness of the prime decomposition of an integer, we can decompose R into prime factors M .p/ M irr Rp W Cg;1 ! Z Z0 ; RD >0 D p : prime

p : prime

where Z.p/ 0 is a copy of Z0 , the monoid of non-negative integers whose product is irr given by sum. We can check that fRp W Cg;1 ! Z0 j p : primeg contains infinitely many non-trivial homomorphisms that are linearly independent as homomorphisms irr to Z0 . from Cg;1 It was also observed in [41] that the above homomorphisms Rp are not homology cobordism invariants. As seen in Example 8.3, the -torsion generally changes under homology cobordism. However, Cha–Friedl–Kim [21] found a way to extract homology cobordant invariants of homology cylinders from the torsion

qC2 W Cg;1 ! KH =.˙H /;

586

Takuya Sakasai

which is a crossed homomorphism, as follows. defined by First they consider the subgroup A KH ; ' 2 Sp.2g; Z/g; A WD ff 1 '.f / j f 2 KH

by which we can obtain a homomorphism

qC2 W Cg;1 ! KH =.˙H A/:

(8.2)

=.˙H A/ since I2g 2 Sp.2g; Z/. Second, they Note that f D fN holds in KH observe that if .M; iC ; i /; .N; jC ; j / 2 Cg;1 are homology cobordant, then there exists q 2 KH such that =.˙H /

qC2 .M / D qC2 .N / q qN 2 KH

(8.3)

by using torsion duality. Note that a similar formula treating general situations had been obtained by Turaev [117], Theorem 1.11.2. From this, we see that if we put N WD ff fN j f 2 KH g;

then we can finally obtain a homomorphism

qC2 W Hg;1 ! KH =.˙H A N /: =.˙H A N /. Note that f 2 D 1 holds for any f 2 KH We can see the structure of KH =.˙H A N / as follows. Recall that KH D ZŒH .ZŒH f0g/1 . The ring ZŒH is a Laurent polynomial ring of 2g variables and it is a unique factorization domain. Thus every Laurent polynomial f is factorized into irreducible polynomials uniquely up to multiplication by a unit in ZŒH . In particular, for every irreducible polynomial , we can count the exponent of in the . Under factorization of f . This counting naturally extends to that for elements in KH the identification by ˙H A N , an element in KH =.˙H A N / is determined by the exponents of all Sp.2g; Z/-orbits of irreducible polynomials (up to multiplication by a =.˙H AN / is isomorphic to .Z=2Z/1 . Finally unit in ZŒH ) modulo 2. Hence KH by using .Z=2Z/-torsion of the knot concordance group, they show the following:

Theorem 8.13 (Cha–Friedl–Kim [21]). The homomorphism =.˙H A N /

qC2 W Hg;1 ! KH

is not surjective but its image is isomorphic to .Z=2Z/1 . Remark 8.14. Cha–Friedl–Kim showed that the same statement holds for Hg;0 . Moreover, they considered abelian quotients of the other Hg;n and showed that a similar construction gives an epimorphism Hg;n .Z=2Z/1 ˚ Z1 if n 2 (when g 1) or n 3 (when g D 0). Now we return to our discussion on applications of Magnus representations. We use the above Cha–Friedl–Kim’s idea. Since Magnus representations are homology


587

cobordism invariant, we have two maps rq2

det

! KH =.˙H /; rOq2 W Hg;1 ! GL.2g; KH / ! KH rOq2

=.˙H / ! KH =.˙H A/: rQq2 W Hg;1 ! KH

While rOq2 is a crossed homomorphism, its restriction to Hg;1 Œ2 and rQq2 are homo morphisms. Note that both KH =.˙H / and KH =.˙H A/ are isomorphic to Z1 . Theorem 8.15 ([106]). (1) For g 1 and .M; iC ; i / 2 Cg;1 , the equality =.˙H / rOq2 .M / D qC2 .M / . qC2 .M //1 2 KH

holds. (2) For g 1, the homomorphism rQq2 W Hg;1 ! KH =.˙H A/ is trivial. (3) For g 2, the homomorphism rOq2 W Hg;1 Œ2 ! KH =.˙H / is not surjective 1 but its image is isomorphic to Z . Sketch of proof. .1/ can be shown by using the formula (7.2) and torsion duality. As mentioned above, the action of Sp.2g; Z/ implies that f D fN for any f 2 =.˙H A/. Then our claim .2/ immediately follows from .1/. To show .3/, we KH use the homology cylinder ML 2 C2;1 in Example 7.9. While ML … C2;1 Œ2, we can adjust it by some g1 2 M2;1 so that ML g1 2 C2;1 Œ2. Since rOq2 is trivial on M2;1 , we have 3 C 4 1 2 KH =.˙H /: rOq2 .ML g1 / D rOq2 .ML / D 1 3 C 41 1 Take f 2 M2;1 such that 2 .f / 2 Sp.4; Z/ maps 1 7! 1 C 4 ;

2 7! 2 ;

3 7! 2 C 3 ;

4 7! 4 :

Consider f m ML 2 C2;1 and adjust it by some gm 2 M2;1 so that f m ML gm 2 C2;1 Œ2. Then we have 2m 3 C 4 1 2 KH =.˙H /: 2m 31 C 41 1 o1 n m 3 C4 1 generate an infinitely generated We can check that the values m2 1 1 C 1 rOq2 .f m ML gm / D 2 .f

m/

.rOq2 .ML // D

2

3

4

mD0

=.˙H /. This completes the proof when g D 2. We can use the subgroup of KH above computation for g 3.

Consequently, we obtain a result similar to Theorem 8.11.

8.4 Generalization to higher-dimensional cases We can consider homology cylinders over X for any compact oriented connected k-dimensional manifold X with k 3 by rewriting Definition 5.1 word-by-word.

588

Takuya Sakasai

Let M.X /, C.X/ and H .X/ denote the corresponding diffeotopy group, monoid of homology cylinders and homology cobordism group of homology cylinders. We have natural homomorphisms / C.X /

M.X/

/ / H .X /

and we can apply the argument in Section 6 to C .X / and H .X /. For k 2 and n 1, we put Xnk WD #.S 1 S k1 /: n

Xnk

may be regarded as a generalization of a closed surface since Xn2 D The manifold †n;0 . Suppose k 3. Then 1 .Xnk Int D k / Š 1 Xnk Š Fn , where Int D k is an open k-ball. We have homomorphisms

acy W C .Xnk Int D k / ! Aut .Fnacy /;

acy W C .Xnk / ! Out .Fnacy /

and similarly for H .Xnk Int D k / and H .Xnk /. Consider the composition rq2

det

! KH =.˙H1 A0 / Š Z1 ; rQq2 W Aut .Fnacy / ! GL.n; KH1 / ! KH 1 1 ; ' 2 GL.n; Z/g. The map rQq2 is a homomorwhere A0 WD ff 1 '.f / j f 2 KH 1 phism for the same reason mentioned in the previous subsection.

Theorem 8.16 ([106]). For any k 3 and n 2, we have: (1) acy W H .Xnk Int D k / ! Aut .Fn / and acy W H .Xnk / ! Out .Fn / are surjective. acy

acy

(2) The image of rQq2 is an infinitely generated subgroup of Z1 . In particular, acy H1 .Aut .Fn // and H1 .H .Xnk Int D k // have infinite rank. (3) rQq2 factors through Out .Fn /, so that H1 .Out .Fn // and H1 .H .Xnk // have infinite rank. acy

acy

Sketch of proof. .1/ follows from a construction similar to the one used in the proof of Theorem 6.10. To show .2/, consider 2-connected homomorphisms fm W Fn ! Fn defined by fm .1 / D .1 21 11 21 /m 1 22m ;

fm .i / D i .2 i n/;

acy

which in turn give automorphisms of Fn . We can easily check that rQq2 .fm / D 1 2 C 22 23 C C 22m : Then .2/ follows from the irreducibility of these polynomials when 2m C 1 is prime by a well-known fact on the cyclotomic polynomials. .3/ can be easily checked. Remark 8.17. The statements in Theorem 8.16 do not hold for k D 2 by the symplecticity of the Magnus representation rq2 as seen in the previous subsection. When


589

k D 3, the theorem can be seen as a partial generalization of a theorem of Laudenbach ([68], Theorem 4.3), stating that there exists an exact sequence 1 ! .Z=2Z/n ! M.Xn3 / ! Aut .Fn / ! 1; where the i -th summand of .Z=2Z/n corresponds to the rotation of S 2 in the i -th factor of Xn3 by using 1 .SO.3// Š Z=2Z. In contrast with the case of surfaces, the homomorphism M.X / ! C .X / is not necessarily injective for a general manifold X. In fact, if Œ' 2 Ker.M.X / ! C .X //, the definition of the homomorphism only says that ' is a pseudo isotopy over X, for which we refer to Cerf [19] and Hatcher–Wagoner [54]. Note also that we can argue about homology cylinders in other categories such as piecewise linear and continuous, which would bring us to further different and interesting phenomena.

References [1]

M. Abdulrahim, Complex specializations of the reduced Gassner representation of the pure braid group. Proc. Amer. Math. Soc. 125 (1997), 1617–1624. 551

[2]

J. E.Andersen,A. Bene, J. B. Meilhan, and R. Penner, Finite type invariants and fatgraphs. Adv. Math. 225 (2010), 2117–2161. 559 J. E. Andersen, A. Bene, and R. Penner, Groupoid extensions of mapping class representations for bordered surfaces. Topology Appl. 156 (2009), 2713–2725. 548 S. Andreadakis, On the automorphisms of free groups and free nilpotent groups. Proc. London Math. Soc. 15 (1965), 239–268. 555 E. Artin, Theorie der Zöpfe. Abh. Math. Semin. Univ. Hamburg 4 (1925), 47–72. 544 E. Artin, Theory of braids. Ann. of Math. 48 (1947), 101–126. 544 M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I. Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. 580 M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. II. Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432. 580, 581 M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Philos. Soc. 79 (1976), 71–99. 581 S. Bachmuth, Automorphisms of free metabelian groups. Trans. Amer. Math. Soc. 118 (1965), 93–104. 544, 546 S. Bachmuth and H. Y. Mochizuki, The non-finite generation of Aut .G/, G free metabelian of rank 3. Trans. Amer. Math. Soc. 270 (1982), 693–700. 546 S. Bachmuth and H. Y. Mochizuki, Aut .F / ! Aut .F=F 00 / is surjective for free group for rank 4. Trans. Amer. Math. Soc. 292 (1985), 81–101. 546 J. Birman, An inverse function theorem for free groups. Proc. Amer. Math. Soc. 41 (1973), 634–638. 542 J. Birman, Braids, links and mapping class groups. Ann. of Math. Stud. 82, Princeton University Press, Princeton, N.J., 1974. 534, 544, 545, 547, 551, 553

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

590 [15] [16] [17] [18] [19]

[20] [21] [22] [23] [24] [25] [26]

[27] [28] [29] [30] [31] [32] [33] [34] [35]

Takuya Sakasai A. Bousfield, Homological localization towers for groups and -modules. Mem. Amer. Math. Soc. 10 (1977), no. 186. 564 K. Brown, Cohomology of groups. Grad. Texts in Math. 87, Springer-Verlag, New York 1982. 536, 537, 539 R. M. Bryant and C. K. Gupta, Automorphism groups of free nilpotent groups. Arch. Math. 52 (1989), 313–320. 555 W. Burau, Über Zopfgruppen und gleichsinnig verdrillte Verkettungen. Abh. Math. Sem. Univ. Hamburg 11 (1935), 179–186. 544 J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Inst. Hautes Études Sci. Publ. Math. 39 (1970), 5–173. 589 J. C. Cha, Injectivity theorems and algebraic closures of groups with coefficients. Proc. London Math. Soc. 96 (2008), 227–250. 565 J. C. Cha, S. Friedl, and T. Kim, The cobordism group of homology cylinders. Compos. Math. 147 (2011), no. 3, 914–942. 559, 560, 585, 586 O. Chein, IA automorphisms of free and free metabelian groups. Comm. Pure Appl. Math. 21 (1968), 605–629. 546 T. Church and B. Farb, Infinite generation of the kernels of the Magnus and Burau representations. Algebr. Geom. Topol. 10 (2010), 837–851. 546, 551 T. Cochran, Noncommutative knot theory. Algebr. Geom. Topol. 4 (2004), 347–398. 571, 576, 577 T. Cochran and S. Harvey, Homology and derived series of groups. Geom. Topol. 9 (2005), 2159–2191. 574 T. Cochran, S. Harvey, and P. Horn, Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders. Internat. Math. Res. Notices, to appear; Doi 10.1093/imrn/rnr149. 584 T. Cochran, K. Orr, and P. Teichner, Knot concordance, Whitney towers and L2 signatures. Ann. of Math. 157 (2003), 433–519. 571, 576, 577, 583 T. Cochran, K. Orr, and P. Teichner, Structure in the classical knot concordance group. Comment. Math. Helv. 79 (2004), 105–123. 583 P. M. Cohn, Free rings and their relations. Academic Press, New York 1985. 569, 571 R. Crowell and H. Trotter, A class of pretzel knots. Duke Math. J. 30 (1963), 373–377. 561 B. Farb and D. Margalit, A primer on mapping class groups. Princeton Math. Ser. 49, Princeton University Press, Princeton, N.J., 2012. 547, 585 R. H. Fox, Free differential calculus. Ann. of Math. 57 (1953), 547–560. 534 S. Friedl, Reidemeister torsion, the Thurston norm and Harvey’s invariants. Pacific J. Math. 230 (2007), 271–296. 577 S. Friedl, A. Juhász, and J. Rasmussen, The decategorification of sutured Floer homology. J. Topol. 4 (2011), no. 2, 431–478. 578 S. Garoufalidis and J. Levine, Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism. In Graphs and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc., Providence, R.I., 2005, 173–205. 532, 558, 559, 563, 568

Chapter 9. A survey of Magnus representations [36] [37] [38] [39] [40] [41] [42]

[43] [44] [45]

[46] [47] [48] [49]

[50] [51] [52] [53] [54] [55]

591

B. J. Gassner, On braid groups. Abh. Math. Sem. Univ. Hamburg 25 (1961), 10–22. 544 S. Gersten, A presentation for the special automorphism group of a free group. J. Pure Appl. Algebra 33 (1984), 269–279. 554 S. Gervais and N. Habegger, The topological IHX relation, pure braids, and the Torelli group. Duke Math. J. 112 (2002), 265–280. 553 P. Ghiggini, Knot Floer homology detects genus-one fibred knots. Amer. J. Math. 130 (2008), 1151–1169. 585 H. Goda and T. Sakasai, Homology cylinders and sutured manifolds for homologically fibered knots. Preprint 2008; arXiv:0807.4034v4 [math.GT]. 559, 560, 572 H. Goda and T. Sakasai, Abelian quotients of monoids of homology cylinders. Geom. Dedicata 151 (2011), 387–396. 584, 585 H. Goda and T. Sakasai, Factorization formulas and computations of higher-orderAlexander invariants for homologically fibered knots. J. Knot Theory Ramifications 20 (2011), 1355–1380. 560, 577, 578, 579 M. Goussarov, Finite type invariants and n-equivalence of 3-manifolds. C. R. Math. Acad. Sci. Paris 329 (1999), 517–522. 532, 558, 559 C. K. Gupta, Around automorphisms of relatively free groups. In Algebra, Trends Math., Birkhäuser, Basel 1999, 63–74. 546 C. K. Gupta and V. Shpilrain, Lifting automorphisms: a survey. In Groups ’93 Galway/St. Andrews, Vol. 1, London Math. Soc. Lecture Note Ser. 211, Cambridge University Press, Cambridge 1995, 249–263. 546 N. Habegger, Milnor, Johnson, and tree level perturbative invariants. Preprint 2000; http://www.math.sciences.univ-nantes.fr/~habegger/ 560, 563 N. Habegger and X. S. Lin, The classification of links up to link-homotopy. J. Amer. Math. Soc. 3 (1990), 389–419. 560 K. Habiro, Claspers and finite type invariants of links. Geom. Topol. 4 (2000), 1–83. 532, 558, 559 K. Habiro and G. Massuyeau, From mapping class groups to monoids of homology cobordisms: a survey. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2012, 465–529. 533, 555, 556, 558, 559, 561, 579 J. Harer, The second homology group of the mapping class group of an orientable surface. Invent. Math. 72 (1983), 221–239. 554, 585 S. Harvey, Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm. Topology 44 (2005), 895–945. 571, 576 S. Harvey, Monotonicity of degrees of generalized Alexander polynomials of groups and 3-manifolds. Math. Proc. Cambridge Philos. Soc. 140 (2006), 431–450. 571, 576 S. Harvey, Homology cobordism invariants and the Cochran–Orr–Teichner filtration of the link concordance group. Geom. Topol. 12 (2008), 387–430. 574, 583 A. Hatcher and J. Wagoner, Pseudo-isotopies of compact manifolds. Astérisque 6, Soc. Math. France, Paris 1973. 589 A. Heap, Bordism invariants of the mapping class group. Topology 45 (2006), 851–886. 579

592 [56] [57] [58] [59] [60] [61] [62] [63] [64]

[65] [66] [67] [68] [69] [70] [71] [72] [73] [74]

[75] [76] [77]

Takuya Sakasai J. Hempel, 3-Manifolds. Ann. of Math. Stud, 86, Princeton University Press, Princeton, N.J., 1976. 561 J. Hillman, Algebraic invariants of links. Ser. Knots Everything 32, World Scientific Publishing, River Edge, N.J., 2002. 565 K. Igusa and K. Orr, Links, pictures and the homology of nilpotent groups. Topology 40 (2001), 1125–1166. 580 N. V. Ivanov, Mapping class groups. In Handbook of geometric topology, North-Holland, Amsterdam 2002, 523–633. 547 D. Johnson, A survey of the Torelli group. Contemp. Math. 20 (1983), 165–179. 555 A. Juhász, Holomorphic discs and sutured manifolds. Algebr. Geom. Topol. 6 (2006), 1429–1457. 585 A. Juhász, Floer homology and surface decompositions. Geom. Topol. 12 (2008), 299– 350. 585 N. Kawazumi, Cohomological aspects of Magnus expansions. Preprint 2005; arXiv:math/0505497v3 [math.GT]. 554, 556 N. Kawazumi, Canonical 2-forms on the moduli space of Riemann surfaces. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 217–237. 554, 556 T. Kitano, T. Morifuji, and M. Takasawa, L2 -torsion invariants of a surface bundle over S 1 . J. Math. Soc. Japan 56 (2004), 503–518. 557 P. Kirk, C. Livingston, and Z. Wang, The Gassner representation for string links. Commun. Contemp. Math. 3 (2001), 87–136. 571, 572 M. Korkmaz and A. Stipsicz, The second homology groups of mapping class groups of oriented surfaces. Math. Proc. Cambridge Philos. Soc. 134 (2003), 479–489. 554 F. Laudenbach, Topologie de la dimension trois: homotopie et isotopie. Astérisque 12, Soc. Math. France, Paris 1974. 589 J. Y. Le Dimet, Cobordisme d’enlacements de disques. Mém. Soc. Math. France (N.S.) 32 (1988). 565 J. Y. Le Dimet, Enlacements d’intervalles et représentation de Gassner. Comment. Math. Helv. 67 (1992), 306–315. 564, 569, 570 J. Levine, Link concordance and algebraic closure, II. Invent. Math. 96 (1989), 571–592. 565, 566, 567, 580 J. Levine, Algebraic closure of groups. Contemp. Math. 109 (1990), 99–105. 565 J. Levine, Link invariants via the eta invariant. Comment. Math. Helv. 69 (1994), 82–119. 581 J. Levine, Pure braids, a new subgroup of the mapping class group and finite-type invariants. In Tel Aviv Topology Conference: Rothenberg Festschrift, Contemp. Math. 231, (1999), Amer. Math. Soc., Providence, R.I., 1999, 137–157. 553, 560 J. Levine, Homology cylinders: an enlargement of the mapping class group. Algebr. Geom. Topol. 1 (2001), 243–270. 558, 559 W. Magnus, Über n-dimensionale Gittertransformationen. Acta Math. 64 (1935), 353– 367. 544 W. Magnus, On a theorem of Marshall Hall. Ann. of Math. 40 (1939), 764–768. 536

Chapter 9. A survey of Magnus representations [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88]

[89]

[90]

[91] [92] [93] [94] [95] [96] [97]

[98]

593

W. Magnus and A. Peluso, On a theorem of V. I. Arnol’d. Comm. Pure Appl. Math. 22 (1969), 683–692. 551 G. Massuyeau and J.-B. Meilhan, Characterization of Y2 -equivalence for homology cylinders. J. Knot Theory Ramifications 12 (2003), 493–522. 559 W. Meyer, Die Signatur von lokalen Koeffizientensystemen und Faserbündeln. Bonner Math. Schr. (1972), no. 53. 554, 583 J. Milnor, A duality theorem for Reidemeister torsion. Ann. of Math. 76 (1962), 137–147. 577 J. Milnor, Whitehead torsion. Bull. Amer. Math. Soc. 72 (1966), 358–426. 575 S. Morita, Characteristic classes of surface bundles. Invent. Math. 90 (1987), 551–577. 554 S. Morita, Families of Jacobian manifolds and characteristic classes of surface bundles I. Ann. Inst. Fourier 39 (1989), 777–810. 554 S. Morita, Families of Jacobian manifolds and characteristic classes of surface bundles. II. Math. Proc. Cambridge Philos. Soc. 105 (1989), 79–101. 554 S. Morita, Casson’s invariant for homology 3-spheres and characteristic classes of surface bundles I. Topology, 28 (1989), 305–323. 548, 554, 556 S. Morita, Abelian quotients of subgroups of the mapping class group of surfaces. Duke Math. J. 70 (1993), 699–726. 548, 549, 555 S. Morita, Cohomological structure of the mapping class group and beyond. In Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, R.I., 2006, 329–354. 555 S. Morita, Introduction to mapping class groups of surfaces and related groups, In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume I, EMS Publishing House, Zurich 2007, 353–386. 544, 546, 547, 555 S. Morita, Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces. In Groups of diffeomorphisms, Adv. Stud. Pure Math. 52, Math. Soc. Japan, Tokyo 2008, 443–468. 585 R. Myers, Homology cobordisms, link concordances, and hyperbolic 3-manifolds. Trans. Amer. Math. Soc. 278 (1983), 271–288. 562 Y. Ni, Sutured Heegaard diagrams for knots. Algebr. Geom. Topol. 6 (2006), 513–537. 585 Y. Ni, Knot Floer homology detects fibred knots. Invent. Math. 170 (2007), 577–608. 585 J. Nielsen, Die Isomorphismengruppe der freien Gruppen. Math. Ann. 91 (1924), 169– 209. 544 T. Oda,A lower bound for the graded modules associated with the relative weight filtration on the Teichmüller group. Preprint. 553 K. Orr, Homotopy invariants of links. Invent. Math. 95 (1989) 379–394. 580 C. D. Papakyriakopoulos, Planar regular coverings of orientable closed surfaces. In Knots, groups, and 3-manifolds, Ann. of Math. Stud. 84, Princeton University Press, Princeton, N.J., University of Tokyo Press, Tokyo 1975, 261–292 549, 550 L. Paris, Braid groups and Artin groups. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 389–451. 544

594 [99] [100] [101] [102] [103] [104] [105]

[106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119]

Takuya Sakasai D. Passman, The Algebraic structure of group rings. Pure Appl. Math., John Wiley and Sons, New York 1977. 571 R. Penner, The decorated Teichmüller space of punctured surfaces. Comm. Math. Phys. 113 (1987), 299–339. 548 B. Perron, A homotopic intersection theory on surfaces: applications to mapping class group and braids. Enseign. Math. 52 (2006), 159–186. 549, 551 J. Rosenberg, Algebraic K-theory and its applications. Grad. Texts in Math. 147, Springer-Verlag, New York 1994. 575 T. Sakasai, Homology cylinders and the acyclic closure of a free group. Algebr. Geom. Topol. 6 (2006), 603–631. 568, 569, 580, 584 T. Sakasai, The symplecticity of the Magnus representation for homology cobordisms of surfaces. Bull. Austral. Math. Soc. 76 (2007), 421–431. 574 T. Sakasai, The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces. Algebr. Geom. Topol. 8 (2008), 803–848. 573, 574, 584, 585 T. Sakasai, The Magnus representation and homology cobordism groups of homology cylinders. Preprint 2011, arXiv:1108.6111v1 [math.GT]. 587, 588 T. Satoh, Twisted first homology groups of the automorphism group of a free group. J. Pure Appl. Algebra 204 (2006), 334–348. 554 T. Satoh, The kernel of the Magnus representation of the automorphism group of a free group is not finitely generated. Preprint 2009, arXiv:0910.0386v2 [math.AT]. 546 V. Shpilrain, Automorphisms of F=R0 groups. Internat. J. Algebra Comput. 1 (1991), 177–184. 546 J. Stallings, Homology and central series of groups. J. Algebra 2 (1965), 170–181. 562 R. Strebel, Homological methods applied to the derived series of groups. Comment. Math. Helv. 49 (1974), 302–332. 572, 577 M. Suzuki, The Magnus representation of the Torelli group g;1 is not faithful for g 2. Proc. Amer. Math. Soc. 130 (2002), 909–914. 548, 551 M. Suzuki, Irreducible decomposition of the Magnus representation of the Torelli group. Bull. Austral. Math. Soc. 67 (2003), 1–14. 548, 551 M. Suzuki, Geometric interpretation of the Magnus representation of the mapping class group. Kobe J. Math. 22 (2005), 39–47. 548, 549, 550, 556 M. Suzuki, On the kernel of the Magnus representation of the Torelli group. Proc. Amer. Math. Soc. 133 (2005), 1865–1872. 548, 551 V. G. Turaev, Intersections of loops in two-dimensional manifolds. Mat. Sb. 106 (148) (1978), 566–588. 550, 551, 575 V. G. Turaev, Reidemeister torsion in knot theory. Uspekhi Mat. Nauk 41 (1986), 97–147; English transl. Russian Math. Surveys 41 (1986), 119–182. 586 V. Turaev, Introduction to combinatorial torsions. Lectures Math. ETH Zurich, Birkhäuser, Basel 2001. 557 C. T. C. Wall, Non-additivity of the signature. Invent. Math. 7 (1969), 269–274. 583, 584

Chapter 10

Asymptotically rigid mapping class groups and Thompson’s groups Louis Funar, Christophe Kapoudjian and Vlad Sergiescu

Contents 1 2

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Thompson’s groups to mapping class groups of surfaces . . . 2.1 Three equivalent definitions of Thompson’s groups . . . . . . 2.2 Some properties of Thompson’s groups . . . . . . . . . . . . 2.3 Thompson’s group T as a mapping class group of a surface . . 2.4 Braid groups and Thompson’s groups . . . . . . . . . . . . . 2.5 Extending the Burau representation . . . . . . . . . . . . . . From the Ptolemy groupoid to the Hatcher–Thurston complex . . . 3.1 Universal Teichmüller theory according to Penner . . . . . . . 3.2 The isomorphism between Ptolemy and Thompson groups . . 3.3 A remarkable link between the Ptolemy groupoid and the Hatcher–Thurston complex of S0;1 , following [50] . . . . . . 3.4 The Hatcher–Thurston complex of S0;1 . . . . . . . . . . . . The universal mapping class group in genus zero . . . . . . . . . . 4.1 Definition of the group B . . . . . . . . . . . . . . . . . . . . 4.2 B is finitely presented . . . . . . . . . . . . . . . . . . . . . The braided Ptolemy–Thompson group . . . . . . . . . . . . . . . 5.1 Finite presentation . . . . . . . . . . . . . . . . . . . . . . . 5.2 Asynchronous combability . . . . . . . . . . . . . . . . . . . Central extensions of T and quantization . . . . . . . . . . . . . . 6.1 Quantum universal Teichmüller space . . . . . . . . . . . . . 6.2 The dilogarithmic representation of T . . . . . . . . . . . . . 6.3 The relative abelianization of the braided Ptolemy–Thompson group T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Computing the class of Tab . . . . . . . . . . . . . . . . . . . 6.5 Identifying the two central extensions of T . . . . . . . . . . 6.6 Classification of central extensions of the group T . . . . . . . More asymptotically rigid mapping class groups . . . . . . . . . . 7.1 Other planar surfaces and braided Houghton groups . . . . . . 7.2 Infinite genus surfaces and mapping class groups . . . . . . .

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Cosimplicial extensions for Thompson’s group V . . . . . . . 8.1 Strand doubling maps . . . . . . . . . . . . . . . . . . . 8.2 Cosimplicial S-extensions . . . . . . . . . . . . . . . . 8.3 Dyadic trees and the functor K . . . . . . . . . . . . . . 8.4 Extensions of Thompson’s group V . . . . . . . . . . . 8.5 Main examples . . . . . . . . . . . . . . . . . . . . . . 8.6 The universal mapping class group in genus zero and the Grothendieck–Teichmüller group . . . . . . . . . . . . . 9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

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1 Introduction The purpose of this chapter is to present the recently developed interaction between mapping class groups of surfaces, including braid groups, and Richard J. Thompson’s groups F , T and V . We follow here the present authors’ geometrical approach, while giving some hints to the algebraic developments of Brin and Dehornoy and the quasiconformal approach of de Faria, Gardiner and Harvey. When compared to mapping-class groups, already thoroughly studied by Dehn and Nielsen, Thompson’s groups appear quite recent. Introduced by Thompson in the middle of the 1960s, they originally developed from algebraic logic; however, a PL representation of them was immediately obtained. Recall that Thompson’s group F is the group of PL homeomorphisms of Œ0; 1 whose local pieces are of the form 2n x C 2pq , with breaks in Z 12 . The group T acts in a similar way on the unit circle S 1 . The group V acts by left continuous bijections on Œ0; 1 as a group of affine interval exchanges. This action may be lifted to a continuous one on the triadic Cantor set. By conjugating these groups via the Farey map sending the rationals to the dyadics, one obtains a similar definition as groups of piecewise PSL.2; Z/ maps with rational breakpoints; this definition already has a certain 2-dimensional flavour. Observe that Thompson’s groups act near the boundary of the hyperbolic disk and thus near the boundary of the infinite binary tree. This observation played a basic role in the beginning of the material discussed here. From this point of view Thompson’s group T is a piecewise generalisation of SL.2; Z/; the mapping class group is a multi-handle generalisation of SL.2; Z/. In the same vein SL.n; Z/ is an arithmetic generalization and Aut.Fn / is a non-commutative one. We also note that, following Thurston, the mapping class group g acts on the boundary of the Teichmüller space and preserves its piecewise projective integral structure. Another way to encode these groups is to consider pairs of binary trees which represent dyadic subdivisions. Dually, this data gives a simplicial bijection of the complementary forests, called partial automorphism of the infinite binary tree 2 . Of

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course this does not extend as an automorphism of 2 . However, one observes the following simple but essential fact: if one thickens the infinite tree 2 to a surface S0;1 , then the corresponding partial homeomorphisms extend to the entire surface S0;1 . Thus, two objects appear here: the surface S0;1 and the mapping class group which lifts the elements of a Thompson group. To make definitions precise, we are forced to endow the surface S0;1 with a rigid structure which encodes its tree-like aspect. The homeomorphisms we consider are asymptotically rigid, i.e. they preserve the rigid structure outside a compact subsurface. These homeomorphisms give rise to the asymptotically rigid mapping class groups. We now give some details on the structure of this chapter. We present in Section 2 various constructions of groups and spaces and explain how the group T itself is a mapping class group of S0;1 . Next, we introduce the (historically) first relation between Thompson groups and braid groups, namely the extension: 1 ! B1 ! AT ! T ! 1: In order to avoid working with non-finitely supported braids, the authors chose to build AT from a convenient geometric homomorphism T ! Out.B1 /: However, retrospectively, while having definite advantages, this choice may not have been the best. The main theorem of [68] says that the group AT is almost acyclic – the corresponding group AF 0 being acyclic. The proofs of these theorems are quite involved and far from the geometric-combinatorial topics discussed in the rest of this chapter; this is why we shall present them rather sketchily. However, we do describe the group AT as a mapping class group. It is actually while trying to extend the Burau representation from B1 to AT that the notion of asymptotically rigid mapping class group was formulated. The next two sections, 3 and 4, are of central importance. We show that a group B which is an asymptotically rigid mapping class group of S0;1 and surjects onto V is finitely presented. While the acyclicity theorem mentioned above was formulated on the basis of homotopy-theoretic evidence, the group B and its finite presentability came largely from conformal field theory evidence. The Moore–Seiberg duality groupoid is finitely presented, a fact mathematically established in [4], [5], [49]. We begin by introducing Penner’s Ptolemy groupoid, partly issued from the conformal field theory work of Friedan and Shenker. Its objects are ideal tesselations and its morphisms are compositions of flips. We then explain how Thompson’s groups fit into this setting. A basic observation here is that the Ptolemy groupoid is isomorphic to a sub-groupoid of the Moore–Seiberg stable duality groupoid. This duality groupoid is related in turn to a Hatcher–Thurston type complex for the surface S0;1 . One main result is that this complex is simply connected. Section 4 applies all this to the asymptotically rigid mapping class group B of S0;1 .

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Let us emphasize here that our notion of asymptotically rigid mapping class group is different from the asymptotic mapping class group considered recently by various authors (see the chapter written by Matsuzaki [99] in volume IV of this Handbook). The kernel of the morphism from B onto V is the compactly supported mapping class group of S0;1 . Let us note that the group B contains all genus zero mapping class groups as well as the braid groups. The main theorem states that B is a finitely presented group. A quite compact symmetric set of relations is produced as well. Section 5 is dedicated to the braided Ptolemy–Thompson group T . This is an extension of T by the braid group B1 . It is an asymptotically rigid mapping class group of S0;1 of a special kind. It is a simpler group than AT and will be used in Sections 5 and 6. We prove that T , like B, is a finitely presented group. We note that so far, AT is only known to be finitely generated. In Section 6, we consider a relative abelianisation of T : 1 ! Z D B1 =ŒB1 ; B1 ! T =ŒB1 ; B1 ! T ! 1: We prove that this central extension is classified by a multiple of the Euler class of T that we detect to be 12, where is the Euler class pulled-back to T . This fact eventually allows us to classify the dilogarithmic projective extension of T which arises in the quantization of the Teichmüller theory, as we explain as well. In Section 7 we discuss an infinite genus mapping class group that maps onto V which is proved to be (at least) finitely generated. It also has the property of being homologically equivalent to the stable mapping class group. As already mentioned, the proofs involve as a key ingredient the group T . In Section 8 we introduce a simplicial unified approach to the various extensions of the group V . This includes the extension BV of Brin and Dehornoy coming from categories with multiplication and from the geometry of algebraic laws, respectively. Moreover, one can approach in this way the action of the Grothendieck–Teichmüller y of B, thus getting a quite neat presentation of the entire group on a V -completion B setting. A sample of open questions is contained in the final section. We would like to dedicate these notes to the memory of Peter Greenberg and of Alexander Reznikov. Their work is inspiring us forever. Acknowledgments. The authors are indebted to L. Bartholdi, M. Bridson, M. Brin, J. Burillo, D. Calegari, F. Cohen, P. Dehornoy, D. Epstein, V. Fock, R. Geoghegan, E. Ghys, S. Goncharov, F. González-Acuña, V. Guba, P. Haissinsky, B. Harvey, V. Jones, R. Kashaev, F. Labourie, P. Lochak, J. Morava, H. Moriyoshi, P. Pansu, A. Papadopoulos, B. Penner, C. Pittet, M. Sapir, L. Schneps and H. Short for useful discussions and suggestions concerning this subject during the last few years.


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2 From Thompson’s groups to mapping class groups of surfaces 2.1 Three equivalent definitions of Thompson’s groups Groups of piecewise affine bijections. Thompson’s group F is the group of continuous and nondecreasing bijections of the interval Œ0; 1 which are piecewise dyadic affine. In other words, for each f 2 F , there exist two subdivisions of Œ0; 1, a0 D 0 < a1 < < an D 1 and b0 D 0 < b1 < bn , with n 2 N , such that: (1) ai C1 ai and biC1 bi belong to f 21k ; k 2 Ng; (2) the restriction of f to Œai ; aiC1 is the unique nondecreasing affine map onto Œbi ; bi C1 . Therefore, an element of F is completely determined by the data of two dyadic subdivisions of Œ0; 1 having the same cardinality. Let us identify the circle to the quotient space Œ0; 1=0 1. Thompson’s group T is the group of continuous and nondecreasing bijections of the circle which are piecewise dyadic affine. In other words, for each g 2 T , there exist two subdivisions of Œ0; 1, a0 D 0 < a1 < < an D 1 and b0 D 0 < b1 < bn , with n 2 N , such that: (1) ai C1 ai and biC1 bi belong to f 21k ; k 2 Ng. (2) There exists i0 2 f1; : : : ; ng, such that, pour each i 2 f0; : : : ; n 1g, the restriction of g to Œai ; aiC1 is the unique nondecreasing map onto ŒbiCi0 ; biCi0 C1 . The indices must be understood modulo n. Therefore, an element of T is completely determined by the data of two dyadic subdivisions of Œ0; 1 having the same cardinality, say n 2 N , plus an integer i0 mod n. Finally, Thompson’s group V is the group of bijections of Œ0; 1Œ, which are rightcontinuous at each point, piecewise nondecreasing and dyadic affine. In other words, for each h 2 V , there exist two subdivisions of Œ0; 1, a0 D 0 < a1 < < an D 1 and b0 D 0 < b1 < bn , with n 2 N , such that: (1) ai C1 ai and biC1 bi belong to f 21k ; k 2 Ng; (2) there exists a permutation 2 Sn , such that, for each i 2 f1; : : : ; ng, the restriction of h to Œai1 ; ai Œ is the unique nondecreasing affine map onto Œb.i/1 ; b.i/ Œ. It follows that an element h of V is completely determined by the data of two dyadic subdivisions of Œ0; 1 having the same cardinality, say n 2 N , plus a permutation 2 Sn . Denoting Ii D Œai1 ; ai and Ji D Œbi1 ; bi , these data can be summarized into a triple ..Ji /1in ; .Ii /1in ; 2 Sn /. Such a triple is not uniquely determined by the element h. Indeed, a refinement of the subdivisions gives rise to a new triple defining the same h. This remark also applies to elements of F and T .

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The inclusion F T is obvious. The identification of the integer i0 mod n to the cyclic permutation W k 7! k C i0 yields the inclusion T V . R. Thompson proved that F , T and V are finitely presented groups and that T and V are simple (cf. [26]). The group F is not perfect (F=ŒF; F is isomorphic to Z2 ), but F 0 D ŒF; F is simple. However, F 0 is not finitely generated (this is related to the fact that an element f of F lies in F 0 if and only if its support is included in 0; 1Œ). Historically, Thompson’s groups T and V are the first examples of infinite simple and finitely presented groups. Unlike F , they are not torsion-free. 1

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Groups of diagrams of finite binary trees. A finite binary rooted planar tree is a finite planar tree having a unique 2-valent vertex, called the root, a set of monovalent vertices called the leaves, and whose other vertices are 3-valent. The planarity of the tree provides a canonical labelling of its leaves, in the following way. Assuming that the plane is oriented, the leaves are labelled from 1 to n, from left to right, the root being at the top and the leaves at the bottom. There exists a bijection between the set of dyadic subdivisions of Œ0; 1 and the set of finite binary rooted planar trees. Indeed, given such a tree, one may label its vertices by dyadic intervals in the following way. First, the root is labelled by Œ0; 1. Suppose that a vertex is labelled by I D Œ 2kn ; kC1 , then its two descendant vertices 2n k 2kC1 are labelled by the two halves I : Œ 2n ; 2nC1 for the left one and Œ 2kC1 ; kC1 for the 2n 2nC1 right one. Finally, the dyadic subdivision associated to the tree is the sequence of intervals which label its leaves. As we have just seen, an element of Thompson’s group V is defined by the data of two dyadic subdivisions of Œ0; 1, with the same cardinality n, plus a permutation 2 Sn . This amounts to encoding it by a pair of finite binary rooted trees with the same number of leaves n 2 N , plus a permutation 2 Sn . Thus, an element h of V is represented by a triple .1 ; 0 ; /, where 0 and 1 have the same number of leaves n 2 N , and belongs to the symmetric group Sn . Such a triple will be called a symbol for h. It is convenient to interpret the permutation as the bijection ' which maps the i-th leaf of the source tree 0 to the .i/-th leaf


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of the target tree 1 . When h belongs to F , the permutation , which is the identity, is not represented, and the symbol reduces to a pair of trees .1 ; 0 /. When h belongs to T , the cyclic permutation is graphically materialized by a small circle surrounding the leaf number .1/ of 1 . One introduces the following equivalence relation on the set of symbols: two symbols are equivalent if they represent the same element of V . One denotes by Œ1 ; 0 ; the equivalence class of the symbol. Therefore, V is (in bijection with) the set of equivalence classes of symbols. The composition law of piecewise dyadic affine bijections is pushed out on the set of equivalence classes of symbols in the following way. In order to define Œ10 ; 00 ; 0 Œ1 ; 0 ; , one may suppose, at the price of refining both symbols, that the tree 1 coincides with the tree 00 . The product of the two symbols is Œ10 ; 1 ; 0 Œ1 ; 0 ; D Œ10 ; 0 ; 0 B : The neutral element is represented by any symbol .; ; 1/, for any finite binary rooted planar tree . The inverse of Œ1 ; 0 ; is Œ0 ; 1 ; 1 . It follows that V is isomorphic to the group of equivalence classes of symbols endowed with this internal law. 1

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Partial automorphisms of trees ([84]). The beginning of the article [84] formalizes a change of point of view, consisting in considering, not the finite binary trees, but their complements in the infinite binary tree. Let T2 be the infinite binary rooted planar tree (all its vertices other than the root are 3-valent). Each finite binary rooted planar tree can be embedded in a unique way into T2 , assuming that the embedding maps the root of onto the root of T2 , and

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respects the orientation. Therefore, may be identified with a subtree T2 , whose root coincides with that of T2 . Definition 2.1 ( cf. [84]). A partial isomorphism of T2 consists of the data of two finite binary rooted subtrees 0 and 1 of T2 having the same number of leaves n 2 N , and an isomorphism q W T2 n 0 ! T2 n 1 . The complements of 0 and 1 have n components, each one isomorphic to T2 , which are enumerated from 1 to n according to the labelling of the leaves of the trees 0 and 1 . Thus, T2 n 0 D T01 [ [ T0n and T2 n 1 D T11 [ [ T1n where the Tji ’s are the connected components. Equivalently, the partial isomorphism of T2 is given by a permutation 2 Sn and, for i D 1; : : : ; n, an isomorphism qi W T0i ! T1.i/ . Two partial automorphisms q and r can be composed if and only if the target of r coincides with the source of r. One gets the partial automorphism q B r. The composition provides a structure of inverse monoid on the set of partial automorphisms, which is denoted Fred.T2 /. 0

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1

2

One may construct a group from Fred.T2 /. Let @T2 be the boundary of T2 (also called the set of “ends” of T2 ) endowed with its usual topology, for which it is a Cantor set. The point is that a partial automorphism does not act (globally) on the tree, but does act on its boundary. One has therefore a morphism Fred.T2 / ! Homeo.@T2 /, whose image N is the spheromorphism group of Neretin. Let now FredC .T2 / be the sub-monoid of Fred.T2 /, whose elements are the partial automorphisms which respect the local orientation of the edges. Thompson’s group V can be viewed as the subgroup of N which is the image of FredC .T2 / by the above morphism. Remark 2.1. There exists a Neretin group Np for each integer p 2, as introduced in [106] (with different notation). They are constructed in a similar way as N , by replacing the dyadic complete (rooted or unrooted) tree by the p-adic complete (rooted or unrooted) tree. They are proposed as combinatorial or p-adic analogues of the diffeomorphism group of the circle. Some aspects of this analogy have been studied in [80].


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2.2 Some properties of Thompson’s groups Most readers of this section are probably more comfortable with the mapping class group than with Thompson’s groups. Therefore, we think that it will be useful to gather here some of the classical and less classical properties of Thompson’s groups. There is a fair amount of randomness in our choices and the only thing we would really like to emphasize is their ubiquity. Thompson’s groups became known in algebra because T and V were the first infinite finitely presented simple groups. They were preceded by Higman’s example of an infinite finitely generated simple group in 1951. More recently, Burger and Mozes (see [21]) constructed an example which is also without torsion. Thompson used F and V to give new examples of groups with an unsolvable word problem and also in his algebraic characterisation of groups with a solvable word problem (see [116]) as being those which embed in a finitely generated simple subgroup of a finitely presented group. The group F was rediscovered in homotopy theory, as a universal conjugacy idempotent, and later in universal algebra. We refer to [26] for an introduction from scratch to several aspects of Thompson’s groups, including their presentations, and also their piecewise linear and projective representations. One can find as well an introduction to the amenability problem for F , including a proof of the Brin–Squier–Thompson theorem that F does not contain a free group of rank 2. Last but not least, one can find a list of the merely 25 notations in the literature for F , T and V . Fortunately, after [26] appeared, the notation has almost stabilized. We also mention the survey [115] for various other aspects and [62] and [105] for the general topic of homeomorphisms of the circle. The groups F , T and V are actually FP1 , i.e. they have classifying spaces with finite skeleton in each dimension; this was first proved by Brown and Geoghegan (see [19], [17]). Let us mention what is the rational cohomology of these groups, computed by Ghys and Sergiescu in [63] and Brown in [18]. First, H .F I Q/ is the product between the divided powers algebra on one generator of degree 2 and the cohomology algebra of the 2-torus. The cohomology of T is the quotient Q.; ˛/= ˛, where is the Euler class and ˛ a (discrete) Godbillon–Vey class. For what concerns the group V , its rational cohomology vanishes in each dimension. See [115] for more results with either Z or with twisted coefficients. Here are other properties of these groups involving cohomology. Using a smoothening of Thompson’s group it is proved in [63] that there is a representation 1 .†12 / ! Diff.S 1 / having Euler number 1 and an invariant Cantor set. Reznikov showed that the group T does not have Kazhdan’s property T (see [112]), and later Farley [39] proved that it has Haagerup’s property AT (also called a-Tmenability). Therefore it verifies the Baum–Connes conjecture (see also [38]). Napier and Ramachandran proved that F is not a Kähler group [104]. Cyclic cocycles on T were introduced in [108]. The group T in relation with the symplectic automorphisms of CP 2 was considered by Usnich in [119].

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A theorem of Brin [15] states that the group of outer automorphisms of T is Z=2Z. Furthermore, in [22] the authors computed the abstract commensurator of F . Using the above mentioned smoothening, it is proved in [63] that all rotation numbers of elements in T are rational. New direct proofs were given by Calegari ([25]), Liousse ([90]) and Kleptsyn (unpublished). For the connection of F and T with the piecewise projective C 1 -homeomorphisms, see for instance [66], [67] and [97]. The group F is naturally connected to associativity in various frameworks [32], [57], [41]. See also [13], [14] for the group V . Brin proved that the rank-2 free group is a limit of Thompson’s group F ([16]). Complexity aspects were considered in [9]. Guba ([72]) showed that the Dehn function for F is quadratic. The group F was studied in cryptography in [113], [100], [6]. Thompson’s groups were studied from the viewpoint of C -algebras and von Neumann algebras; see for instance Jolissaint ([79]) and Haagerup–Picioroaga [74]. On the edge of logic and group theory, the interpretation of arithmetic in Thompson’s groups was investigated by Bardakov–Tolstykh ([6]) and Altinel–Muranov ([2]). Let us finally mention the work of Guba and Sapir on Thompson’s groups as diagram groups; see for instance [73]. Let us emphasize here that we avoided to speak on generalisations of Thompson’s groups: this topic is pretty large and we think it would not be at its place here. Let us close this section by mentioning again that our choice was just to mention some developments related to Thompson’s groups from the unique angle of ubiquity.

2.3 Thompson’s group T as a mapping class group of a surface The article [84] is partly devoted to developing the notion of an asymptotically rigid homeomorphism. Definition 2.2 (following [84]). (1) Let S0;1 be the oriented surface of genus zero, which is the following inductive limit of compact oriented genus zero surfaces with boundary Sn : Starting with a cylinder S1 , one gets SnC1 from Sn by gluing a pair of pants (i.e. a three-holed sphere) along each boundary circle of Sn . This construction yields, for each n 1, an embedding Sn ,! SnC1 , with an orientation on SnC1 compatible with that of Sn . The resulting inductive limit (in the topological category) of the Sn ’s is the surface S0;1 : S0;1 D lim Sn : !

n

(2) By the above construction, the surface S0;1 is the union of a cylinder and of countably many pairs of pants. This topological decomposition of S0;1 will be called the canonical pair of pants decomposition. The set of isotopy classes of orientation-preserving homeomorphisms of S0;1 is an uncountable group. The group operation is map composition. By restricting


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to a certain type of homeomorphisms (called asymptotically rigid), we shall obtain countable subgroups. We first need to complete the canonical decomposition to a richer structure. Let us choose an involutive homeomorphism j of S0;1 which reverses the orientation, stabilizes each pair of pants of its canonical decomposition, and has fixed points along lines which decompose the pairs of pants into hexagons. The surface S0;1 can be disconnected along those lines into two planar surfaces with boundary, one of which is called the visible side of S0;1 , while the other is the hidden side of S0;1 . The involution j maps the visible side of S0;1 onto the hidden side, and vice versa. From now on, we assume that such an involution j is chosen, hence a decomposition of the surface into a “visible” and a “hidden” side. Definition 2.3. The data consisting of the canonical pants decomposition of S0;1 together with the above decomposition into a visible and a hidden side is called the canonical rigid structure of S0;1 . The tree T2 may be embedded into the visible side of S0;1 , as the dual tree to the pants decomposition. This set of data is represented in Figure 3.

Figure 3. Surface S0;1 with its canonical rigid structure.

The surface S0;1 appears already in [50], endowed with a pants decomposition (with no cylinder), dual to the regular unrooted dyadic tree. In [84], the notion of asymptotically rigid homeomorphism is defined. It plays a key role in [50], [51] and [52]. Let us introduce some more terminology. Any connected and compact subsurface of S0;1 which is the union of the cylinder and finitely many pairs of pants of the canonical decomposition will be called an admissible subsurface of S0;1 . The type of such a subsurface S is the number of connected components in its boundary. The tree of S is the trace of T2 on S. Clearly, the type of S is equal to the number of leaves of its tree. Definition 2.4 (following [84] and [50]). A homeomorphism ' of S0;1 is asymptotically rigid if there exist two admissible subsurfaces S0 and S1 having the same type,

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such that '.S0 / D S1 and whose restriction S0;1 n S0 ! S0;1 n S1 is rigid, meaning that it maps each pants (of the canonical pants decomposition) onto a pants, preserving the canonical rigid structure. The asymptotically rigid mapping class group of S0;1 is the group of isotopy classes of asymptotically rigid homeomorphisms. Though the proof of the following theorem is easy, this theorem is seminal as it is the starting point of a deeper study of the links between Thompson’s groups and mapping class groups. Theorem 2.2 ([84], Theorem 3.3). Thompson’s group T can be embedded into the group of isotopy classes of orientation-preserving homeomorphisms of S0;1 . An isotopy class belongs to the image of the embedding if it may be represented by an asymptotically rigid homeomorphism of S0;1 which globally preserves the decomposition into visible/hidden sides. We denote by SC 0;1 the visible side of S0;1 . This is a planar surface which inherits from the canonical decomposition of S0;1 a decomposition into hexagons (and one rectangle, corresponding to the visible side of the cylinder). We could restate the above definitions by replacing pairs of pants by hexagons and the surface S0;1 by its visible side SC 0;1 . Then Theorem 2.2 states that T can be embedded into the mapping class group of the planar surface SC 0;1 . In fact T is the asymptotically rigid mapping C class group of S0;1 , namely the group of mapping classes of those homeomorphisms of SC 0;1 which map all but finitely many hexagons onto hexagons.

2.4 Braid groups and Thompson’s groups A seminal result, which is the starting point of the article [84], is a theorem of Greenberg and the third author ([68]). It states that there exists an extension of the derived subgroup F 0 of Thompson’s group F by the stable braid group B1 (i.e. the braid group on a countable set of strands), 1 ! B1 ! A ! F 0 ! 1;

(Gr–Se)

where the group A is acyclic, i.e. its integral homology vanishes. The existence of such a relation between Thompson’s group and the braid group was conjectured by comparing their homology types. On the one hand, it is proved in [63] that F 0 has the homology of S 3 , the space of based loops on the three-dimensional sphere. More precisely, the +-construction of the classifying space BF 0 is homotopically equivalent to S 3 . On the other hand, F. Cohen proved that B1 has the homology of 2 S 3 , the double loop space of S 3 . It turns out that both spaces (S 3 and 2 S 3 ) are related by the path fibration 2 S 3 ,! P .S 3 / ! S 3 ;


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where P .S 3 / denotes the space of based paths on S 3 . The total space of this fibration, P .S 3 /, is contractible. Therefore, the existence of this natural fibration has led the authors of [68] to conjecture the existence of the short exact sequence (Gr–Se). The construction of A amounts to giving a morphism F 0 ! Out.B1 /. In [68] one is lead to consider an extended binary tree and the braid group relative to its vertices. The group F 0 acts on that tree by partial automorphisms and therefore induces the desired morphism. Let us give a hint on how the acyclicity of A is proved in [68]. Via direct computations, one shows that H1 .A/ D 0. One then proves that the fibration BB1C ! BAC ! BFC0 can be delooped to a fibration S 3 ! E ! S 3 : Using the fact that A is perfect one concludes that the space E is contractible and so A is acyclic. As a matter of fact, it is also proved in [68] that the short exact sequence (Gr–Se) extends to the Thompson group T . Indeed, there exists a short exact sequence 1 ! B1 ! AT ! T ! 1 whose pull-back via the embedding F 0 ,! T is (Gr–Se). At the homology level, it corresponds to a fibration 2 S 3 ! S 3 CP 1 ! LS 3 ; where LS 3 denotes the free non-parametrized loop space of S 3 . This fact should not to be considered as anecdotic for the following reason. Let us divide the groups B1 and AT by the derived subgroup of B1 . One obtains a central extension of T by Z D H1 .B1 /, which may be identified in the second cohomology group H 2 .T; Z/ to the discrete Godbillon–Vey class of Thompson’s group T . Let us emphasize that a simpler version of AT , namely the braided Ptolemy– Thompson group T , will be presented later. Retrospectively, AT could be called then the marked braided Ptolemy–Thompson group. One of the motivations of [84] is to pursue the investigations about the analogies between the diffeomorphism group of the circle Diff.S 1 / and Thompson’s group T . A remarkable aspect of this analogy concerns the Bott–Virasoro–Godbillon–Vey class. The latter is a differentiable cohomology class of degree 2. Recall that the Lie algebra of the group Diff.S 1 / is the algebra Vect.S 1 / of vector fields on the circle. There is a map H .Diff.S 1 /; R/ ! H .Vect.S 1 /; R/, where the right hand-side denotes the Gelfand–Fuchs cohomology of Vect.S 1 /, which is simply induced by the differentiation of cocycles. The image of the Bott–Virasoro–Godbillon–Vey class is a generator of H 2 .Vect.S 1 /; R/ corresponding to the universal central extension of Vect.S 1 /, known by the physicists as the Virasoro Algebra.

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Let us explain the analogies between the cohomologies of T and Diff.S 1 /. By the cohomology of T we mean Eilenberg–McLane cohomology, while the cohomology under consideration on Diff.S 1 / is the differentiable one (as very little is known about its Eilenberg–McLane cohomology). The striking result is the following: the ring of cohomology of T (with real coefficients) and the ring of differentiable cohomology of Diff.S 1 / are isomorphic. Both are generated by two classes of degree 2: the Euler class (coming from the action on the circle), and the Bott–Virasoro–Godbillon–Vey class. In the cohomology ring of T , the Bott–Virasoro–Godbillon–Vey class is called the discrete Godbillon–Vey class. The isomorphism between the two cohomology rings does not seem to be induced by known embeddings of T into Diff.S 1 / (such embeddings have been constructed in [63]). A fundamental aspect of the Godbillon–Vey class concerns its relations with the projective representations of Diff.S 1 /, especially those which may be derived into highest weight modules of the Virasoro Algebra. Pressley and Segal ([111]) introduced some representations of Diff.S 1 / in the restricted linear group GLres of the Hilbert space L2 .S 1 /. Pulling back by a certain cohomology class (which we refer to as the Pressley–Segal class) of GLres , one obtains on Diff.S 1 / some multiples of the Godbillon–Vey class (cf. [84], §4.1.3 for a precise statement).

2.5 Extending the Burau representation In [84], we show that an analogous scenario exists for the discrete Godbillon–Vey class gv N of T . We first remark that the Pressley–Segal extension of GLres is itself a pull-back of GL.H/ GL.H/ 1 ! C ! ! 1; ! T1 T where GL.H/ denotes the group of bounded invertible operators of the Hilbert space H, T the group of operators having a determinant, and T1 the subgroup of operators having determinant 1. The first step is to reconstruct the group AT , not in a combinatorial way as in [68], but as a mapping class group of a surface St0;1 . The latter is obtained from S0;1 , by gluing, on each pair of pants of its canonical decomposition, an infinite cylinder or “tube”, marked with countably many punctures (cf. Figure 4). The precise definition of the group AT being rather technical, we refer for that the reader to [84]. This new approach provides a setting that is convenient for an easy extension of the Burau representation of the braid group to AT . We proceed as follows. The group AT acts on the fundamental group of the punctured surface St0;1 , which is a free group of infinite countable rank. Moreover, the action is index-preserving, i.e. it induces the identity on H1 .F1 /. Let Autind .F1 / be the group of automorphisms of F1 which are index-preserving. The Magnus representation of Aut ind .Fn / extends to an infinite-dimensional representation of Aut ind .F1 / in the Hilbert space `2 on the set of punctures of St0;1 . Composing with the map AT ! Autind .F1 /, one obtains a repre-


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fv0

e0∗

v0

Figure 4. Decomposition of St0;1 into pants with tubes.

t W AT ! GL.`2 / which extends the classical Burau representation of the sentation 1 braid group Bn . The scalar t 2 C parameterizes a family of such representations.

Theorem 2.3 ([84], Theorem 4.7). For each t 2 C , the Burau representation t t 1 W B1 ! T extends to a representation 1 of the mapping class group AT in t 2 the Hilbert space ` on the set of punctures S0;1 . There exists a morphism of extensions / B1 / AT /T /1 1

1

/T

/ GL.`2 / /

/1

GL.`2 / T

which induces a morphism of central extensions 1

/ H1 .B1 /

1

/ C

/

AT ŒB1 ;B1

/

GL.`2 / T1

/

/T

/1

/ 1.

GL.`2 / T

The vertical arrows are injective if t 2 C is not a root of unity.

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3 From the Ptolemy groupoid to the Hatcher–Thurston complex 3.1 Universal Teichmüller theory according to Penner In [109] (see also [110]), R. Penner introduced his version of a universal Teichmüller space, together with an associated universal group. Unexpectedly, this group happens to be isomorphic to the Thompson group T . This connection between Thompson groups and Teichmüller theory plays a key role in [50], [51] and [52]. It is therefore appropriate to give some insight into Penner’s approach. The universal Teichmüller space according to Penner is a set Tess of ideal tessellations of the Poincaré disk, modulo the action of PSL.2; R/ (cf. Definition 3.1 below). The space Tess is homogeneous under the action of the group HomeoC .S 1 / of orientation-preserving homeomorphisms of the circle: Tess D HomeoC .S 1 /=PSL.2; R/: Denoting by Diff C .S 1 / the diffeomorphism group of S 1 and Homeoqs .S 1 / the group of quasi-symmetric homeomorphisms of S 1 (a quasi-symmetric homeomorphism of the circle is induced by a quasi-conformal homeomorphism of the disk) one has the following inclusions Diff C .S 1 /=PSL.2; R/ ,! Homeoqs .S 1 /=PSL.2; R/ ,! HomeoC .S 1 /=PSL.2; R/; which justify that Tess is a generalization of the “well known” universal Teichmüller spaces, namely Bers’space Homeoqs .S 1 /=PSL.2; R/, and the physicists’space Diff C .S 1 /=PSL.2; R/. Moreover, Penner introduced some coordinates on Tess, as well as a “formal” symplectic form, whose pull-back on Diff C .S 1 /=PSL.2; R/ is the Kostant–Kirillov– Souriau form. Definition 3.1 (following [109] and [110]). Let D be the Poincaré disk. A tessellation of D is a locally finite and countable set of complete geodesics on D whose endpoints 1 lie on the boundary circle S1 D @D and are called vertices. The geodesics are called arcs or edges, forming a triangulation of D. A marked tessellation of D is a pair made of a tessellation plus a distinguished oriented edge (abbreviated d.o.e.) aE . One denotes by Tess0 the set of marked tessellations. p 1 Consider the basic ideal triangle having vertices at 1; 1; 1 2 S1 in the unit disk model D. The orbits of its sides by the group PSL.2; Z/ is the so-called Farey tessellation 0 , as drawn in Figure 5. Its ideal vertices are the rational points of @D. The marked Farey tessellation has its distinguished oriented edge aÆ0 joining 1 to 1. The group HomeoC .S 1 / acts on the left on Tess0 in the following way. Let be an arc of a marked tessellation , with endpoints x and y, and f be an element of HomeoC .S 1 /; then f . / is defined as the geodesic with endpoints f .x/ and f .y/. If is oriented from x to y, then f . / is oriented from f .x/ to f .y/. Finally,


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Figure 5. Farey tessellation and its dual tree.

f . / is the marked tessellation ff . /; 2 g. Viewing PSL.2; R/ as a subgroup of HomeoC .S 1 /, one defines Tess as the quotient space Tess0 =PSL.2; R/. For any 2 Tess0 , let us denote by 0 its set of ideal vertices. It is a countable and dense subset of the boundary circle, so that it may be proved that there exists a unique f 2 HomeoC .S 1 / such that f .0 / D . One denotes this homeomorphism f . The resulting map Tess0 ! HomeoC .S 1 /; 7! f is a bijection. It follows that Tess D HomeoC .S 1 /=PSL.2; R/. Since the action of PSL.2; R/ is 3-transitive, each element of Tess can be uniquely represented by its normalized marked triangulation containing the basic ideal triangle and whose d.o.e. is aÆ0 . The marked tessellation is of Farey-type if its canonical marked triangulation has the same vertices and all but finitely many triangles (or sides) as the Farey triangulation. Unless explicitly stated otherwise all tessellations considered in the sequel will be Farey-type tessellations. In particular, the ideal triangulations have the same vertices as 0 and coincide with 0 for all but finitely many ideal triangles.

3.2 The isomorphism between Ptolemy and Thompson groups Definition 3.2 (Ptolemy groupoid). The objects of the (universal) Ptolemy groupoid Pt are the marked tessellations of Farey-type. The morphisms are ordered pairs of marked tessellations modulo the common PSL.2; R/ action. We now define particular elements of Pt called flips. Let e be an edge of the marked tessellation represented by the normalized marked triangulation .; aE /. The result of

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the flip Fe on is the triangulation Fe ./ obtained from by changing only the two neighboring triangles containing the edge e, according to the picture below:

e

e0

This means that we remove e from and then add the new edge e 0 in order to get Fe . /. In particular there is a natural correspondence W ! Fe . / sending e to e 0 and being the identity on all other edges. The result of a flip is the new triangulation together with this edge correspondence. a/ D aE . If e is the d.o.e. of then Fe .E a/ D eÅ0 , If e is not the d.o.e. of then Fe .E e ; eÅ0 / is positively oriented. where the orientation of eÅ0 is chosen so that the frame .E a//. We define the flipped tessellation Fe ..; aE // to be the tessellation .Fe . /; Fe .E It is proved in [109] that flips generate the Ptolemy groupoid, i.e. any element of Pt is a composition of flips. There is also a slightly different version of the Ptolemy groupoid which is quite useful in the case where we consider Teichmüller theory for surfaces of finite type. Specifically, we should assume that the tessellations are labelled, namely that their edges are indexed by natural numbers. Definition 3.3 (Labelled Ptolemy groupoid). The objects of the labelled (universal) are the labelled marked tessellations. The morphisms between Ptolemy groupoid Pt two objects .1 ; aÆ1 / and .2 ; aÆ2 / are eventually trivial permutation maps (at the labels level) W 1 ! 2 such that .aÆ1 / D aÆ2 . When marked tessellations are represented by their normalized tessellations, the latter coincide for all but finitely many triangles. Recall that is said to be eventually trivial if the induced correspondence at the level of the labelled tessellations is the identity for all but finitely many edges. Indeed Now flips make sense as elements of the labelled Ptolemy groupoid Pt. the flip Fe is endowed with the natural eventually trivial permutation W ! Fe . / sending e to e 0 and being the identity for all other edges. There is a standard procedure for converting a groupoid into a group, by using an a priori identification of all objects of the category. Here is how this goes in the case of the Ptolemy groupoid. For any marked tessellation .; aE / there is a characteristic map Q W Q f1; 1g ! . Assume that is the canonical triangulation representing this tessellation. We first label by Q [ 1 the vertices of , by induction: p (1) 1 is labelled by 0=1, 1 is labelled by 1 D 1=0 and 1 is labelled by 1=1. (2) If we have a triangle in having two vertices already labelled by a=b and c=d then its third vertex is labelled .a C c/=.b C d /. Notice that vertices in the upper half-plane are labelled by negative rationals and those from the lower half-plane by positive rationals.


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As it is well known this labeling produces a bijection between the set of vertices of and Q [ 1. Let now e be an edge of , which is different from aE . Let v.e/ be the vertex opposite to e of the triangle of containing e in its frontier and lying in the component of D e which does not contain aE . We then associate to e the label of v.e/. We also give aE the label 0 2 Q. In this way one obtains a bijection Q W Q f1; 1g ! . Remark that if .1 ; aÆ1 / and .2 ; aÆ2 / are marked tessellations then there exists a unique map f between their vertices sending triangles to triangles and marking on marking. Then f B Q1 D Q2 . The role played by Q is to allow flips to be indexed by the rationals and not by the edges of . Definition 3.4 (Ptolemy group [109]). Let T be the set of marked tessellations of Farey-type. Define the action of the free monoid M generated by Q f1; 1g on T by means of q .; aE / D FQ .q/ .; aE /

for q 2 Q f1; 1g; .; aE / 2 FT:

We set f f 0 on M if the two actions of f and f 0 on T coincide. Then the induced composition law on M= is a monoid structure for which each element has an inverse. This makes M= a group, which is called the Ptolemy group T (see [109] for more details). In particular it makes sense to speak of flips in the present case. It is clear that flips generate the Ptolemy group. The notation T for the Ptolemy group is not misleading because this group is isomorphic to the Thompson group T and for this reason, we preferred to call it the Ptolemy–Thompson group. Given two marked tessellations .1 ; aÆ1 / and .2 ; aÆ2 / the above combinatorial isomorphism f W 1 ! 2 provides a map between the vertices of the tessellations, 1 . This map extends continuously to a homewhich are identified with P 1 .Q/ S1 1 omorphism of S1 , which is piecewise-PSL.2; Z/. This establishes an isomorphism between the Ptolemy group and the group of piecewise-PSL.2; Z/ homeomorphisms of the circle. An explicit isomorphism with the group T in the form introduced above was provided by Lochak and Schneps (see [93]). In order to understand this isomorphism we will need another characterization of the Ptolemy groupoid, as follows. Definition 3.5 (Ptolemy groupoid, second definition [109], [110]). The universal Ptolemy groupoid Pt 0 is the category whose objects are the marked tessellations. As for the morphisms, they are composed of morphisms of two types, called elementary moves: (1) A-move: it is the data of a pair of marked tessellations .1 ; 2 /, where 1 and 2 only differ by the d.o.e. The d.o.e. aÆ1 of 1 is one of the two diagonals of a

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quadrilateral whose 4 sides belong to 1 . Let us assume that the vertices of this quadrilateral are enumerated in the cyclic direct order by x, y, z, t , in such a way that aÆ1 is the edge oriented from z to x. Let aÆ2 be the other diagonal, oriented from t to y. Then, 2 is defined as the marked tessellation 1 n faÆ1 g [ faÆ2 g, with oriented edge aÆ2 . (2) B-move: it is the data of a pair of marked tessellations .1 ; 2 /, where 1 and 2 have the same edges, but only differ by the choice of the d.o.e. The marked edge aÆ1 is the side of the unique triangle of the tessellation 1 with ideal vertices x, y, z, enumerated in the direct order, in such a way that aÆ1 is the edge from x to y. Let aÆ2 be the edge oriented from y to z. Then, aÆ2 is the d.o.e. of 2 . Relations between morphisms: if 1 and 2 are two marked tessellations such that there exist two sequences of elementary moves .M1 ; : : : ; Mk / and .M10 ; : : : ; Mk0 0 / connecting 1 to 2 , then the morphisms Mk B B M1 and Mk0 0 B B M10 are equal. Remark 3.1. Given two marked tessellations 1 and 2 with the same sets of endpoints, there is a (non-unique) finite sequence of elementary moves connecting 1 to 2 if and only if 1 and 2 only differ by a finite number of edges. From the above remark, it follows that Pt 0 is not a connected groupoid. Let Pt D Pt 0Q be the connected component of the Farey tessellation. It is the full sub-groupoid of Pt0 obtained by restricting to the tessellations whose set of ideal vertices are the rationals of the boundary circle @D, and which differ from the Farey tessellation by only finitely many edges, namely the Farey-type tessellations. Then it is not difficult to prove that the two definitions of Pt are actually equivalent. However the second definition makes the Lochak–Schneps isomorphism more transparent. Construction of the universal Ptolemy group. Let W be a symbol A or B. For any 2 Ob.Pt 0 /, let us define the object W ./, which is the target of the morphism of type W , whose source is . For any sequence W1 ; : : : ; Wk of symbols A or B, let us use the notation Wk W2 W1 ./ for Wk .:::W2 .W1 . //:::/. Let M be the free group on fA; Bg. Let us fix a tessellation (the construction will not depend on this choice). Let K be the subgroup of M made of the elements Wk W2 W1 such that Wk W2 W1 ./ D (it can be easily checked that this implies Wk W2 W1 . 0 / D 0 for any 2 Ob.Pt0 /, and that K is a normal subgroup of M ). Definition 3.6 ([109], [110]). The group G D M=K is called the universal Ptolemy group. Theorem 3.2 (Imbert–Kontsevich–Sergiescu, [78]). The universal Ptolemy group G is anti-isomorphic to the Thompson group T , which will be henceforth also called the Ptolemy–Thompson group in order to emphasize this double origin.


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Let us indicate a proof that relies on the definition of T as a group of bijections of the boundary of the dyadic tree. Let 2 Pt, and let T be the regular (unrooted) dyadic tree which is dual to the tessellation . Let e be the edge of T which is transverse to the oriented edge aE of . The a ; eE / is directly oriented in the disk. For edge e is oriented in such a way that .E each pair .; 0 / of marked tessellations of Pt, let '; 0 2 Isom.T ; T 0 / be the unique isomorphism of planar oriented trees which maps the oriented edge eE onto the oriented edge eE 0 . As a matter of fact, the planar trees T and T 0 coincide outside two finite subtrees t and t 0 respectively, so that their boundaries @T and @T 0 may be canonically identified. Therefore, '; 0 induces a homeomorphism of @T0 , denoted @'; 0 . Clearly, @'; 0 belongs to T , as it is induced on the boundary of the dyadic planar tree by a partial isomorphism which respects the local orientation of the edges. The map g 2 G 7! @'0 ;g.0 / 2 Homeo.@T0 / has T as image, and is an antiisomorphism onto T . An explanation for the anti-isomorphy is the following. One has '0 ;gh.0 / D 'h.0 /;g.h.0 // '0 ;h.0 / . Now 'h.0 /;g.h.0 // is the conjugate of '0 ;g.0 / by '0 ;h.0 / , hence '0 ;gh.0 / D '0 ;h.0 / '0 ;g.0 / . Following [78], it is also possible to construct an anti-isomorphism between G and T , when the latter is realized as a subgroup of HomeoC .S 1 /, viewing the circle as the boundary of the Poincaré disk. For each g 2 G, there exists a unique f 2 HomeoC .S 1 / such that f .0 / D g.0 /. It is denoted by fg . This provides a map f W G ! HomeoC .S 1 /, g 7! fg , which is an anti-isomorphism. Indeed, for all h and g in G, the effect of h on D g.0 / is the same as the effect of the conjugate fg Bfh Bfg1 , so that .hg/.0 / D fg Bfh Bfg1 . / D .fg B fh /.0 /. The morphism is injective, since fg D id implies that g.0 / D 0 , hence g D 1. It is worth mentioning that a new presentation of T has been obtained in [93], derived from the anti-isomorphism of G and T . It uses only two generators ˛ and ˇ, defined as follows. Let ˛ 2 T be the element induced by '0 ;A:0 , and ˇ 2 T induced by '0 ;B:0 . Theorem 3.3 ([93]). The Ptolemy–Thompson group T is generated by two elements ˛ and ˇ, with relations ˛ 4 D 1; Œˇ˛ˇ; ˛ 2 ˇ˛ˇ˛ 2 D 1;

ˇ 3 D 1;

.ˇ˛/5 D 1;

Œˇ˛ˇ; ˛ 2 ˇ 2 ˛ 2 ˇ˛ˇ˛ 2 ˇ˛ 2 D 1:

(The relation .ˇ˛/5 D 1 is called the pentagon relation in T .) Let us make explicit the relation between the Cayley graph of T , for the above presentation, and the nerve of the category Pt. Definition 3.7. Let Gr.Pt/ be the graph whose vertices are the objects of Pt, and whose edges correspond to the elementary moves of type A and B.

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From the anti-isomorphism between G D M=K and T , it follows easily that Gr.Pt/ is precisely the Cayley graph of Thompson’s group T , for its presentation on the generators ˛ and ˇ. We can use the same method to derive a labelled Ptolemy group Tz out of the It is not difficult to obtain therefore the following: labelled Ptolemy groupoid Pt. Proposition 3.4. We have an exact sequence 1 ! S1 ! Tz ! T ! 1; where S1 is the group of eventually trivial permutations of the labels. Moreover, the group Tz is generated by the obvious lifts ˛Q and ˇQ of the generators ˛, ˇ of T . The pentagon relation now reads .ˇQ ˛/ Q 5 D 12 , where 12 is the transposition exchanging the labels of the diagonals of the pentagon. Remark 3.5. Let us mention that the image of G in HomeoC .S 1 / by the anti-isomorphism f W g 7! fg does not correspond to the piecewise dyadic affine version of T , as recalled in the preliminaries. Let us view here the circle S 1 as the real projective line, and not as the quotient space Œ0; 1=0 1. Under this identification, f .G/ is the group P PSL.2; Z/ of orientation preserving homeomorphisms of the projective line, which are piecewise PSL.2; Z/, with rational breakpoints. This version of T is the starting point of a detailed study of the piecewise projective geometry of Thompson’s group T , led in [97] and [98].

3.3 A remarkable link between the Ptolemy groupoid and the Hatcher–Thurston complex of S0;1 , following [50] In [50], we give a generalization of the Ptolemy groupoid which uses pairs of pants decompositions of the surface S0;1 . The surface S0;1 appears in [84] with its “canonical rigid structure” (see also Section 2.3). The constructions involved in [50] require to handle not only the canonical rigid structure of S0;1 , but also a set of rigid structures. Definition 3.8. A rigid structure on S0;1 consists of the data of a pants decomposition of S0;1 together with a decomposition of S0;1 into two connected components, called the visible and the hidden side, which are compatible in the following sense. The intersection of each pair of pants with the visible or hidden sides of the surface is a hexagon. The choice of a reference rigid structure defines the canonical rigid structure (cf. Figure 6). The dyadic regular (unrooted) tree T is embedded onto the visible side of S0;1 , as the dual tree to the canonical decomposition (into hexagons). A rigid structure is marked when one of the circles of the decomposition is endowed with an orientation. The choice of a circle of the canonical decomposition and of an orientation of this circle defines the canonical marked rigid structure.


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Figure 6. Surface S0;1 with its canonical rigid structure.

A rigid structure is asymptotically trivial if it coincides with the canonical rigid structure outside a compact subsurface of S0;1 . The set of isotopy classes of (resp. marked) asymptotically trivial rigid structures is denoted Rig.S0;1 / (resp. Rig0 .S0;1 /). In [50], we define the stable groupoid of duality Ds0 , which generalizes Pt, since it contains a full sub-groupoid isomorphic to Pt. We first recall the definition of this sub-groupoid, which will be denoted Ds0;Q . Definition 3.9. The objects of the groupoid Ds0;Q are the asymptotically rigid marked structures of S0;1 whose underlying decomposition into visible and hidden sides is the canonical one. The morphisms are composed of elementary morphisms, called moves, of two types, A and B. (1) A-move: Let r1 be an object of Ds0;Q . The distinguished oriented circle

separates two adjacent pairs of pants, whose union is a 4-holed sphere †0;4 . Up to isotopy, there exists a unique circle contained in †0;4 , whose geometric intersection number with is equal to 2, and which is invariant by the involution j interchanging the visible and hidden sides. Otherwise stated, the circle 0 is the image of by the rotation of angle C 2 described in Figure 7 which stabilizes both sides of S0;1 and †0;4 . Let r2 D r1 n f g [ f 0 g. By definition, the pair .r1 ; r2 / is the A-move on the rigid marked structure r1 . Its source is r1 while r2 is its target. (2) B-move: Let r1 be an object of Ds0;Q . Let P be the pair of pants of r1 bounded by , which is on the left when one moves along following its orientation. Let 00 be the oriented circle of the boundary of P , which is the image of the described in Figure 7 oriented circle by the rotation of order 3 and angle C 2 3 (it stabilizes both sides of S0;1 and P ). Let r2 be the pants decomposition whose circles are the same as those of r1 , but whose distinguished oriented circles is 00 .

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By definition, the pair .r1 ; r2 / is the B-move on r1 . Its source is r1 while its target is r2 . Relations among morphisms: if r1 and r2 are two objects of Ds0;Q such that there exist two sequences of moves .M1 ; : : : ; Mk / and .M10 ; : : : ; Mk0 0 / transforming r1 into r2 , then Mk B B M1 D Mk0 0 B B M10 .

A

B

Figure 7. Moves in the groupoid Ds0;Q .

Remark 3.6. There is a bijection between the set of objects of Pt and the set of objects of Ds0;Q , which maps the marked Farey tessellation onto the canonical marked rigid structure of S0;1 . This bijection extends to a groupoid isomorphism Pt ! Ds0;Q . Via this isomorphism, the generators ˛ and ˇ may be viewed as isotopy classes of asymptotically rigid homeomorphisms (which preserve the visible/hidden sides decomposition) of S0;1 . The generator ˛ corresponds to the mapping class such that ˛.r / D A.r /, and ˇ to the mapping class such that ˇ.r / D B.r /. This gives a new proof of the existence of an embedding of T into the mapping class group of S0;1 , obtained in [84].

3.4 The Hatcher–Thurston complex of S0;1 The Hatcher–Thurston complex of pants decompositions is first mentioned in the appendix of [77]. It is defined again in [76], for any compact oriented surface, possibly with boundary, where it is proved that it is simply connected. We extend its definition to the non-compact surface S0;1 . Definition 3.10 ([50]). The Hatcher–Thurston complex HT .S0;1 / is a cell 2-complex. (1) Its vertices are the asymptotically trivial pants decompositions of S0;1 . (2) Its edges correspond to pairs of decompositions .p; p 0 / such that p 0 is obtained from p by a local A-move, i.e. by replacing a circle of p by any circle 0 whose geometric intersection number with is equal to 2 (and does not intersect the other circles of p). (3) Its 2-cells fill in the cycles of moves of the following types: triangular cycles, pentagonal cycles (cf. Figure 8), and square cycles corresponding to the commutation of two A-moves with disjoint supports.


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Figure 8. Triangular cycle and pentagonal cycle in HT .S0;1 /.

The Hatcher–Thurston complex HT .S0;1 / is an inductive limit of Hatcher–Thurston complexes of compact subsurfaces of S0;1 . It is therefore simply connected. The following proposition establishes a fundamental relation between the Cayley graph of Thompson’s group T (generated by ˛ and ˇ) and the Hatcher–Thurston complex of S0;1 . The presentation of T will be exploited to prove some useful properties of the Hatcher–Thurston complex. Proposition 3.7 (following [50]). The forgetful map Ob.Ds0;Q / ! HT .S0;1 /, which maps an asymptotically rigid marked structure onto the underlying pants decomposition, extends to a cellular map W Gr.Ds0;Q / ! HT .S0;1 / from the graph of the groupoid onto the 1-skeleton of the Hatcher–Thurston complex. It maps an edge corresponding to an A-move onto an edge of type A of HT .S0;1 /, and collapses an edge corresponding to a B-move onto a vertex. Under the isomorphisms Gr.Ds0;Q / Gr.Pt/ Cayl.T /, where Cayl.T / is the Cayley graph of T with generators ˛ and ˇ, may be identified with a morphism from Cayl.T / to HT .S0;1 /. One can easily check that: (1) the image by of the cycle of ten moves associated to the relation .˛ˇ/5 D 1 is a pentagonal cycle of the Hatcher–Thurston complex; (2) the image by of the cycle associated to the relation Œˇ˛ˇ; ˛ 2 ˇ˛ˇ˛ 2 D 1 is a square cycle .DC 1 /, corresponding to the commutation of two A-moves supported by two adjacent 4-holed spheres;

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(3) the image by of the cycle associated to the relation Œˇ˛ˇ; ˛ 2 ˇ 2 ˛ 2 ˇ˛ˇ˛ 2 ˇ˛ 2 D 1 is a square cycle .DC 2 /, corresponding to the commutation of two A-moves supported by two 4-holed spheres separated by a pair of pants. Definition 3.11 (Reduced Hatcher–Thurston complex). Let HT red .S0;1 / be the subcomplex of HT .S0;1 /, which differs from the latter by the set of square 2-cells: a square 2-cell of HT .S0;1 / belongs to HT red .S0;1 / if and only if it is of type .DC 1 / (corresponding to the commutation of A-moves supported by two adjacent 4-holed spheres), or of type .DC 2 / (corresponding to the commutation of A-moves supported by two 4-holed spheres separated by a pair of pants). Proposition 3.8 ([50], Proposition 5.5). The subcomplex HT red .S0;1 / is simply connected. We refer to [50] for the proof. It is based on the existence of the morphism of complexes , and consists in proving, using the presentation of Thompson’s group T , that any square cycle of HT .S0;1 / may be expressed as a product of conjugates of at most three types of cycles: the squares of types .DC 1 / and .DC 2 /, and the pentagonal cycles.

4 The universal mapping class group in genus zero 4.1 Definition of the group B We have seen that T is isomorphic to the group of mapping classes of asymptotically rigid homeomorphisms of S0;1 which globally preserve the decomposition of the surface into visible/hidden sides. It turns out that if one forgets the last condition, one obtains an interesting larger group, which is the main object of the article [50]. Definition 4.1 ([50]). The universal mapping class group in genus zero B is the group of isotopy classes of (orientation-preserving) homeomorphisms of S0;1 which are asymptotically rigid, namely the asymptotically rigid mapping class group of S0;1 (see also Definition 2.4). From what precedes, T imbeds into B. As a matter of fact, B is an extension of Thompson’s group V . Proposition 4.1 ([50], Proposition 2.4). Let K1 be the pure mapping class group of the surface S0;1 , i.e. the group of mapping classes of homeomorphisms which are compactly supported in S0;1 . There exists a short exact sequence 1 ! K1 ! B ! V ! 1

Moreover, the extension splits over T V .


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Proof. For the comfort of the reader, we recall the proof given in [50]. Let us define the projection B ! V . Consider ' 2 B and let † be a support for '. We introduce the symbol .T'.†/ ; T† ; .'//, where T† (resp. T'.†/ ) denotes the minimal finite binary subtree of T which contains q.†/ (resp. q.'.†//), and .'/ is the bijection induced by ' between the set of leaves of both trees. The image of ' in V is the class of this triple, and it is easy to check that this correspondence induces a well-defined and surjective morphism B ! V . The kernel is the subgroup of isotopy classes of homeomorphisms inducing the identity outside some compact set, and hence is the direct limit of the pure mapping class groups. Denote by T the subgroup of B consisting of mapping classes represented by asymptotically rigid homeomorphisms preserving the whole visible side of . The image of T in V is the subgroup of elements represented by symbols .T1 ; T0 ; /, where is a bijection preserving the cyclic order of the labeling of the leaves of the trees. Thus, the image of T is Ptolemy–Thompson’s group T V . Finally, the kernel of the epimorphism T ! T is trivial. In the following, we shall identify T with T . As the kernel of this extension is not finitely generated, there is no evidence that B should be finitely generated. The main theorem of [50] asserts a stronger result.

4.2 B is finitely presented Theorem 4.2 ([50], Theorem 3.1). The group B is finitely presented. The proof is geometric, and inspired by the method of Hatcher and Thurston for the presentation of mapping class groups of compact surfaces. It relies on the Bass–Serre theory, as generalized by K. Brown in [17], which asserts the following. Let a group G act on a simply connected 2-dimensional complex X, whose stabilizers of vertices are finitely presented, and whose stabilizers of edges are finitely generated. If the set of G-orbits of cells is finite (otherwise stated, the action is cocompact), then G is finitely presented. Clearly, the group B acts cellularly on the Hatcher–Thurston complex of S0;1 . However, the idea consisting in exploiting this action must be considerably improved if one wishes to prove the above theorem. Indeed, the complex HT .S0;1 / is simply connected, but it has infinitely many orbits of B-cells. This is due to the existence of the square cycles, corresponding to the commutation of A-moves on disjoint supports. Let be a 2-cell filling in such a square cycle; the A-moves which commute are supported on two 4-holed spheres, separated by a certain number of pairs of pants n . Clearly, this integer is an invariant of the B-orbit of , which can be arbitrarily large. The interest for the reduced Hatcher–Thurston HT red .S0;1 / appears now clearly: it is both simply connected and finite modulo B. Unfortunately, the stabilizers of the vertices or edges of HT red .S0;1 / (which are the same as those of HT .S0;1 /) under the action of B are not finitely generated. The idea, in order to overcome this difficulty,

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is to “rigidify” the pants decompositions so that the size of their stabilizers become more reasonable. This leads us to introduce a complex DP .S0;1 /, whose definition is rather technical (cf. [50], §5), which is a sort of mixing of the Hatcher–Thurston complex, and a certain V -complex, called the “Brown–Stein complex”, defined in [18]. The latter has been used in [18] to prove that V has the FP1 property. Therefore, our B-complex DP .S0;1 / encodes simultaneously some finiteness properties of the mapping class groups M.0; n/ as well as of the Thompson group V . With the right complex in hand it is not difficult to find the explicit presentation for B, by following the method described in [17].

5 The braided Ptolemy–Thompson group 5.1 Finite presentation In the continuity of our investigations on the relations between Thompson groups and mapping class groups of surfaces we introduced and studied a group (in fact two groups which are quite similar) called the braided Ptolemy–Thompson group ([51]) T , which might appear as a simplified version of the group AT of [68], and studied from a different point of view in [84]. Indeed, T , like AT , is an extension of T by the stable braid group B1 . Its definition is simpler than that of AT , and is essentially topological. Definition 5.1 (from [51]). (1) Let D be the planar surface with boundary obtained by thickening the dyadic complete (unrooted) planar tree. The decomposition into hexagons of D, which is dual to the tree, is called the canonical decomposition. By a separating arc of the decomposition we mean a connected component of the boundary of a hexagon which is not included in the boundary of D. (2) Let D ] be the surface D with punctures corresponding to the vertices of the tree, and D the surface D whose punctures are the middles of the separating arcs of the canonical decomposition (cf. Figure 9). A connected subsurface of D ] or D is admissible if it is the union of finitely many hexagons of the canonical decomposition. (3) Let D ˘ denote D ] or D . An orientation-preserving homeomorphism g of ˘ D is asymptotically rigid if it preserves globally the set of punctures, and if there exist two admissible subsurfaces S0 and S1 such that g induces by restriction a “rigid” homeomorphism from D ˘ n S0 onto D ˘ n S1 , i.e. a homeomorphism that respects the canonical decomposition and the punctures. Note that D may be identified with the visible side of the surface S0;1 of [84] and [50]. Its canonical decomposition into hexagons is the trace on the visible side of S0;1 of the canonical pants decomposition of the latter.


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Figure 9. D ] and D with their canonical rigid structures.

Definition 5.2. The braided Ptolemy–Thompson group T ˘ (where the symbol ˘ may denote either or ]) is the group of isotopy classes of asymptotically rigid homeomorphisms of D ˘ . It is not difficult to see that there exists a short exact sequence 1 ! B1 ! T ˘ ! T ! 1: Unlike the extension of T by B1 which defines AT , the above extensions, producing respectively the groups T ] and T , are not related to the discrete Godbillon–Vey class. The main result of [51] is a theorem concerning the group presentations. Theorem 5.1 ([51], Theorem 4.5). The groups T ] and T are finitely presented. Moreover, an explicit presentation for T ] is given, with 3 generators. We show that T is generated by 2 elements. By comparing their associated abelianized groups, one proves that T ] and T , though quite similar, are not isomorphic. As in [50], we prove the above theorem by making T ] and T act on convenient simply-connected 2-complexes, The results of §4 are used once again, especially the reduced Hatcher–Thurston complex, by introducing braided versions of the Hatcher– Thurston complex of the surface D ] and D (the pairs of pants being replaced by hexagons). In short, a vertex of these two complexes is a decomposition into hexagons which coincides with the canonical decomposition outside a compact subsurface D, such that:

(1) in the T ] -complex each hexagon contains a puncture of D ] in its interior; (2) in the T -complex each separating arc passes through a puncture of D . There are two types of edges: an A-move of Hatcher–Thurston, and a braiding move B (cf. [51], §3). Forgetting the punctures, one obtains fibrations from the complexes onto the Hatcher–Thurston complex of D, whose fibers over the vertices are isomorphic to the Cayley complex of the stable braid group B1 . The presentation of B1 that is

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convenient exploits the distribution of the punctures on a tree or a graph. It is given by a more general theorem of the third author (cf. [114]). The groups T ] and T share a number of properties which make them quite different from B. For instance the cyclic orderability of T together with the left orderability of B1 leads to a cyclic order on T . Using a result from [24] we obtain: Proposition 5.2 ([51], Proposition 2.13). The group T can be embedded into the group of orientation-preserving homeomorphisms of the circle. Adapting one of the Artin solutions of the word problem in the braid group, we also prove Proposition 5.3 ([51], Proposition 2.16). The word problem for the group T is solvable. The group T is also used in the study of an asymptotically rigid mapping class group of infinite genus, whose rational homology is isomorphic to the “stable homology of the mapping class group”.

5.2 Asynchronous combability The aim of this section is to show that T ? has strong finiteness properties. Although it was known that one can generate Thompson groups using automata ([69]), very little was known about the geometry of their Cayley graphs. Recently, Farley proved ([38]) that Thompson groups (and more generally picture groups, see [73]) act properly by isometries on CAT(0) cubical complexes (and hence are a-T-menable), and Guba (see [71], [72]) obtained that the smallest Thompson group F has a quadratic Dehn function while T and V have polynomial Dehn functions. It is known that automatic groups have quadratic Dehn functions on one side and Niblo and Reeves ([107]) proved that any group acting properly discontinuously and cocompactly on a CAT(0) cubical complex is automatic. One might therefore wonder whether Thompson groups are automatic. We approach this problem from the perspective of mapping class groups, since one can view T and T as mapping class groups of a surface of infinite type. One of the far reaching results in this respect is the Mosher theorem ([103]) stating that mapping class groups of surfaces of finite type are automatic. Our main result in [53] shows that, when shifting to surfaces of infinite type, a slightly weaker result still holds. We will follow below the terminology introduced by Bridson in [1], [11], [12], in particular we allow very general combings. We refer the reader to [35] for a thorough introduction to the subject. Let G be a finitely generated group with a finite generating set S , such that S is closed with respect to taking inverses, and C.G; S / be the corresponding Cayley graph. This graph is endowed with the word metric in which the distance d.g; g 0 /


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between the vertices associated to the elements g and g 0 of G is the minimal length of a word in the generators S representing the element g 1 g 0 of G. A combing of the group G with generating set S is a map which associates to any element g 2 G a path g in the Cayley graph associated to S from 1 to g. In other words g is a word in the free group generated by S that represents the element g in G. We can also represent g .t/ as an infinite edge path in C.G; S / (called combing path) that joins the identity element to g, moving at each step to a neighboring vertex and which becomes eventually stationary at g. Denote by jg j the length of the path g , i.e. the smallest t for which g .t/ becomes stationary. Definition 5.3. The combing of the group G is synchronously bounded if it satisfies the synchronous fellow traveler property defined as follows. There exists K such that the combing paths g and g 0 of any two elements g, g 0 at distance d.g; g 0 / D 1 are at most distance K apart at each step, i.e. d.g .t/; g 0 .t// K

for any t 2 RC .

A group G having a synchronously bounded combing is called synchronously combable. In particular, combings furnish normal forms for group elements. The existence of combings with special properties (like the fellow traveler property) has important consequences for the geometry of the group (see [1], [11]). We will also introduce a slightly weaker condition (after Bridson and Gersten) as follows: Definition 5.4. The combing of the group G is asynchronously bounded if there exists K such that for any two elements g, g 0 at distance d.g; g 0 / D 1 there exist ways to travel along the combing paths g and g 0 at possibly different speeds so that corresponding points are at most distance K apart. Thus, there exists continuous increasing functions '.t / and ' 0 .t/ going from zero to infinity such that d.g .'.t//; g 0 .' 0 .t/// K

for any t 2 RC .

A group G having an asynchronously bounded combing is called asynchronously combable. The asynchronously bounded combing has a departure function D W RC ! RC if for all r > 0, g 2 G and 0 s; t jg j, the assumption js t j > D.r/ implies that d.g .s/; g .t// > r. The main result of [53] can be stated as follows: Theorem 5.4 ([53]). The group T ? is asynchronously combable. In particular, in the course of the proof we also prove that: Corollary 5.5. The Thompson group T is asynchronously combable.

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The proof is largely inspired by the methods of Mosher. The mapping class group is embedded into the Ptolemy groupoid of some triangulation of the surface, as defined by Mosher and Penner. It suffices then to provide combings for the latter. Remark 5.6. There are known examples of asynchronously combable groups with a departure function: asynchronously automatic groups (see [35]), the fundamental group of a Haken 3-manifold ([11]), or of a geometric 3-manifold ([12]), semi-direct products of Zn by Z ([11]). Gersten ([58]) proved that asynchronously combable groups with a departure function are of type FP3 and announced that they should actually be FP1 . Recall that a group G is FPn if there is a projective ZŒG-resolution of Z which is finitely generated in dimensions at most n (see [56], Chapter 8 for a thorough discussion on this topic). Notice that there exist asynchronously combable groups (with departure function) which are not asynchronously automatic, for instance the Sol and Nil geometry groups of closed 3-manifolds (see [10]); in particular, they are not automatic.

6 Central extensions of T and quantization 6.1 Quantum universal Teichmüller space The goal of the quantization is, roughly speaking, to obtain non-commutative deformations of the action of the mapping class group on Teichmüller space. It appears that the Teichmüller space of a surface has a particularly nice global system of coordinate charts whenever the surface has at least one puncture, the so-called shearing coordinates introduced by Thurston (see [47] for a survey). Each coordinate chart corresponds to fixing the isotopy class of a triangulation of the surface with vertices at the puncture. The mapping class group embeds into the labelled Ptolemy groupoid of the surface and there is a natural extension of the mapping class group action to an action of this groupoid on the set of coordinate charts. The necessity of considering labelled triangulations comes from the existence of triangulations with non-trivial automorphism groups. This theory extends naturally to the universal setting of Fareytype tessellations of the Poincaré disk D, which behaves naturally as an infinitely punctured surface. Since there are no automorphisms of the binary tree which induce eventually trivial permutations it follows that we do not need labelled tessellations. The analogue of the mapping class group is therefore the Ptolemy–Thompson group T . We will explain below (see Section 6.2) how one obtains by quantization a projective representation of T , namely a representation into the linear group modulo scalars, which is called the dilogarithmic representation. One of the main results of [55] (see also Sections 6.4 and 6.5) is the fact that the dilogarithmic representation comes from a central extension of T whose class is 12 times the Euler class generator. This result is very similar to the case of a finite type surface where the dilogarithmic representations come from a central extension of the mapping class group of a punctured surface


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having extension class 12 times the Euler class plus the puncture classes (see [54] for details). Here and henceforth, for the sake of brevity, we will use the term tessellation instead of marked tessellation. For each tessellation let E./ be the set of its edges. We associate further a skew-symmetric matrix ". / with entries "ef , for all e; f 2 E./, as follows. If e and f do not belong to the same triangle of or e D f then "ef D 0. Otherwise, e and f are distinct edges belonging to the same triangle of and thus have a common vertex. We obtain f by rotating e in the plane along that vertex such that the moving edge is sweeping out the respective triangle of . If we rotate clockwise then "ef D C1 and otherwise "ef D 1. The pair .E./; ".// is called a seed in [45]. Observe that in this particular case seeds are completely determined by tessellations. Let .; 0 / be a flip Fe in the edge e 2 E. /. Then the associated seeds .E. /; ". // and .E. 0 /; ". 0 // are obtained one from another by a mutation in the direction e. Specifically, this means that there is an isomorphism e W E./ ! E. 0 / such that

0

". /e .s/e .t/

8 ˆ 0

X

". /et a t

:

t I". /et 0, and (iii) f ./ is bounded on the half-plane Im C for any positive constant C . Since the lattice ZC Z is the same as ZC. C1/Z, the function f is periodic with period 1. Therefore there exists a function '.q/, holomorphic on the open punctured unit disc, such that f ./ D '.e 2 i /. This function is bounded at the neighborhood of the origin, therefore it can be extended to a holomorphic function on the whole unit

Chapter 11. An introduction to moduli spaces of curves and their intersection theory 689

disc and expanded into a power series in q at 0, which is the usual way to represent a modular form. Since L D L for every lattice L, we see that there are no nonzero modular forms of odd weight. On the other hand, there exists a nonzero modular form of any even weight k 4, given by X 1 Ek .L/ D : zk z2Lnf0g

(For odd k this sum vanishes, while for k D 2 it is not absolutely convergent.) The value of the corresponding function 'k .q/ at q D 0 is equal to X 1 'k .0/ D lim Ek .Z C Z/ D D 2.k/: Im !1 zk z2Znf0g

The relation between modular forms and the following proposition.

x 1;1 comes from the -class on M

Proposition 2.20. The space of modular forms of weight k is naturally identified with x the space of holomorphic sections of L˝k 1 over M1;1 . Proof. Let F be a modular form of weight k. We claim that F .L/dz k is a well-defined x holomorphic section of L˝k 1 over M1;1 . First of all, if C=L is any elliptic curve, then the value of F .L/dz k at the marked point (the image of 0 2 C) is indeed a differential k-form, that is, an element of the fiber of L˝k 1 . If we apply a homothety z 7! cz, replacing L by cL, we obtain an isomorphic elliptic curve. However, the k-form F .L/dz k does not change, because F .L/ is divided by c k , while dz k is multiplied by c k . Thus F .L/dz k is a well-defined section of L˝k 1 . The fact that this section is holomorphic over M1;1 follows from the fact that f ./ is holomorphic. The fact that it is also holomorphic at the boundary point follows from the fact the function '.q/ is holomorphic at q D 0. Conversely, if s is a holomorphic section of L˝k 1 , then taking the value of s over k the curve C=L and dividing by dz , we obtain a function on lattices L. The same argument as above shows that it is a modular form of weight k. Proposition 2.21. We have

Z x 1;1 M

1

D

1 : 24

Proof. We are going to give three similar computations leading to the same result. Denote by fk ./ and 'k .q/ the functions associated with the modular form Ek . One can check (see, for instance [32], chapter VII) that inpthe modular figure (i.e., on M1;1 ) the function f4 has a unique simple zero at D 12 ˙ 23 i , while f6 has a unique simple zero at D i . (The fact that these are indeed zeroes is an easy exercise for the reader.)

690 Further, the function

Dimitri Zvonkine

'4 2.4/

3

'6 2.6/

2

x 1;1 are has a unique zero at q D 0. The stabilizers of the corresponding points in M Z=6Z, Z=4Z, and Z=2Z respectively (see Example 1.25). Thus the first Chern class ˝6 ˝12 equals 1=2. In every case of L˝4 1 equals 1=6, that of L1 equals 1=4, that of L1 we find that the first Chern class of L1 equals 1 D 1=24. x 1;n the divisors Proposition 2.22. Denote by ı.irr/ ; ı.1/ M

1 ı.irr/

ı.1/

In other words, the points of ı.irr/ encode curves with at least one nonseparating node; the points of ı.1/ encode curves with a separating node dividing the curve into a stable curve of genus 1 and a stable curve of genus 0 containing the marked point number 1. x 1;n equals Then the class 1 on M 1

D

1 Œı.irr/ C Œı.1/ : 12

The proof follows from the computation of

1

x 1;1 and from Proposition 2.17. on M

2.3 Other tautological classes All cohomology classes we consider are with rational coefficients. 2.3.1 The classes on the universal curve. On the universal curve we define the following classes. • Di is the divisor given by the i th section of the universal curve. In other words, the intersection of Di with a fiber C of Cxg;n is the i th marked point on C . By abuse of notation we denote by Di 2 H 2 .Cxg;n / the cohomology class Poincaré dual to the divisor. P • D D niD1 Di . • ! D c1 .L/. • K D c1 .Llog / D ! C D 2 H 2 .Cxg;n /, where Llog is the line bundle L twisted by the divisor D. • is the codimension 2 subvariety of Cxg;n consisting of the nodes of the singular fibers. By abuse of notation, 2 H 4 .Cxg;n / will also denote the Poincaré dual cohomology class.


• Let N be the normal vector bundle to in Cxg;n . Then we set k;l D .c1 .N //k lC1 : To simplify the notation, we introduce two symbols 1 and 2 with the convention 1 C 2 D c1 .N /, 1 2 D c2 .N /. Since c2 .N / D 2 , we also identify 1 2 with . Thus, even though the symbols 1 and 2 separately are meaningless, every symmetric polynomial in 1 and 2 divisible by 1 2 determines a well-defined cohomology class. For instance, we have k;l D .1 C 2 /k .1 2 /l D .1 C 2 /k .1 2 /lC1 : Since is the set of nodes in the singular fibers of Cxg;n , it has a natural 2-sheeted z ! whose points are couples (node C choice of a (unramified) covering p W z we can define two natural line bundles L˛ and Lˇ cotangent, branch). Over respectively, to the first and to the second branch at the node. The pull-back p N _ z is naturally identified with L_ of N to ˛ ˚ Lˇ . Thus, if P .1 ; 2 / is a symmetric z P .c1 .L˛ /; c1 .Lˇ //. polynomial, we have p . P .1 ; 2 // D 2.3.2 Intersecting classes on the universal curve Proposition 2.23. For all 1 i; j n, i 6D j we have KDi D Di Dj D K D Di D 0 2 H .Cxg;n /: Proof. The divisors Di and Dj do not intersect, so the intersection of the corresponding classes vanishes. Similarly, the divisor Di does not meet , so their intersection vanishes. The restriction of the line bundle Llog to Di is trivial. Indeed, the sections of Llog are 1-forms with simple poles at the marked points, and the fiber at the marked point is the line of residues, so it is canonically identified with C. The intersection KDi is the first Chern class of the restriction of Llog to Di . Therefore it vanishes. The restriction of Llog to is not necessarily trivial. However its pull-back to the z is trivial (because the fiber is the line of residues identified double-sheeted covering with C). Alternatively, one can say that .Llog /˝2 is trivial. Therefore K D 0. Remark 2.24. A line bundle whose tensor power is trivial is called rationally trivial. Although it is not necessarily trivial itself, all it characteristic classes over Q vanish. This is the case of Lj D Llog j . Corollary 2.25. Every polynomial in the classes Di , K, k;l on Cxg;n can be written in the form n X Pi .Di / C P .1 ; 2 /; PK .K/ C iD1

while PK and Pi , 1 i n are arbitrary polynomials, while P is a symmetric polynomial with the convention 1 2 D , 1 C 2 D c1 .N /.

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Proof. Given a polynomial in Di ; K; k;l , we can, according to the proposition, cross out the “mixed terms”, that is, the monomials containing products Di Dj , Di K, Kk;l or Di k;l . We end up with a sum of powers of Di , powers of K, and products of k;l . Now, by definition, k1 ;l1 k2 ;l2 D k1 Ck2 ;l1 Cl2 C1 . Therefore a polynomial in the variables k;l can be rewritten in the form P .1 ; 2 /, where P is a symmetric polynomial. x g;n be the universal curve. 2.3.3 The classes on the moduli space. Let p W Cxg;n ! M x On the moduli space Mg;n we define the following classes. x g;n /. • m D p .K mC1 / 2 H 2m .M •

i

x g;n /. D p .Di2 / 2 H 2 .M

x g;n /. • ık;l D p .k;l / 2 H kC2lC1 .M x g;n /, where ƒ is the Hodge bundle and ci the i th Chern • i D ci .ƒ/ 2 H 2i .M class. Note that this definition of -classes coincides with Definition 2.12 by Proposition 2.18. Also note that our definition of -classes follows the convention of Arbarello and Cornalba [1]. x g;n are Thus, with the exception of the -classes, the tautological classes on M x push-forwards of tautological classes on Cg;n and their products. For example, ı0;0 is x g;n n Mg;n . x g;n , i.e., ı0;0 D M the boundary divisor on M x 1;1 the line bundles Example 2.26. As an exercise, the reader can show that over M ƒ and L1 are isomorphic. Hence Z 1 1 D : 24 x 1;1 M

Theorem 2.27. The classes sense of Definition 2.5.

i,

m , ık;l , and i lie in the tautological ring in the

x g;nC1 ! M x g;n be the forgetful map. Then i D p .ı 2 Proof. Let p W M /, .i;nC1/ mC1 where ı.i;nC1/ is defined in Propositions 2.17, while m D p . nC1 /. The class ık;l is the sum of push-forwards under the attaching maps of the class . nC1 C k l nC2 / . nC1 nC2 / . Thus all these classes lie in the tautological ring. The class i is expressed via the -, -, and ı-classes in Theorem 3.16. It follows that it too lies in the tautological ring.


3 Algebraic geometry on moduli spaces In the previous section we introduced a wide range of tautological classes on the x g;n , namely, the -, -, ı-, and -classes. Now we would like to learn moduli space M how to compute all possible intersection numbers between these classes. This is done in three steps. First, by applying the Grothendieck–Riemann–Roch (GRR) formula we express -classes in terms of -, -, and ı-classes. This gives us an opportunity to introduce the GRR formula and to give an example of its application in a concrete situation. Secondly, by studying the pull-backs of the -, -, and ı-classes under attaching and forgetful maps, we will be able to eliminate one by one the - and ı-classes from intersection numbers. The remaining problem of computing intersection numbers of the -classes is much more difficult. The answer was first conjectured by E. Witten [35]. It is formulated below in Theorems 4.4 and 4.5. Witten’s conjecture now has at least 5 different proofs (the most accessible to a non-specialist is probably [22]), and all of them use nontrivial techniques. In this note we will not prove Witten’s conjecture, but give its formulation and say a few words about how it appeared. Witten’s conjecture is also discussed in Mondello’s chapter of this Handbook [29].

3.1 Characteristic classes and the GRR formula In this section we present the Grothendieck–Riemann–Roch (GRR) formula. But first we recall the necessary information on characteristic classes of vector bundles, mostly without proofs. 3.1.1 The first Chern class Definition 3.1. Let L ! B be a holomorphic line bundle over a complex manifold B. Let s be a nonzero meromorphic section of L and Z P the associated divisor: the set of zeroes minus the set of poles of s. Then ŒZ ŒP 2 H 2 .B; Z/ is called the first Chern class of L and denoted by c1 .L/. The first Chern class is well-defined, i.e., it does not depend on the choice of the section. Moreover, c1 .L/ is a topological invariant of L. In other words, it only depends on the topological type of L and B, but not on the complex structure of B nor on the holomorphic structure of L. Actually, there exists a different definition of first Chern classes (which we will not use) that does not involve the holomorphic structure at all. 3.1.2 Total Chern class, Todd class, Chern character. Let V ! B be a vector bundle of rank k.

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Definition 3.2. We say that V can be exhausted by line bundles if we can find a line subbundle L1 of V , then a line subbundle L2 of the quotient V1 D V =L1 , then a line subbundle L3 of the quotient V2 D V1 =L2 , and so on, until the last quotient is itself a line bundle Lk . This is equivalent to asking that V has a completeL flag of subbundles, Li . with graded pieces L1 ; : : : ; Lk . The simplest case is when V D If V is exhausted by line bundles, the first Chern classes ri D c1 .Li / are called the Chern roots of V . Definition 3.3. Let V be a vector bundle with Chern roots r1 ; : : : ; rk . Its total Chern class is defined by c.V / D

k Y

.1 C ri /I

iD1

its Todd class is defined by Td.V / D

k Y iD1

ri I 1 e ri

its Chern character is defined by ch.V / D

k X

e ri :

iD1

The homogeneous parts of degree i of these classes are denoted by ci , Tdi , and chi respectively. If we know the total Chern class of a vector bundle, we can compute its Todd class and Chern character (except ch0 , which is equal to the rank of the bundle). For instance, let us compute ch3 . We have 1X 3 ri 6 1 X 1 X 3 1 X X ri D ri rj C ri rj rk ri 6 2 2

ch3 D

i<j

i<j dim B. We will use only two properties of the cohomology groups. First, H 0 .V / is the space of holomorphic sections of V . This follows directly from the definition. Secondly, let K be the canonical line bundle over B, in other words the highest exterior power of the cotangent vector bundle. Let V _ be the dual vector bundle of V . Then H k .B; V / and H dim Bk .B; K ˝ V _ / are dual vector spaces. This property is called the Serre duality. These definitions, which we gave in the case of a smooth complex manifold B, can be generalized to any projective algebraic variety. In particular, the Serre duality holds on a stable curve C if we replace the canonical line bundle K by LjC . 3.1.4 K 0 , p , and pŠ . Let X be a complex manifold. Consider the free additive group generated by the vector bundles over X. To every short exact sequence 0 ! V1 ! V2 ! V3 ! 0 we assign the relation V1 V2 C V3 D 0 in this group. The Grothendieck group K 0 .X / is the factor of the free group by all such relations. We would obtain the same group if the vector bundles were replaced by coherent sheaves, because every coherent sheaf has a finite resolution by vector bundles. The Chern character determines a group morphism ch W K 0 .X / ! H .X; Q/ from the Grothendieck group to the cohomology group of X. Let p W X ! Y be a proper morphism of complex manifolds (that is, the inverse images of compact sets are compact). Then p induces a morphism p W H .X; Q/ ! H .Y; Q/ of cohomology groups (the fiberwise integration). It is defined by hp ˛; C i D h˛; p 1 .C /i for any cycle C Y in general position. On the other hand, p also determines a morphism pŠ W K 0 .X / ! K 0 .Y /, that we now describe. Denote by Xy the fiber over a point y 2 Y . Then we can consider the cohomology spaces H k .Xy ; V / (where V actually stands for the restriction of V to Xy ). For each k, the vector spaces H k .Xy ; V / form a sheaf over Y . This sheaf is denoted by Rk .p; V /. The morphism pŠ of Grothendieck groups is now defined by pŠ .V / D R0 .p; V / R1 .p; V / C : Now we have the following diagram of morphisms.


0

K .X/

ch -

Td.p/

H .X; Q/ p

pŠ ? 0

K .Y /

ch -

?

H .Y; Q/

The question is: is it commutative? The answer, given by the GRR formula, is that it is not, but can be made to commute if we add a multiplicative factor Td.p/. 3.1.5 The Grothendieck–Riemann–Roch formula. Let p W X ! Y be a morphism of complex manifolds with compact fibers. Let V be a vector bundle over X. Denote by Td.TX / : Td.p/ D Td.p .T Y // Theorem 3.6 (GRR, see [13], Chapter 9). We have ch.pŠ V / D p Œch.V /Td.p/ : Example 3.7. Apply the GRR formula to the situation F ! X ! point, where X is a (compact) Riemann surface and F is a sheaf. We obtain the Riemann–Roch formula h0 .F / h1 .F / D c1 .F / C 1 g; where h0 and h1 are the dimensions of H 0 and H 1 . Example 3.8. Let p W X ,! Y be an embedding of smooth manifolds and N ! X the normal vector bundle to X in Y . Let n be the rank of N . Then we have p .˛/ D cn .N / ˛. Indeed, cn .N / ŒX is the self-intersection of X in the total space of N . Applying GRR to the situation OX ! X ,! Y we get cn .N / D Td.N /

n X

.1/k ch

Vk

N_ :

kD0

As an exercise, the reader can express both sides of this equality in terms of Chern roots of N and check that they coincide. 3.1.6 The Koszul resolution. The Koszul resolution (see [7], Chapter 17) is a tool that we will need in order to apply the GRR formula to the universal curve, so let us introduce it here. Let p W V ! X be a vector bundle over a smooth complex manifold X. Then X is embedded in the total space of V by the zero section, X V . On the total space V , consider the sheaf OX supported on X. Its sections over an open set U V are the holomorphic functions on U \ X. Thus the fiber of OX over a point of X is C, while its fiber over a point outside X is 0.

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It is always unpleasant to work with sheaves that are not vector bundles, therefore we would like to construct a resolution of OX , in other words, an exact sequence of sheaves that contains OX , but whose all other terms are vector bundles. Example 3.9. Suppose V is a line bundle. Then over the total space of V there are two sheaves: the sheaf OV of holomorphic functions and the sheaf OV .X / of holomorphic functions vanishing on X. The short exact sequence 0 ! OV .X/ ! OV ! OX ! 0 is a resolution of OX . Our aim is to generalize this example. Denote by V the pull-back of the vector bundle V to the total space of V : V

- V

? V

? - X

Thus V is a vector bundle over the total space of V . Denote by Ak .V/ the sheaf of skew-symmetric k-forms on V. In particular, A1 .V/ is the sheaf of sections of V_ , while A0 .V/ D OV . Let x be a point in X , v 2 Vx a point in the fiber Vx over x, and ˛ be a skewsymmetric k-form on Vx . Then the form iv ˛ obtained by substituting v as the first entry of ˛ is a skew-symmetric .k 1/-form on Vx . Thus we obtain a natural sheaf morphism d W Ak .V/ ! Ak1 .V/ by substituting v into ˛. Denote by p the rank of V . Theorem 3.10. The following sequence of sheaves in exact. 0 ! Ap .V/ ! Ap1 .V/ ! ! A1 .V/ ! OV ! OX ! 0: This exact sequence is called the Koszul resolution of the sheaf OX . Proof of Theorem 3.10 in the case of a rank 2 vector bundle. For simplicity we restrict ourselves to the case p D 2, since it is the only case that we will need. Choose an open chart in X and a trivialization of the vector bundle V over this chart. Let t D .t1 ; t2 ; : : : / be the local coordinates on the chart and .x; y/ the coordinates in the fibers of V . The maps d of the Koszul resolution can be explicitly written out as 0 7! 0 dx ^ dy; f .t I x; y/ dx ^ dy 7! yf .t I x; y/ dx C xf .t I x; y/ dy; gx .t I x; y/ dx C gy .t I x; y/ dy 7! xgx .t I x; y/ C ygy .t I x; y/;

(1) (2) (3)


h.t I x; y/ 7! h.t I 0; 0/;

(4)

h.tI 0; 0/ 7! 0:

(5)

We must check that the image of each map coincides with the kernel of the next map. The image of the map (1) and the kernel the map (2) vanish. The kernel of the map (3) is given by the condition xgx C ygy D 0. In particular, this implies that gx is divisible by y, while gy is divisible by x. Hence an element of the kernel lies in the image of the first map, since we can take f D gx =y D gy =x. Conversely, if gx D yf and gy D xf , then xgx C ygy D 0, hence the image of the map (2) is included in the kernel of the map (3). The kernel of map (4) and the image of the map (3) are composed of maps that vanish at x D y D 0. Finally, the map (4) is surjective and the map (5) takes everything to zero. We leave the generalization to the arbitrary rank as an exercise to the reader.

3.2 Applying GRR to the universal curve The aim of this section is to apply the GRR formula to the case where the morphism of x g;n , while the vector bundle complex manifolds is the universal curve p W Cxg;n ! M x over Cg;n is the relative cotangent line bundle L from Definition 2.6. We follow Mumford’s paper [31]. A careful reader may wonder whether the GRR formula is applicable to orbifolds. Such worries are well-founded, because in general it is not. As an example, consider x 1;1 . As we saw in Section 2.2.2, the holomorphic sections the line bundle L1 over M k of L1 are the modular forms of weight k. It is well-known (see, for instance, [32], VII, 3, Theorem 4) that modular forms are homogeneous polynomials in E4 and E6 , which allows one to find the dimension of H 0 .Lk1 / and to work out the necessary modifications of the Riemann–Roch formula in this example. However, the GRR formula applies without changes to a morphism between two orbifolds if every fiber of the morphism is a compact manifold, i.e., the orbifold structure of the fibers is trivial. This is, of course, true for the universal curve, because its fibers are stable curves. If the fibers have a nontrivial orbifold structure, the GRR formula must be modified to take into account the stabilizers of different points of the fibers. Taking a look at the GRR formula, we see that we must compute three things: ch.pŠ L/, ch.L/, and Td.p/. Here “compute” means express either via the -, -, ı-, and -classes (on the moduli space) or via the classes Di , K, and 1 , 2 (on the universal curve). Proposition 3.11. We have pŠ L D ƒ C, where ƒ is the Hodge bundle and C the x g;n . trivial line bundle over M

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Proof. Let C be a stable curve. Then H 0 .C; L/ is the space of abelian differentials on C . This means that R0 .p; L/ is the Hodge bundle ƒ (Definition 2.11). By Serre’s duality, the space H 1 .C; L/ is dual to H 0 .C; L ˝ L_ / D H 0 .C; C/ D C. (In Section 3.1.3 we only described the Serre duality over smooth manifolds, but it can be extended to stable curves by using the line bundle L instead of the cotangent line bundle.) Thus the space H 1 .C; L/ is naturally identified with C, so R1 .p; L/ D C is the trivial line bundle. Corollary 3.12. We have ch.pŠ L/ D ch.ƒ/ 1: Thus the left-hand side of the GRR formula involves -classes. Once we have computed the right-hand side we will be able to express the -classes via other classes. Proposition 3.13. We have

ch.L/ D e ! :

Proof. Obvious. 3.2.1 Computing Td.p/. Computing the first two ingredients of the GRR formula was easy, but the computation of Td.p/ requires more work. If all the fibers of the universal curve were smooth, there would have been an exact sequence relating the x g;n , T Cxg;n and L. The Todd class Td.p/ would then be equal vector bundles T M to the Todd class of L. In reality, however, some fibers of p are singular and the singularity locus is . Therefore we will get a more complicated expression for Td.p/, involving L and the sheaf O . We will then compute the Todd class of O using the Koszul resolution. For practical reasons, it is easier to write an exact sequence involving the cotangent x g;n and T _ Cxg;n to the moduli space and the universal curve. Let O bundles T _ M be the sheaf over Cxg;n whose sections are functions on (so the sheaf is supported on ). Proposition 3.14. The sequence _

.dp/ x g;n / ! T _ Cxg;n ! L ! O ˝ Lj ! 0 0 ! p .T _ M

is an exact sequence of sheaves over Cxg;n . Proof. The maps in this exact sequence are (i) the adjoint map of the differential of p, (ii) the restriction of a 1-form on the tangent space to Cxg;n to the tangent space of a fiber, and (iii) restricting a section of L to . First consider the map p at the neighborhood of a point z 2 Cxg;n n and let x g;n at t x g;n be its image in the moduli space. The cotangent space to M t D p.z/ 2 M is injected in the cotangent space to Cxg;n at z by dp _ . The cokernel is the cotangent


space to the fiber, i.e., L. The fiber of O is equal to 0, so we obtain a short exact sequence. x g;n be its image in the moduli space. Now take a point z 2 and let t D p.z/ 2 M Consider the model local picture where p has the form p W .X; Y / 7! T D X Y , where z is the point X D Y D 0 and t is the point T D 0. According to Section 1.4.4, in the general case the local picture is the direct product of our model local picture with a trivial map p W B ! B. x g;n is the line bundle generated In the model local picture, the vector bundle T _ M by d T . The vector bundle T _ Cxg;n is generated by dX and d Y . Finally, the line bundle L is generated by the sections dX and dYY modulo the relation dX C dYY D 0. X X The maps of the sequence can be explicitly written out as follows: f .T /d T 7! Y f .X Y / dX C X f .X Y / d Y; (i) dX gX .X; Y / dX C gY .X; Y / d Y 7! XgX .X; Y / Y gY .X; Y / X (ii) dY ; D XgX .X; Y / C Y gY .X; Y / Y dX h.X; Y / 7! h.0; 0/: (iii) X These maps are very close to the maps of the Koszul resolution from Theorem 3.10 and the proof of the exactness is literally the same. Define Td.p _ / D

Td.T _ Cxg;n / : x g;n / Td.p T _ M

Then Td.p _ / is obtained from Td.p/ by changing the signs of the odd degree terms. Thus computing Td.p/ is equivalent to computing Td.p _ /. Corollary 3.15. We have Td.p _ / D

Td.L/ : Td.O /

Proof. Using Proposition 3.14, we obtain Td.p _ / D

Td.L/ ; Td.O ˝ L/

because the Todd class is multiplicative. On the other hand, we have Td.O ˝ L/ D Td.O /. Indeed, as we know, the restriction of the line bundle L to is rationally trivial. Hence all characteristic classes of L on are the same as for a trivial line bundle. Td.L/ is immediately evaluated to be Td.L/ D

! : 1 e !

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Dimitri Zvonkine

Evaluating the Todd class of O will be the last step of our computation. To do that we will use the Koszul resolution. Let N be the normal vector bundle to in Cxg;n . Rather than evaluating Td.O / on Cxg;n we can evaluate Td.O / for the zero section of the vector bundle N . (As we mentioned, the characteristic classes are topological invariants of vector bundles. Therefore the characteristic classes of a sheaf supported on depend only on its resolution in the neighborhood of in Cxg;n , which is topologically the same as the neighborhood of in the total space of N .) The Koszul resolution gives us the following exact sequence of sheaves: V 0 ! 2 N _ ! N _ ! ON ! O ! 0; where ON is the sheaf of holomorphic functions on the total space of the vector bundle N . Hence we have 1 Td.N _ / 1 2 1 e .1 C2 / D V2 _ D Td.O / 1e 1 1e 2 1 C 2 Td. N / 1 e .1 C2 / .1 e 1 /.1 e 2 / 1 e 1 C e 1 e .1 C2 / .1 e 1 /.1 e 2 / e 1 1 C 1 e 2 1 e 1 1 1 C 1 1 e 2 1 e 1 X B2k 2k1 C 2k1 1 2 D1C ; .2k/Š 1 C 2

1 2 1 C 2 1 2 D 1 C 2 1 2 D 1 C 2 1 2 D 1 C 2 D

k1

where B2n are the Bernoulli numbers (see Remark 3.5). 3.2.2 The right-hand side of GRR. Now we can assemble the right-hand side of GRR from the results of our computations. We have

X B2k 2k1 C 2k1 ! 2 1 1 C : Td.p / D 1 e ! .2k/Š 1 C 2 _

k1

We must change the signs of the odd degree terms to obtain Td.p/. But the part in brackets is even, while the only odd term of !=.1 e ! / is !=2. Thus

X B2k 2k1 C 2k1 ! 1 2 Td.p/ D ! 1C : e 1 .2k/Š 1 C 2 k1


Further,

X B2k 2k1 C 2k1 e! ! 2 1 1C ch.L/Td.p/ D ! e 1 .2k/Š 1 C 2 k1

X B2k 2k1 C 2k1 ! 2 1 D 1 C : 1 e ! .2k/Š 1 C 2 k1

(By a coincidence, this is also equal to Td.p _ /, but there is no reason to be confused by that.) Now, taking into account that ! D 0, we obtain

ch.L/Td.p/ D 1 C

2k1 C 22k1 ! X B2k C ! 2k C 1 2 .2k/Š 1 C 2 k1

n

X 2k1 C 22k1 ! X B2k D1C C K 2k Di2k C 1 : 2 .2k/Š 1 C 2 iD1

k1

Finally, according to the GRR formula, the push-forward p of this class is equal to the class ch.ƒ/ 1. For clarity, we separate this equality into homogeneous parts. Theorem 3.16. We have ch0 .ƒ/ 1 D g 1I ch2k .ƒ/ D 0I

n

X B2k ch2k1 .ƒ/ D 2k1 .2k/Š

2k1 i

C

ƒ ı2k1

;

iD1

ƒ is the push-forward p of the class .12k1 C 22k1 /=.1 C 2 /. where ı2k1

A curious reader can check that 1p

C

2p

D

lD0

and hence the class

ƒ ı2k1

ƒ ı2k1

D

pl p .1 2 /l .1 C 2 /p2l ; .1/ pl l

Œp=2 X

l

is expressed via the standard classes ık;l as

k1 X lD0

2k 1 l 2k 1 ı2k22l;l : .1/ 2k 1 l l l

Theorem 3.16 expresses the Chern characters of the Hodge bundle via the -, -, and ı-classes. To conclude this section we show how to obtain similar expressions for the classes i D ci .ƒ/.

704

Dimitri Zvonkine

Proposition 3.17. We have

ch3 ch4 C 2Š 3Š ch5 ch3 C C ; D exp ch1 C 2Š 4Š

1 C 1 C 2 C C g D exp ch1 ch2 C

where chk D chk .ƒ/. We leave the proof as an exercise to the reader. Corollary 3.18. Every class i is a polynomial in the

-, -, and ı-classes.

Corollary 3.19. We have c.ƒ/c.ƒ_ / D 1. Proof. We have c.ƒ/c.ƒ_ / D .1 C 1 C 2 C 3 C /.1 1 C 2 3 C / ch3 ch4 ch3 ch4 C exp ch1 ch2 D exp ch1 ch2 C 2Š 3Š 2Š 3Š ch3 ch3 ch5 ch5 D exp ch1 C C C exp ch1 C D 1: 2Š 4Š 2Š 4Š Example 3.20. Applying Theorem 3.16 and Proposition 3.17, we obtain n

X 1 1 D .1 12

i

C ı0;0 /I

iD1

1 2 I 2 1 n X 1 1 .3 3 D 31 6 360

2 D

3 i

C ı2;0 3ı0;1 /:

iD1

3.3 Eliminating - and ı-classes In the previous section we expressed the -classes in terms of -, -, and ı-classes. x g;n involving -, -, and ı-classes to a Now we will reduce any integral over M combination of integrals involving only -classes. This does not mean that the - and ı-classes can be expressed in terms of -classes. Indeed, the integrals that we will x g 0 ;n0 with g 0 and n0 not necessarily equal to g and n. obtain involve moduli spaces M To do that, we will use computations with attaching and forgetful maps. Before going into our computations, recall the following basic property that we will use.


Let f W X ! Y be a morphism of smooth compact manifolds. Then f induces two maps in cohomology: the pull-back f W H .Y; Q/ ! H .X; Q/ and the fiberwise integration f W H .X; Q/ ! H .Y; Q/. If the cohomology classes are represented by generic Poincaré dual cycles, then f and f are the geometric preimage and image of cycles. Let ˛ 2 H .X; Q/ and ˇ 2 H .Y; Q/. In general f .f .˛// ¤ ˛ (and these classes have different degrees). Similarly, in general f .f .ˇ// ¤ ˇ (and these classes also have different degrees). Instead, the following equality is true: f .˛f .ˇ// D f .˛/ˇ and therefore

Z

˛f .ˇ/ D

X

Z Y

f .˛/ˇ:

This property is called the projection formula. x g;nC1 and Cxg;n are x g;nC1 and Cxg;n . Recall that M 3.3.1 Equivalence between M x g;n (see Proposition 2.2). Let us make a small naturally isomorphic as families over M dictionary between the tautological classes defined on this family viewed as a moduli space and as a universal curve. Consider the set of stable curves that contain a spherical component with exactly three special points: a node and the marked points number i and n C 1. i nC1

x g;nC1 . The points encoding such curves form a divisor ı.i;nC1/ of M Further, consider the set of stable curves that contain a spherical component with exactly three special points: two nodes and the marked point number n C 1. nC1

x g;nC1 . The points encoding such curves form a codimension 2 subvariety ı.nC1/ of M x g;nC1 with Cxg;n , the divisor ı.i;nC1/ is Lemma 3.21. Under the identification of M identified with the divisor Di , the subvariety ı.nC1/ is identified with , while the class nC1 is identified with K. Proof. The first two identifications follow immediately from the proof of Proposition 2.2: they correspond to the cases (ii) and (iii). The identification of nC1 with K is more delicate. Recall that K is the first Chern x g;nC1 is naturally identified class of L.D/. In the case (i), the fiber of LnC1 over M with the fiber of L over Cxg;n .

706

Dimitri Zvonkine

This identification fails in the cases (ii) and (iii). However, the case (iii) is not important, since it only occurs on a codimension 2 subvariety, hence does not influence the first Chern class. Case (ii), on the other hand, must be inspected more closely. A holomorphic section of L at the neighborhood of xO i in the figure is a holomorphic 1-form on the fibers of Cxg;n . It is naturally extended by 0 on the genus 0 component containing xi and xnC1 (cf. Proposition 2.17). Thus the sections of LnC1 acquire an additional zero whenever we are in case (ii). Thus LnC1 is identified not with L but with L.D/. 3.3.2 Eliminating -classes: the forgetful map. The contents of this and the next section are a reformulation of certain results of [2]. x g;n be the forgetful map. Our aim is to compare the taux g;nC1 ! M Let p W M x g;nC1 and the pull-backs of analogous tautological class from tological classes on M x g;nC1 as the x g;n . It turns out that the difference is easily expressed if we consider M M x universal curve over Mg;n . Theorem 3.22. We have D K;

(1)

p id D .Di /d 1 Di ; m p m D K m ;

(2) (3)

nC1 d i

ık;0 p ık;0 D .D/k D C .1 C 2 /kC1 1kC1 2kC1 ;

(4)

ık;l p ık;l D .1 C 2 /kC1 l for l 6D 0:

(5)

Proof. We are going to prove Equalities (1), (2), and (3), leaving the last two equalities as an exercise1 . (1) According to Lemma 3.21, we have nC1 D K. (2) According to Proposition 2.17, we have i p i D ı.i;nC1/ () i ı.i;nC1/ D p i . On the other hand, the line bundle Li restricted to ı.i;nC1/ is trivial, hence i ı.i;nC1/ D 0. Thus .

i

ı.i;nC1/ /d D p

d i

() ()

d i d i

C .ı.i;nC1/ /d D p p

d i

d i

D .ı.i;nC1/ /d :

It remains to note that, according to Lemma 3.21, ı.i;nC1/ is identified with Di . (3) If pQ W Cxg;nC1 ! Cxg;n is the forgetful map for universal curves, we have pQ K D K DnC1 . On the other hand, KDnC1 D 0. Hence mC1 .K DnC1 /mC1 D pQ K mC1 () K mC1 DnC1 D pQ K mC1 : 1 The last two equalities are not more complicated than then first three, but require some juggling between xg;n , C xg;nC1 , and M x g;nC1 identified with C xg;n , which makes the text of the proof rather ugly. C


The push-forward of this equality to the moduli spaces gives m p m D

m nC1

D K m:

Corollary 3.23. Let Q be a polynomial in the variables m , ık;l , be the polynomial obtained from Q by the substitution i 7! i Z Z mC1 z m Q D nC1 Q: x g;n M

z

1 ; : : : ; n . Let Q i nC1 . Then we have

x g;nC1 M

Proof. By definition, m D p .K mC1 /. By the projection formula, we obtain Z Z mC1 /Q D K mC1 p Q: p .K xg;n DM x g;nC1 C

x g;n M

According to Theorem 3.22, the pull-back p modifies each term of the polynomial Q. However most of these modifications play no role, because they vanish when we multiply them by K mC1 . Indeed, the difference between id and p id is a multiple of Di , and we know that Di K D 0. Similarly, the difference between ık;l and p ık;l is either a multiple of (if l 6D 0), or a sum of a multiple of and a multiple of D (if l D 0). But we know that K D KD D 0. The only remaining terms are i , and for these the difference is important. According to Theorem 3.22, we have p i D i K i . Recalling that K (on Cxg;n ) is the same x g;nC1 ), we obtain that we must replace every i in Q by i i . as nC1 (on M nC1 z and the assertion of the corollary. Thus we obtain exactly the polynomial Q Corollary 3.23 allows us to express an integral involving at least one -class as a combination of integrals with fewer -classes. Example 3.24. We have Z Z 2 2 1 D 6 .1 x 0;5 M

x 0;6 M

Z 6/

D x 0;7 M

2 7

2 6

Z

3 6

D 6 1 D 5:

x 0;6 M

3.3.3 Eliminating ı-classes: the attaching map. Recall that Cxg;n is the set of nodes of the singular fibers. Take a singular stable curve, choose a node, unglue the branches of the curve at the node and number the two preimages of the node (there are two ways of doing that). We obtain a new stable curve that has two new marked points. It can either be connected of genus g 1, or composed of two connected x split the disjoint union components of genera g1 and g2 , g1 C g2 D g. Denote by M F F x g ;I [fnC1g M x g1;nC2 : x split D x g ;I [fnC2g M M M 1 1 2 2 I1 [I2 Df1;:::;ng g1 Cg2 Dg

708

Dimitri Zvonkine

We have actually proved the following statement. x split is a 2-sheeted covering of . Lemma 3.25. M z The same 2-sheeted covering was previously denoted by . p c x split ! M x g;n the composition M x split ! ! x g;n . The image Denote by j W M M x of j is the boundary ı0;0 D Mg;n n Mg;n of the moduli space. x split is defined by taking the same tautological class on A tautological class on M x split . every component of M x split and Our aim is now to study the difference between a tautological class on M x the pull-back of the analogous class from Mg;n . Theorem 3.26. We have m j m D 0; d i

j

d i

(1)

D 0;

ık;l j ık;l D .

(2) nC1

C

kC1

nC2 /

.

nC1

nC2 /

l

:

(3)

Proof. Equalities (1) and (2) follow from the obvious fact that |Q K D K and |Q Di D Di , where |Q is the map of universal curves associated with j . For Equality (3) we only sketch the argument. First consider the case k D l D 0. Rewrite the equality as j ı0;0 D ı0;0 .

nC1

C

nC2 /:

The class j ı0;0 is almost the same as the self-intersection of the boundary ı0;0 x split are of the moduli space. More precisely, different connected components of M degree 2 coverings of irreducible components of ı0;0 , and we are interested in the pull-backs of the intersections of these components with ı0;0 . Some of the intersection points encode stable curves with two nodes. Such intersections are transversal and they give rise to the term ı0;0 in the right-hand side. But when we try to intersect a component of ı0;0 with itself we obtain a nontransversal intersection. To determine the intersection in cohomology we need to x g;n . It turns out that know the first Chern class of the normal line bundle to ı0;0 in M this first Chern class equals j . nC1 C nC2 /. This can be seen using a local model of the smoothening of a node. Let B be a base manifold and T1 , T2 two line bundles over B. Let T D T1 ˝ T2 . Then we have a natural map .x; y/ 7! t D xy between the total spaces of the bundles T1 ˚ T2 and T . In this local model the zero section of T corresponds to ı0;0 , the total space of T x g;n , and T is the normal line bundle to ı0;0 represents the neighborhood of ı0;0 in M x in Mg;n . The line bundles T1 and T2 are the tangent line bundles to the branches of the universal curve at the node. Thus we see that c1 .T / D c1 .T1 / C c1 .T2 /. Thus the nontransversal intersection gives us the correction term . nC1 C nC2 / in the right-hand side of (3).


The case of general k and l is similar, since ık;l is the intersection of ı0;0 with x split . .1 C2 /k .1 2 /l , which is transformed into . nC1 C nC2 /k . nC1 nC2 /l on M

Corollary 3.27. Let Q be a polynomial in variables m , ık;l , the polynomial obtained from Q by the substitution ıi;j 7! ıi;j . Then we have

Z ık;l

x g;n M

1 QD 2

iC1 . nC2 /

nC1

C

.

nC1

nC1

C

Z C

nC2 /

k

nC1

.

1; : : : ;

nC2 /

nC1

j

n.

z be Let Q

:

nC2 /

l

z Q:

x split M

Proof. By definition, ık;l D

1 j . 2

nC2 /

k

.

nC1

l nC2 /

;

x split is a double covering of . The projection where the factor 1=2 appears because M formula implies that Z 1 j . nC1 C nC2 /k . nC1 nC2 /l Q 2 x g;n M

1 D 2

Z .

nC1

C

nC2 /

k

.

nC1

nC2 /

l

j Q:

x split M

According to Theorem 3.26, every term of Q remains unchanged after a pull-back by j , except for the terms ık;l , that are transformed according to the rule j ık;l D ık;l .

nC1

C

nC2 /

kC1

.

nC1

nC2 /

l

:

z so we obtain the claim of the corollary. In other words, j Q D Q,

Corollary 3.27 allows us to express an integral involving at least one ı-class as a combination of integrals with fewer ı-classes. Applying Corollaries 3.23 and 3.27 several times allow us to reduce any integral involving -, -, and ı-classes to a combination of integrals involving only -classes. These integrals are the subject of the next section.

710

Dimitri Zvonkine

4 Around Witten’s conjecture Now our aim is to compute the integrals Z

d1 1

dn n :

x g;n M

4.1 The string and dilaton equations Proposition 4.1. For 2 2g n < 0 we have Z Z n X d1 dn 1 ::: n D

:::

di 1 i

:::

dn n I

iD1 x Mg;n

x g;nC1 M

Z

d1 1

d1 1

dn n

:::

Z nC1

D .2g 2 C n/

x g;nC1 M

d1 1

:::

dn n :

x g;n M

These equations are called the string and the dilaton equations. We borrowed their proof from Witten’s paper [35]. x g;n and on M x g;nC1 . Proof. We will need to consider the line bundles Li both on M 0 We momentarily denote the former by Li , 1 i n, the latter retaining the notation Li , 1 i n C 1. Further, denote by i0 the first Chern class of the line bundle L0i x g;n and, by abuse of notation, its pull-back to M x g;nC1 by the map forgetting the on M x g;nC1 . .n C 1/st marked point. Let i be the first Chern class of Li on M According to Proposition 2.17, we have 0 i

D

i

C ı.i;nC1/ :

(1)

Moreover, we have i

ı.i;nC1/ D

nC1

ı.i;nC1/ D 0

(2)

(because the line bundles Li and LnC1 are trivial over ı.i;nC1/ ) and ı.i;nC1/ ı.j;nC1/ D 0

for i 6D j

(because the divisors have an empty geometric intersection). Let us first prove the string relation. We have .2/ d 0 d .1/ D ı.i;nC1/ id 1 C C . i0 /d 1 D ı.i;nC1/ . i . i/ where we set by convention . d i

0 1 i/

D.

D 0. Thus

0 d i/

C ı.i;nC1/ .

0 d 1 : i/

(3)

0 d 1 ; i/


It follows that Z x g;nC1 M

d1 1

Z

D .3/

::: h

x g;nC1 M

Z

D

.

.

dn n 0 d1 1/

0 d1 1/

C ı.1;nC1/ .

:::.

0 dn n/

0 d1 1 1/

Z

n X

C

i

.

h ::: . 0 d1 1/

0 dn n/

C ı.n;nC1/ .

: : : ı.i;nC1/ .

0 di 1 i/

0 dn 1 n/

:::.

i

0 dn n/ :

iD1 x Mg;nC1

x g;nC1 M

x g;n . As The first integral is equal to 0, because the integrand is a pull-back from M for the integrals composing the sum, we integrate the class ı.i;nC1/ over the fibers of x g;nC1 ! M x g;n . This is equivalent to restricting the integral to the the projection M x g;n . Finally, we obtain divisor ı.i;nC1/ , which is naturally isomorphic to M Z n Z X d1 dn . 10 /d1 : : : . i0 /di 1 : : : . n0 /dn : 1 ::: n D x g;nC1 M

iD1

x g;n M

This proves the string relation. Now let us prove the dilaton relation. We have Z d1 dn nC1 n 1 ::: x g;nC1 M Z 0 d1 .1/ ::: D 1 C ı.1;nC1/ x M Z g;nC1 .2/ D . 10 /d1 : : : . n0 /dn nC1 x g;nC1 M Z D .2g 2 C n/ . 10 /d1 : : : . x g;n M

0 n

C ı.n;nC1/

dn

nC1

0 dn n/ :

The last equality is obtained by integrating the factor nC1 over the fibers of the x g;nC1 ! M x g;n . This proves the dilaton relation. projection M Proposition 4.2. The string and the dilaton equations, together with the initial conditions Z Z 1 1 D 1 and 1 D 24 x 0;3 M

x 1;1 M

allow one to compute all integrals

Z

x g;n M

for g D 0; 1.

d1 1

dn n

712

Dimitri Zvonkine

We leave the proof as an exercise to the reader. Example 4.3. Recall the first equality of Example 3.20: X 1 .1 1 D i C ı0;0 /: 12 This equality is true for all g and n, but let us consider the case g D n D 1. We have (cf. Example 2.26) Z Z Z Z Z 1 1 1 2 I I 1 D 1 D ı0;0 D : 1 D 2 D 24 24 2 x 1;1 M

x 1;1 M

x 1;2 M

x 1;1 M

x 1;1 M

x 1;1 is indeed correct: 1=24 D Thus we have checked that the expression for 1 over M 1=12 .1=24 1=24 C 1=2/.

4.2 KdV and Virasoro It looks like we are very close to our goal of computing all intersection numbers of -classes and thus of all tautological classes. However the last step is actually quite hard; therefore we only formulate the theorems that make it possible to compute the remaining integrals, but do not give the proofs. Introduce the following generating series: Z X X d1 dn td1 : : : tdn : F .t0 ; t1 ; : : : / D 1 ::: n nŠ x g;n g0;n1 d1 ;:::;dn M 22gnd BSO.d /I Q Œd H .1 MTSO.d /I Q/ Š ƒ H

where Œd denotes shift in grading by d . For d D 2, H .1 MTSO.2/I Q/ Š QŒ1 ; 2 ; : : : ;

deg i D 2i:

The class i corresponds to the .i C 1/st power of the Euler class.

1.3 Smooth fiber bundles and the spectra MTO.d/ Consider an embedded smooth fiber bundle with closed manifold fibers e W E kCd ,! B k Rd Cn ;

D pr 1 W E ! B;

equipped with a fiberwise tubular neighborhood eO W int D.N E/ ,! B Rd Cn as in §1.1. The Pontryagin–Thom map cE W BC ^ S d Cn ! Th.N E/

(1.14)

726

Ib Madsen

is defined to be the identity on eO int D.N E/ and constant at 1 outside the open disk bundle. Hence cE is the map of one point compactifications induced by the embedding e. O We remember that T E D ker.d W TE ! TB/ is the vertical tangent bundle with complement N E. The fiber T Ez at z 2 E is a d -dimensional subspace of f.z/g Rd Cn . This defines a map t from E to the Grassmannian G.d; n/. Vectors in N Ez are mapped into the orthogonal complement, so we get a fiberwise map of vector bundles tO / U? N E d;n

(1.15) E

t

/ G.d; n/

with t and tO defined to be t .z/ D de.T Ez / Rd Cn ;

tO.z; v/ D v 2 de.T Ez /? :

Since tO is a fiberwise isometry it is proper and induces a map of Thom spaces, which composed with cE above gives ? /: ˛Q E W BC ^ S d Cn ! Th.Ud;n

This works for every n, and yields a map of spectra ˛Q E W †1 .BC / ! MTO.d /: For the universal bundles of (1.7) and n 1 this amounts to a map ˛Q n W †1 Bnf .M d /C ! MTO.d /: For n D 1, (1.8) leads to

˛Q M W †1 BDiff.M d /C ! MTO.d /

with adjoint

˛M W BDiff.M d / ! 1 MTO.d /:

(1.16)

Suppose next that the embedded fiber bundle is oriented (in the sense that the fiberwise tangent bundle is oriented). Then ˛Q E above defines a map of spectra ˛Q E W †1 .BC / ! MTSO.d /: Universally we get ˛Q M W †1 BDiff C .M d /C ! MTSO.d / with adjoint

˛M W BDiff C .M d / ! 1 MTSO.d /: C

d

(1.17)

Here, we remember that BDiff .M / denotes the topological group of orientation preserving diffeomorphisms of M d .

Chapter 12. Homology of the open moduli space of curves

727

For each closed oriented d -manifold M d we get potentially interesting cohomology classes of the classifying space BDiff C .M d / via the composition Š ˛QM ˇQM W H kCd BSO.d / ! H k MTSO.d / ! H k BDiff C .M d / :

(1.18)

For cohomology with rational coefficients, Theorem 1.4 implies that the image of C d ˛M W H .1 MTSO.d /I Q/ ! H .BDiff .M /I Q/

is the free graded commutative algebra generated by the image of ˇQM . Remark 1.8. The integral cohomology of 1 MTSO.d / contains an abundance of torsion classes, and with Z coefficients the image of ˛M is much bigger than the algebra generated by the image of ˇQM . Given an oriented embedded fiber bundle together with a fiberwise tubular neighborhood, one defines the push-forward map (“integration along fibers”) by the diagram Š / H kCn Th.N E/ H k .E/ Š

H kd .B/

Š

cE

/ H kCn .B ^ S d Cn / C

with cE the Pontryagin–Thom map of (1.14). The map cE has degree one so that hŠ ."/; ŒBi D h"; ŒEi Let us consider a map

for " 2 H kCd .E/.

(1.19)

f W B k ! BDiff C .M d /

with domain a k-dimensional smooth manifold. It follows from remark 1.3 that f induces an embedded fiber bundle E d Ck B k Rd Cn for n sufficiently large. Let T E be its tangent bundle along the fibers and 2 Cd BSO.d / a universal characteristic class. Then H (1.20) f ˇQM ./ D Š .T E/ :

Indeed, the formula is a consequence of (1.15) and naturality of the Thom isomorphism.

2 The cobordism category and MTO.d/ This section gives a geometric interpretation of the space ? 11 MTO.d / D colim d Cn1 Th.Ud;n /

728

Ib Madsen

as the classifying space of the category Cobd of d -dimensional cobordisms embedded in a euclidian space of arbitrarily high dimension, similar in spirit to the interpretation of the homotopy groups of the spectrum MO D fTh.Un;1 /gn2N as cobordism classes of manifolds.

2.1 The embedded cobordism category For 0 < n 1, let Cobd;n denote the following category of embedded manifolds. An object .M; a/ is a closed .d 1/-dimensional submanifold M d 1 fag Rd Cn1 . A non-identity morphism from .M0 ; a0 / to .M1 ; a1 / has a0 < a1 and consists of a compact d -dimensional submanifold W d Œa0 ; a1 Rd Cn1 with the further requirement that for some " > 0: (i) W \ Œa0 ; a0 C " Rd Cn1 D Œa0 ; a0 C " M0 , (ii) W \ Œa1 "; a1 Rd Cn1 D Œa1 "; a1 M1 ,

(2.1)

(iii) @W D W \ fa0 ; a1 g Rd Cn1 . Composition is union of subsets of R Rd Cn1 . In the notation of §1.1, G ob Cobd;n D R Bn .M /;

(2.2)

where the disjoint union runs over all diffeomorphism types of closed .d 1/-dimensional manifolds which embed in codimension n. The set of morphisms can be described in a similar fashion. Let (W; h0 ; h1 / be an abstract cobordism from M0 to M1 , that is, a compact d -dimensional manifold together with embeddings or collars h0 W Œ0; 1/ M0 ! W;

h1 W .0; 1 M1 ! W

with @W D h0 .0; M0 / t h1 .1; M1 /. Assuming that W embeds in codimension n, let Emb" .W; Œ0; 1 Rd Cn1 / denote the space of embeddings e W W ! Œ0; 1 Rd Cn1 such that e B h0 .t0 ; x0 / D t0 ; e.x0 / ; e B h1 .t1 ; x1 / D t1 ; e.x1 / for t0 2 Œ0; "/ and t1 2 .1 "; 1. Let Diff " .W / be the group of diffeomorphisms that restrict to product diffeomorphisms on the "-collars. It acts freely on the embedding space. The size of the collars is not part of the structure so we let Emb.W; Œ0; 1 Rd Cn1 / D colim" Emb" .W; Œ0; 1 Rd Cn1 / Diff.W / D colim" Diff " .W /; and define Bn .W / D Emb.W; Œ0; 1; Rd Cn1 / = Diff.W /:


Then the space of morphisms is mor Cobd;n D ob Cobd;n t

G

R2>0 Bn .W /;

729

(2.3)

where R2>0 D f.a0 ; a1 / j a0 < a1 g, and the disjoint union extends over abstract cobordism classes of d -dimensional cobordisms. An element .a0 ; a1 ; e/ of R2>0 Bn .W / gives a morphism of Cobd;n , namely the image of the composition e

W ! Œ0; 1 Rd Cn1 ! Œa0 ; a1 Rd Cn1 ; where the second map is the affine isomorphism Œ0; 1 ! Œa0 ; a1 . The identifications (2.2) and (2.3) turn Cobd;n into a topological category in the sense that the structure maps d0

mor Cobd;n

d1

// ob Cobd;n ;

s0

ob Cobd;n ! mor Cobd;n

and the composition mor Cobd;n ob Cobd;n mor Cobd;n ! mor Cobd;n are continuous. The category Cobd D Cobd;1 is our main concern. We note that for objects .a0 ; M0 / and (a1 ; M1 / in Cobd there is the homotopy equivalence G BDiff.W; @W /; (2.4) Cobd .a0 ; M0 /; .a1 ; M1 / ' where W runs over diffeomorphism classes of abstract cobordisms from M0 to M1 , and Diff.W; @W / denotes the group of diffeomorphisms of W that keeps a collar neighborhood of the boundary pointwise fixed. Indeed, (2.4) is a consequence of the relative Whitney embedding theorem: the embedding space of W into Œ0; 1 R1 that restricts to a fixed embedding of the boundary is contractible. Remark 2.1. One can define a version Cobfd;n of Cobd;n of cobordisms equipped with a tubular neighborhood. Uniqueness of tubular neighborhoods implies that the map that forgets the normal tube provides homotopy equivalences '

ob Cobfd;n ! ob Cobd;n ;

'

mor Cobfd;n ! mor Cobd;n :

In particular the two categories have homotopy equivalent classifying spaces, cf. §2.2 below. of the category above where the objects are There is an oriented version CobC d;n oriented closed .d 1/-dimensional manifolds and the morphisms W from M0 to M1 are oriented d -manifolds such that the induced orientation of @W is .M0 / t M1 . The oriented version of (2.4) is G BDiff C .W; @W /; (2.5) .a0 ; M0 /; .A1 ; M1 / ' CobC d

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where Diff C .W; @W / Diff.W; @W / is the subgroup of orientation preserving diffeomorphisms, and the union is over diffeomorphism classes as above.

2.2 Abstract transversality and BCobd;n Let C be a topological category. We remember that its nerve N C is the simplicial space defined to be N0 C D ob C ; N1 C D mor C and for k 2, Nk C D N1 C N0 C N1 C N0 C N0 C N1 C fk

f1

D fck ! ck1 ! : : : ! c0 j fi 2 N1 C g; topologized as a subspace of .N1 C/k . Following [34], the classifying space of C is the “fat” realization of N C, G Nk C k = ; BC D kN Ck D where the equivalence relation is generated from the face relations only: .di xk ; tk1 / .xk ; d i tk1 /I

xk 2 Nk C ; tk1 2 k1 :

If the simplicial space N is good, i.e. if for each k the subspace of degenerate simplices of Nk embeds as a cofibration ([34], Appendix A), then kN k is homotopy equivalent to the standard realization jN j. The main result of this section is the following Theorem 2.2. For n 1 there are homotopy equivalences ? /, (i) BCobd;n ' d Cn1 Th.Ud;n ? ' d Cn1 Th.p Ud;n /, (ii) BCobC d;n

where p W G C .d; n/ ! G.d; n/ is the orientation cover of the Grassmannian. Corollary 2.3. (i) BCobd ' 11 MTO.d /, (ii) BCobC ' 11 MTSO.d /. d The corollary was proved in [15] using the setup of sheaves from [23], §4. The proof of Theorem 2.2 for finite n was sketched in [16] and further developed in [31]. I will indicate a proof of Theorem 2.2 that uses an “abstract transversality theorem” that can be found in [3]. The setup for this transversality is a metrizable space X together with a closed subspace Z X R for which Z.x/ D Z \ .fxg R/ R

(2.6)


731

has measure zero for all x 2 X. Given such a pair of spaces, [3] defines a simplicial space K .X; Z/. Its space of q-simplices is the following subspace of X RqC1 , Kq .X; Z/ D fx; a0 ; : : : ; aq j a0 aq ; ai … Z.x/g

(2.7)

The face and degeneracy operators are by omitting some ai and by repeating an ai . There are two ways to topologize Kq .X; Z/, namely either as a subspace of X RqC1 or as a subspace of X .Rı /qC1 where Rı denotes R with the discrete topology. In the second case we write Kı .X; Z/. It is proved in [3] that both K .X; Z/ and Kı .X; Z/ are good simplicial spaces so that the fat and the standard geometric realizations are homotopy equivalent. Moreover, the map of realizations '

jKı .X; Z/j ! jK .X; Z/j

(2.8)

is proved to be a homotopy equivalence. If X D fxg is a single point, then jKı fxg; Z j is weakly contractible. Indeed, given a finite set of points a0 ; : : : ; aN in R n Z, one can always find a point aN C1 2 R n Z that is larger than all the ai . This implies that a finite simplicial subspace of jKı fxg; Z j is contained in a cone, and hence that the space is weakly contractible. The abstract transversality theorem we need is Theorem 2.4 (from [3]). The projection map p

jKı .X; Z/j ! X is a weak homotopy equivalence. Following [16], we define a space of d -dimensional submanifolds of RN . Its underlying set is ‰d .RN / D fW d RN j @W d D ;;

W closed as a subsetg:

In order to topologize ‰d .RN /, let C RN be compact and N W RN a tubular neighborhood. For each section s W W ! N W , define the norm kskC D sup js.x/j C Tx W; ds.Tx W / ; x2W \C

where is the standard metric on the Grassmannian G.d; N d /, and j j the norm on N W . The neighborhoods of W 2 ‰d .RN / are given by C;" .W / D fs.W / j s W W ! N W section, kskC < "g: The empty set ; 2 ‰d .RN / is the base point. Note that a sequence of manifolds Wi that leaves every compact subset converges to ;. Consider the subspace Dd;n D fW d 2 ‰d .Rd Cn / j W R .0; 1/d Cn1 g

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of ‰d .Rd Cn /, and note that for W 2 Dd;n the projection pr 1 W W ! R on the first factor is a proper map (since W Rd Cn is a closed subset). Theorem 2.5 (from [16]). There is a homotopy equivalence ? /: Dd;n ' d Cn1 Th.Ud;n

The proof of Theorem 2.5, sketched in [16], is a consequence of Gromov’s general h-principle, [17]. For n D 1 the result is Theorem 3.4 of [15]. We can apply Theorem 2.4 with X D Dd;n and Zd;n Dd;n R the set of pairs .W; a/ where a 2 R is a critical value of pr 1 W W ! R . For each W 2 Dd;n , the intersection of Zd;n with fW g R has measure zero by Sard’s theorem. Moreover, it is not hard to see that Zd;n is a closed subset of Dd;n R. Combining Theorem 2.4 and Theorem 2.5 one has Corollary 2.6. There is a weak homotopy equivalence ? /: jK .Dd;n ; Zd;n /j ' d Cn1 Th.Ud;n

The next result completes the proof of Theorem 2.2. Theorem 2.7. There is a simplicial embedding f W N Cobd;n ! K .Dd;n ; Zd;n / which induces a weak homotopy equivalence of geometric realizations. Proof. The embedding

fq W Nq Cobd;n ! Kq Dd;n ; Zd;n

is defined as follows. We choose once and for all a diffeomorphism of R with .0; 1/ so that R Rd Cn1 is identified with R .0; 1/d Cn1 . For q D 0 and M d 1 fag Rd Cn1 D fag .0; 1/d Cn1 , f0 .M d 1 ; a/ D .R M d 1 ; a/;

R M d 1 R .0; 1/d Cn1 :

For q D 1 and W d Œa0 ; a1 .0; 1/d Cn1 , f1 .W d / D .1; a0 M.a0 / [ W d [ Œa1 ; 1/ M.a1 / with M.ai / D W \ fai g Rd Cn1 . Note that f1 .M d / is an element of Dd;n by the requirements (2.1), and that .f1 .W d /; a0 ; a1 / 2 K1 Dd;n ; Zd;n . For general q > 1, the construction is analogous, extending the outside collars at the extreme walls a0 Rd Cn1and aq Rd Cn1 . Write Kq D Kq Dd;n ; Zd;n and consider the subspaces Kq?? Kq? Kq :


733

Kq? is the subspace of .W; a0 ; : : : ; aq / for which W \ .ai "; ai C "/ .0; 1/d Cn1 D M.ai / .ai "; ai C "/

(2.9)

for i D 0; : : : ; q and some " > 0. The subspace Kq?? Kq? consists of .W; a0 ; : : : ; aq / 2 Kq? with W \ .1; a0 .0; 1/d Cn1 D M.a0 / .1; a0 W \ Œaq ; 1/ .0; 1/d Cn1 D M.aq / Œaq ; 1/: The embedding fq of Nq Cobd;n into Kq has image Kq?? , so we must show that the inclusions Kq? Kq and Kq?? Kq? are homotopy equivalences. This is geometrically rather obvious. For Kq? Kq it is the statement that one can deform a transverse intersection to an orthogonal intersection as required in (2.9). The full argument is almost identical to the argument which proves Proposition 4.4 of [15]. The difference between Kq?? and Kq? is that for .W; a0 ; : : : ; aq / 2 Kq?? , W is constant on each component of .R n Œa0 ; aq / .0; 1/d Cn1 but for an element of Kq? , W is only constant on the "-collars .a0 "; a0 .0; 1/d Cn1 and Œaq ; aq C "/ .0; 1/d Cn1 for some " > 0. The idea is to extend the collars as follows. Choose a diffeomorphism (depending continuously on ", a0 , a1 ) '" D '" .a0 ; a1 / W .a0 "; aq C "/ ! R with '" .s/ D s if s is in the subinterval .a0 "=2; aq C "=2/. Let t W R ! R the affine isomorphism .a0 t; aq C t / ! .a0 "; aq C "/;

t ":

The map t D 1 t B '" B t is the identity on a subinterval of .a0 t; aq C t / that tends to .1; 1/ for t ! 1. If O t D t .0; 1/d Cn1 , then W 7! O t .W / 2 Kq?? for t ! 1. To complete the proof we use that a map f between “good” simplicial spaces induces a (weak) homotopy equivalence of topological realizations if each fq is a (weak) homotopy equivalence, [34].

3 Diffeomorphism groups and moduli spaces This section applies the results of §1 and §2 to study the moduli space of Riemann surfaces and its relationship with the moduli space of embedded surfaces. Of particular interest is the generalized Mumford conjecture.

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Ib Madsen

3.1 Riemann’s moduli space Given a closed oriented differentiable surface F of genus g, let JC .F / denote the set of maximal holomorphic atlases on F , compatible with the oriented smooth structure. The group Diff C .F / acts on JC .F / by pulling back a holomorphic atlas along a diffeomorphism. Riemann’s moduli space (the open moduli space of curves) is the orbit space M.F / D JC .F / = Diff C .F /: (3.1) It is often convenient to replace the space JC .F / of complex structures by the space of almost complex structures: JC .TF / D fJ W TF ! TF j J 2 D 1; fv; Jx vg oriented for v 2 Tx F n f0gg: p The obvious map JC .F / ! JC .TF /, induced by multiplication with 1, is a bijection by a theorem that goes back to Gauss, cf. [5], [30]. The group Diff C .F / acts on JC .TF / by conjugating an almost complex structure by the differential of the diffeomorphism, and (3.1) can be replaced by M.F / D JC .TF / = Diff C .F /:

(3.2)

The almost complex structures on the euclidian plane can be identified as JC .R2 / D GLC 2 .R/ = GL1 .C/; and JC .TF / can be viewed as the space of sections in the fiber bundle P .F / GLC .R/ JC .R2 / ! F; 2

where P .F / is the frame bundle of TF . This gives JC .TF / and hence M.F / a 2 topology. Moreover, since GL1 .C/ ! GLC 2 .R/ is a homotopy equivalence, JC .R / is contractible. It follows that the space of sections JC .TF / is contractible. C Let Diff C 1 .F / Diff .F / denote the connected component of the identity. By [10], it acts freely on JC .TF / and the orbit map JC .TF / ! JC .TF / = Diff 1 .F /

(3.3)

is a principal fiber bundle. The base space in (3.3) is the Teichmüller space T .F /, homeomorphic to R6g6 . Since both base and total space in (3.3) are contractible, so is Diff C 1 .F /. This implies that the mapping class group C .F / D 0 Diff C .F / is homotopy equivalent to the topological group Diff C .F /. Hence BDiff C .F / ! B C .F /

(3.4)

is a homotopy equivalence. The mapping class group C .F / acts on Teichmüller space with finite isotropy groups, so M.F / Š T .F / = C .F /


735

is a .6g 6/-dimensional orbifold. In particular the rational cohomology groups of M.F / are isomorphic to the rational Eilenberg–Maclane cohomology of C .F /, i.e. H .M.F /I Q/ Š H .B C .F /I Q/ Š ExtQŒ C .F / .Q; Q/: We know from Remark 1.3 that BDiff C .F / is homotopy equivalent to the space of embedded surfaces C B1 .F / D f† R1 j † oriented; † Š F g: C For † in B1 .F /, each tangent fiber Tp † is oriented and it receives an inner product from the surrounding euclidian space. Let J W Tp † ! Tp † be rotation by +90°. This defines an almost complex structure J 2 JC .TF / and gives a map C B1 .F / ! M.F /

(3.5)

that induces isomorphism on H .I Q/. For oriented surfaces with non-empty boundary we set M.F; @F / D JC .TF / = Diff C .F; @F /; where Diff C .F; @F / is the group of orientation preserving diffeomorphisms of F that restrict to the identity on @F . The corresponding Teichmüller space T .F; @F / is homeomorphic to R6g6C2r with r the number of boundary circles. For @F ¤ ; the action of C .F; @F / on T .F; @F / is free, and consequently H .M.F; @F /I Z/ Š H .BDiff C .F; @F /I Z/: Remark 3.1. The universal cover of a Riemann surface of genus g 2 may be identified with the upper half plane H C. The group of complex automorphisms of H agrees with group of isometries when H is equipped with the standard hyperbolic metric, ds 2 D y12 jdzj2 . Thus a Riemann surface may also be viewed as a hyperbolic space form, and M.F / can be defined in terms of hyperbolic structures. This viewpoint is given a detailed treatment in [33]. Theorem 3.2. (i) Let F be a closed surface of genus g. There is a map j W BDiff C .F / ! M.F / that induces an isomorphism on homology with rational coefficients. (ii) If @F ¤ ;, j W BDiff C .F; @F / ! M.F; @F / induces an isomorphism on integral homology.

3.2 The generalized Mumford conjecture This section relates the embedded cobordism category to the classifying space of diffeomorphism groups. Throughout the section we work in the oriented setting. We remember that ' 11 MTSO.d /: BCobC d

736

Ib Madsen

Given a pair of objects c0 ; c1 in a topological category C , the space of morphisms C.c0 ; c1 / is a subspace of N1 C. The obvious map from C .c0 ; c1 / 1 into BC adjoins to a map (3.6) C .c0 ; c1 / ! c0 ;c1 BC into the space of curves in BC from c0 to c1 . Assuming C .c0 ; c1 / ¤ ;, c0 and c1 lie in the same connected component of BC and c0 ;c1 BC is homotopy equivalent to the based loop space of that component. . Given an abstract oriented cobordism W d between We apply (3.6) to C D CobC d two closed oriented .d 1/-manifolds M0 and M1 , we get from (2.4) a map ˛W W BDiff C .W d ; @W d / ! BCobC d which we compose with the homotopy equivalence of Corollary 2.3 to get ˛W W BDiff C .W d ; @W d / ! 1 MTSO.d /:

(3.7)

Remark 3.3. If @W d D ;, then (3.7) is homotopic to the map of (1.16). we have made no connectivity assumpIn the cobordism categories Cobd and CobC d tion on morphisms; they might have connected components that are closed manifolds. This can be avoided by introducing the positive boundary subcategory Cob@d of Cobd . The two categories have the same space of objects but different spaces of morphisms: W d Œa0 ; a1 Rd 1C1 is a morphism in Cob@d if each connected component of W has a non-empty boundary in fa1 g Rd 1C1 . Note in particular that if M1d 1 D W d \ fa1 g Rd 1C1 is connected then so is W d . Theorem 3.4 (from [15]). For d 2, the inclusions BCob@d ! BCobd ;

BCobC;@ ! BCobC d d

are homotopy equivalences. We now restrict attention to d D 2 where one can make use of Harer type stability theorems, [4], [18], [19]. Let Fg;r denote the oriented surface of genus g with r boundary circles. Since FgC1;r D Fg;r [ F1;2 ;

Fg;r1 D Fg;r [ F0;1

there are group homomorphisms Diff C .Fg;r ; @Fg;r / ! Diff C .FgC1;r ; @FgC1;r /; Diff C .Fg;r ; @Fg;r / ! Diff C .Fg;r1 ; @Fg;r1 /;

(3.8)


737

upon extending a diffeomorphism of Fg;r by the identity on F1;2 and F0;1 , respectively. The homological properties of (3.8) were originally studied in [18] where Harer proved that the maps induce isomorphism in homology in a certain stable range depending on the genus. The range was improved first by Ivanov [19], [20] and most recently by S. Boldsen in [4]. The improved range is given in the next theorem. Theorem 3.5. The induced maps Hk .BDiff C .Fg;r ; @Fg;r /I Z/ ! Hk .BDiff C .FgC1;r ; @FgC1;r /I Z/; Hk .BDiff C .Fg;r ; @Fg;r /I Z/ ! Hk .BDiff C .Fg;r1 ; @Fg;r1 /I Z/ are isomorphisms for 3k 2g 2. Remark 3.6. The proof of Theorem 3.5 is based upon the homotopy equivalence (3.4) and the action of the mapping class group on various curve complexes. The connection between diffeomorphism groups of surfaces and BCobC;@ is 2 based upon a version of Quillen’s Theorem B, which I now turn to. Given a functor F W C op ! Spaces, F o C is the category with ob.F o C / D N0 .F o C/ D f.x; c/ j c 2 ob C ; x 2 F .c/g; mor.F o C/ D N1 .F o C/ D f.x; f / j f 2 mor C ; x 2 F .d0 f /g; where d0 f denotes the target and d1 f the source for f W c0 ! c1 . There is a Cartesian diagram N1 .F o C/ N1 C

d0

d0

/ N0 .F o C / / N0 C .

If C is a topological category and F a continuous functor, then F o C becomes a topological category: G F .c/; c 2 N0 C N1 .F o C/ D has the obvious topology and N1 .F o C/ N1 C N0 .F o C / the subspace topology. The version of the group completion theorem we need is the following: Theorem 3.7 (from [15], [25]). Let F W C op ! Spaces be a continuous functor. Suppose (1) N0 .F o C/ ! N0 C is a Serre fibration, (2) B.F o C/ is (weakly) contractible,

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Ib Madsen

(3) each f 2 C.c0 ; c1 / induces an isomorphism F .f / W H F .c1 /I Z ! H F .c0 /I Z : Then for each c 2 C , the map F .c/ ! c BC induces an isomorphism on integral homology. We define BDiff C .F1;r ; @F1;r / to be the (homotopy theoretic) union of the spaces BDiff C .Fg;r ; @Fg;r / as g ! 1 in (3.8), i.e. BDiff C .F1;r ; @F1;r / D hocolim BDiff C .Fg;r ; @Fg;r /: g!1

By Theorem 3.5, Hk BDiff C .Fg;r ; @Fg;r /I Z Š Hk BDiff C .F1;r ; @F1;r /I Z for 3k 2g 2, and the right-hand side is independent of the number r of boundary circles. Theorem 3.8 (from [23]). There is a homology equivalence ˛ W Z BDiff C .F1;r ; @F1;r / ! 1 MTSO.2/ of infinite loop spaces. I will sketch out a proof of Theorem 3.8 based on Theorem 3.7, referring the reader to §7 of [15] for more details. We have already established the homotopy equivalences ' BCobC;@ ' 1 MTSO.d /: BCobC d d is homology equivalent to Z Therefore it is sufficient to argue that BCobC;@ s BDiff C .F1;r ; @F1;r /. We apply Theorem 3.7 to the full subcategory C CobC;@ of objects (M; a/ with 2 C;@ 1 1C1 a < 0. Clearly BC ' BCob2 . Let S R be a fixed embedded circle, bi D S 1 fig and ˇi W bi ! biC1 a fixed embedded surface, diffeomorphic to F1;2 . The functors Gi W C op ! Spaces; Gi .c/ D C .c; bi / are connected by the natural transformations ˇi W Gi .c/ ! GiC1 .c/; and we can take the homotopy-theoretic colimit G.c/ D hocolim Gi .c/: This functor satisfies the three conditions of Theorem 3.7. The first condition is a consequence of the embedded smooth fiber bundles considered in §2.1. The second condition follows from the equivalence B.G o C/ ' hocolim B.Gi o C /


739

together with the observation that .bi ; id/ 2 N0 .Gi o C / is a terminal object, so that B.Gi o C/ ' . The critical third condition is a consequence of Theorem 3.5. By (1.8), G Gi .c/ ' BDiff.Fg;rC1 ; @Fg;rC1 /; r D #0 .c/: g0

The map .ˇi / enlarges g by 1, so in the limit G.c/ ' Z BDiff.F1;r ; @F1;r /: , then the induced map G.f / is, up to If f W c0 ! c1 is any morphism in C CobC;@ d homotopy, given by Z BDiff.F1;r1 C1 ; @F1;r1 C1 / ! Z BDiff.F1;r0 C1 ; @F1;r0 C1 / This is a homology isomorphism by Theorem 3.5. Apply Theorem 3.7 to complete the proof. The spectrum homology of MTSO.2/ is ´ Z for k 0 .mod 2/, k 2, k H .MTSO.2/I Z/ D 0 otherwise: This follows from the Thom isomorphism theorem for oriented vector bundles. Apply Corollary 1.7 to get Theorem 3.9. H BDiff.F1;r ; @F1;r /I Q Š QŒ1 ; 2 ; : : : with deg i D 2i . Remark 3.10. The tautological ring Rg is the subring of H .Mg I Q/ generated by the classes i under the isomorphism (3.5). Morita proved in [29] that i ; : : : ; Œg=3 generates Rg , verifying a part of Faber’s conjecture [13]. In particular there is a relation Œg=3C1 D pol.i ; : : : ; Œg=3 / in Rg : It follows from Theorem 3.9 that H 2Œg=3C2 .BDiff.Fg /I Q/ 6Š H 2Œg=3C2 .BDiff.F1 /I Q/; Thus the best possible stability range for the maps in Theorem 3.5 is k 2Œg=3.

3.3 Non-orientable surfaces and higher dimensions Most of §3.1 and §3.2 works equally well for non-orientable surfaces S . The moduli space M.S / is the space of conformal structures and by [11], H .M.S/I Q/ Š H .BDiff.S /I Q/: Moreover, by the unoriented part of Corollary 2.3, BCob2 ' 11 MTO.2/

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Ib Madsen

and we want to apply Theorem 3.7 to study the relationship between BDiff.S / and BCob2 . The key question is the analogue of the stability Theorem 3.5. This was addressed by N. Wahl. Let Sn;r be the connected sum of n copies of the real projective plane with r disjoint open disks removed. Theorem 3.11 (from [37]). For r 1 the homomorphisms Hk .BDiff.Sn;r ; @Sn;r /I Z/ ! Hk .BDiff.SnC1;r ; @SnC1;r /I Z/; Hk .BDiff.Sn;r ; @Sn;r /I Z/ ! Hk .BDiff.Sn;r1 ; @Sn;r1 /I Z/ are isomorphisms in the range 4k n 5. We define BDiff.S1;r ; @S1;r / D hocolim BDiff.Sn;r ; @Sn;r /; n!1

and notice that Theorem 3.7 gives Theorem 3.12. For each r > 0 there is an integral homology equivalence Z BDiff.S1;r ; @S1;r / ! 1 MTO.2/: The formula (1.11) proves the unoriented Mumford conjecture: Theorem 3.13 (from [37]). For r > 0, H BDiff.S1;r ; @S1;r /I Q Š QŒ1 ; 2 ; : : : : In higher dimensions d 3 there is no known analogue of the stability theorems, and the precise relationship between diffeomorphism groups of d -manifolds and BCobd is not well understood. However the following result of J. Ebert shows that the obvious generalization of Theorem 3.8 is false. Theorem 3.14 (from [12]). Let M 3 be an oriented closed 3-manifold. The map of (1.17), ˛ W BDiff C .M 3 / ! 1 MTSO.3/ induces the zero map in rational cohomology. Proof. Serre’s theorem that the stable homotopy groups of spheres are finite has the geometric interpretation that rational homology classes are represented by stably framed manifolds. So given a non-zero cohomology class

2 H k .BDiff C .M 3 /I Q/; there exists a stably framed closed k-manifold B k and a map f W B ! BDiff C .M / with hf . /; ŒBi ¤ 0:


741

By §1.3, the map f gives rise to an embedded fiber bundle with fiber M , E kC3 B k R3Cn ;

W E kC3 ! B k :

If is a cohomology class in the image of ˛ , then Corollary 1.7 tells us that is a product of cohomology classes in the image of ˇQM W H .BSO.3/I Q/ ! H .BDiff C .M /I Q/; with ˇQM defined in (1.18). The domain is a polynomial algebra in the first Pontryagin class p1 , so it suffices to take D ˇQM .p1j /. By (1.19) and (1.20), f ˇQM .p1j / D Š p1 .T E/j D Š p1 .TE/j ; where the second equality is a consequence of the vector bundle equation TE Š T E ˚ .TB/ and the vanishing P of p1 .TB/. Let L D Lj be the Hirzebruch L-class of BSO.3/; Lj is a non-zero multiple of p1j , so f ˇQM .Lj / D Š Lj .TE/ in H 4j 3 .BI Q/: Apply (1.19) to get

hLj .TE/; ŒEi D hŠ Lj .TE/ ; ŒBi:

The left hand side is the signature of E by Hirzebruch’s index theorem, so sign.E/ D hŠ Lj .TE/ ; ŒBi: Finally we appeal to the theorem that sign.E/ D 0 for a smooth oriented fiber bundle W E 4j ! B 4j d with odd-dimensional fibers, [26].

4 Two applications The results of §3.2 on the open moduli space was generalized in [8], [9] to the moduli space of surfaces in a background space K. This section describes this generalization together with two applications from [2] and [7], respectively. Let g .K/ be the moduli space of embedded surfaces of genus g together with a continuous map to the space K: g .K/ D f.†; f / j † R1 genus g; f W † ! K continuousg D Emb.Fg ; R1 / Diff C .Fg / Map.Fg ; K/: For .†; f / 2 g .K/, and † RnC2 equipped with a tubular neighborhood, consider the composition S nC2 ! Th.N †/ ! Th.N † K/;

742

Ib Madsen

where the first arrow is the Pontryagin–Thom map and the second is induced from the ? , graph of f W † ! K. Composing with the classifying map N † ! U2;n ? ? K/ D Th.U2;n / ^ KC ; Th.N † K/ ! Th.U2;n

leads to a well-defined weak homotopy class ˛g;K W g .K/ ! 1 .MTSO.2/ ^ KC /

(4.1)

Remark 4.1. Homotopy classes of maps from a manifold into g .K/ are in one to one correspondence with concordance classes of pairs .E; f / with E B R1 an embedded smooth bundle as in §1.1 and f W E ! K a continuous map. The ˛g;K of (4.1) is the obvious generalization of the corresponding map for K D pt. Theorem 4.2 (from [8], [9]). If K is simply connected, then ˛g;K induces isomorphism in homology with rational coefficients in degrees k with 3k 2g 2. Remark 4.3. A recent preprint [32] asserts that ˛g;K induces isomorphism of integral homology groups in the stable range. See also [14] for a statement when K is not assumed simply connected. I will discuss two applications of Theorem 4.2. The first one from [7], is an extension of results from [1] and concerns the moduli space of flat connections on principal G-bundles over Riemann surfaces for a connected, compact, semi-simple Lie group G. To simplify, I consider only the case G D SU.m/, where a principal G-bundle E over a Riemann surface is a product E D † SU.m/. Let Aflat .E/ denote the space of flat connections on E and A.E/ the space of all connections. It was proved in [1] that the inclusion Aflat .E/ ! A.E/

(4.2)

is 2.g 1/.m 1/ connected, where g is the genus of † (and G D SU.m/). In the moduli space studied in [7], the Riemann surface is allowed to vary. More precisely, define Aut G g to be the group of fiber bundle diagrams E Fg

'O

'

/E / Fg

with ' 2 Diff C .Fg / and 'O G-equivariant. The moduli space in question is MgG D JC .Fg / Aflat .E/ == AutG g; where the double bar indicates the Borel (or stack) quotion: X == G D X G EG when X is a space with an action of G. By (4.2), MgG ! JC .Fg / A.E/ == AutG g

743


is 2.g 1/.m 1/-connected. But both A.E/ and JC .Fg / are contractible, so the Borel quotient is homotopy equivalent to the classifying space BAutG g . On the other G hand, Aut g is a semi-direct product / Map.Fg ; G/

1

/ Aut G l g

/ Diff C .Fg /

/1

so that C BAut G g ' EDiff .Fg / Diff C .Fg / Map.Fg ; BG/

' Emb.Fg ; R1 / Diff C .Fg / Map.Fg ; BG/: An application of Theorem 4.2 therefore gives: Theorem 4.4 (from [7]). There is an isomorphism H .MgG I Q/ Š H .1 .MTSO.2/ ^ BGC /I Q/ for 3 2g 2. We remember that Theorem 1.4 describes the cohomology ring of the right-hand side to be the free graded commutative, unital algebra generated by the positive part of the spectrum cohomology which is given by H .MTSO.2/ ^ KC I Q/ Š H .BSO.2/I Q/ ˝ H .KI Q/ ˝ hu2 i;

(4.3)

where hu2 i is a single copy of Q, placed in degree 2. The second application of Theorem 4.2 is due to D. Ayala. It is based in part upon [35] which describes the relationship between holomorphic maps and continuous maps from a Riemann surface † into the complex projective m-space CPm . Let c 2 H 2 .CPm / be the first Chern class of the Hopf bundle H , the dual of the canonical complex line bundle. Given a map f W † ! CPm , its degree is deg.f / D hf .c/; Œ†i: Let Holk .†; CPm / be the space of holomorphic maps of degree k, and W Holk .†; CPm / ! Mapk .†; CPm /

(4.4)

the forgetful map. Segal proves that induces isomorphisms in homology in dimensions < .k 2g/.2m 1/. The moduli space Mg .CPm I k/ consists of pairs .†; '/ of a genus g Riemann surface † and a holomorphic map ' W † ! CPm of degree k. It maps into Mgtop .CPm I k/ D JC .Fg / Diff C .Fg / Mapk .Fg I CPm /; where the holomorphic map is relaxed to be just a continuous map. The subspace g .CPm I k/ D Emb.Fg ; R1 / Diff C .Fg / Mapk .Fg ; CPm / of g .CPm / maps into Mg .CPm I k/ by a map which is a rational homology equivalence. top

744

Ib Madsen

Conjecture 4.5 (from [2]). The map Mg .CPm I k/ ! Mgtop .CPm I k/ induces isomorphism on rational homology in degrees less than .k 2g/.2m 1/. The conjecture is stated as a theorem in [2], but the author has informed me that there is a gap in his proof. If true, the conjecture has the following consequence. Corollary 4.6. For < minf.k 2g/.2m 1/; 2=3gg, m H .Mg .CPm I k/I Q/ Š H .1 .MTSO.2/ ^ CPC /I Q/:

Once again the right-hand side above is the free graded commutative algebra with unit generated by the positive degree elements of the graded vector space H .BSO.2/I Q/ ˝ H .CPm I Q/ ˝ hu2 i:

References [1]

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. 742

[2]

D. Ayala, Homological stability among moduli spaces of holomorphic curves in complex projective space. Preprint 2008; arXiv:0811.2274v1 [math.AT]. 741, 744

[3]

M. Bökstedt and I. Madsen, The cobordism category and Waldhausen’s K-theory. Preprint, 2011; arXiv:1102.4155v1 [math.AT]. 730, 731

[4]

S. K. Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients. Preprint, 2009; arXiv:0904.3269v1 [math.AT]. 736, 737

[5]

S. S. Chern, An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc. 6 (1955), 771–782. 734

[6]

F. R. Cohen, T. J. Lada, and J. P. May, The homology of iterated loop spaces. Lecture Notes in Math. 533, Springer-Verlag, Berlin 1976. 722

[7]

R. L. Cohen, S. Galatius, and N. Kitchloo, Universal moduli spaces of surfaces with flat bundles and cobordism theory. Adv. Math. 221 (2009), no. 4, 1227–1246. 741, 742, 743

[8]

R. L. Cohen and I. Madsen, Surfaces in a background space and the homology of mapping class groups. In Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math. 80, Amer. Math. Soc., Providence, R.I., 2009, 43–76. 741, 742

[9]

R. L. Cohen and I. Madsen, Stability for closed surfaces in a background space. Homology Homotopy Appl. 13 (2011), no. 2, 301–313. 741, 742

[10] C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory. J. Differential Geom. 3 (1969), 19–43. 734 [11] C. J. Earle and A. Schatz, Teichmüller theory for surfaces with boundary. J. Differential Geom. 4 (1970), 169–185. 739


745

[12] J. Ebert, A vanishing theorem for characteristic classes of odd-dimensional manifold bundles. J. Reine Angew. Math., to appear; preprint, 2009; arXiv:0902.4719v2 [math.AT]. 740 [13] C. Faber, A conjectural description of the tautological ring of the moduli space of curves. In Moduli of curves and abelian varieties, Aspects Math. E33, Vieweg, Braunschweig 1999, 109–129. 739 [14] S. Galatius and O. Randal-Williams, Monoids of moduli spaces of manifolds. Geom. Topol. 14 (2010), no. 3, 1243––1302. 742 [15] S. Galatius, U. Tillmann, I. Madsen, and M. Weiss, The homotopy type of the cobordism category. Acta Math. 202 (2009), no. 2, 195–239, . 718, 724, 730, 732, 733, 736, 737, 738 [16] S. Galatius, Stable homology of automorphism groups of free groups. Ann. of Math., to appear. 730, 731, 732 [17] M. Gromov, Partial differential relations. Ergeb. Math. Grenzgeb. (3) 9, Springer-Verlag, Berlin 1986. 732 [18] J. L. Harer, Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2) 121 (1985), no. 2, 215–249. 736, 737 [19] N. V. Ivano, On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients. In Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), Contemp. Math. 150, Amer. Math. Soc., Providence, R.I., 1993, 149–194. 736, 737 [20] N. V. Ivanov, Mapping class groups. In Handbook of geometric topology, North-Holland, Amsterdam 2002, 523–633. 737 [21] A. Kriegl and P. W. Michor, The convenient setting of global analysis. Math. Surveys Monogr. 53, Amer. Math. Soc., Providence, R.I., 1997. 719, 721 1 [22] I. Madsen and U. Tillmann, The stable mapping class group and Q.CPC /. Invent. Math. 145 (2001), no. 3, 509–544. 725

[23] I. Madsen and M. Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. of Math. (2) 165 (2007), no. 3, 843–941. 730, 738 [24] I. Madsen. Moduli spaces from a topological viewpoint. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Vol. I, EMS Publishing House, Zürich 2006, 385–411. 718 [25] D. McDuff and G. Segal, Homology fibrations and the “group-completion” theorem. Invent. Math. 31 (1975/76), no. 3, 279–284. 737 [26] W. Meyer, Die Signatur von lokalen Koeffizientensystemen und Faserbündeln. Bonn. Math. Schr. 53, 1972. 741 [27] J. W. Milnor and J. C. Moore, On the structure of Hopf algebras. Ann. of Math. (2) 81 (1965), 211–264. 723 [28] J. Milnor, On spaces having the homotopy type of a CW-complex. Trans. Amer. Math. Soc. 90 (1959), 272–280. 720 [29] S. Morita, Generators for the tautological algebra of the moduli space of curves. Topology 42 (2003), no. 4, 787–819. 739 [30] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds. Ann. of Math. (2) 65 (1957), 391–404. 734

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[31] O. Randal-Williams, Embedded cobordism. PhD thesis, University of Oxford, 2009. 730 [32] O. Randal-Williams, Resolutions of moduli spaces and homological stability. Preprint, 2009; arXiv:0909.4278v3 [math.AT]. 742 [33] J. G. Ratcliffe, Foundations of hyperbolic manifolds. Grad. Texts in Math. 149, SpringerVerlag, New York 1994. 735 [34] G. Segal, Categories and cohomology theories. Topology 13 (1974), 293–312. 730, 733 [35] G. Segal, The topology of spaces of rational functions. Acta Math. 143 (1979), no. 1–2, 39–72. 743 [36] N. Steenrod, The topology of fibre bundles. Princeton Math. Ser. 14, Princeton University Press, Princeton, N.J., 1951. 719 [37] N. Wahl, Homological stability for the mapping class groups of non-orientable surfaces. Invent. Math. 171 (2008), no. 2, 389–424. 740

Chapter 13

On the Lp -cohomology and the geometry of metrics on moduli spaces of curves Lizhen Ji and Steven Zucker

Contents 1 Introduction and the setting . . . . . . . . . . . . . . . . . 2 Lp -cohomology . . . . . . . . . . . . . . . . . . . . . . . 3 Hodge decomposition . . . . . . . . . . . . . . . . . . . . 4 Intersection (co)homology . . . . . . . . . . . . . . . . . . 5 Metrics on Mg;n and the proofs of Theorems 1.1 and 1.2 . . 6 Some geometric questions concerning the complete metrics References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

747 750 758 760 763 769 771

1 Introduction and the setting Let Mg;n be the moduli space of algebraic curves over C of genus g with n punctures. There has been much work devoted to parallels between Mg;n and locally symmetric spaces of finite volume, in particular, Hermitian locally symmetric spaces nD, where D is a bounded symmetric domain in some C d , and Aut.D/ is a lattice subgroup of the group of holomorphic automorphisms of D. This is our point of departure. Let Tg;n be the Teichmüller space that parametrizes the complex structures on a compact orientable surface Sg;n of genus g with n punctures, together with a marking on Sg;n (i.e., the choice of a set of generators of 1 .Sg;n /). Then Tg;n is a complex manifold biholomorphic to a bounded domain in C 3g3Cn (when 3g 3 C n > 0). The mapping class group Modg;n D Diff C .Sg;n /=Diff 0 .Sg;n / acts properly and biholomorphically on Tg;n , where Diff C .Sg;n / denotes the group of all orientationpreserving diffeomorphisms of Sg;n and Diff 0 .Sg;n / is the identity component of Diff C .Sg;n /. The quotient Modg;n nTg;n is the moduli space Mg;n . We note that the Modg;n -quotient nullifies the role of the marking on Sg;n , so Mg;n parametrizes the complex structures alone.1 By [19], Mg;n has a natural structure of a quasi-projective algebraic variety. The moduli space Mg;n D Modg;n nTg;n is noncompact. It admits the Deligne–Mumford 1 For

a good “early” survey of the above, see [10].

748

Lizhen Ji and Steven Zucker

x DM , which parametrizes stable curves [19]. This is a complex compactification M g;n projective algebraic variety, in fact a compact complex orbifold. The mapping class group admits normal subgroups of finite index that act freely on Tg;n , for which Mg;n WD nTg;n is a manifold. For such , Mg;n is the quotient of Mg;n by the finite DM x is the quotient by G of the smooth compactification group G D nModg;n , and M g;n of nTg;n . For convenience, sometimes we will formulate and prove our assertions in this chapter for such Mg;n . One can regard Tg;n as a sort of analogue of a bounded symmetric domain D, and Modg;n as an analogue of a lattice subgroup of Aut.D/, starting with the following. The space D has a canonical complete Kähler metric that is invariant under the action of Aut.D/, and it descends to nD. When is arithmetically-defined and nD is noncompact,2 the latter admits the rather canonical Baily–Borel Satake compactification nD BB [6] (a normal projective variety), showing that nD is a quasi-projective DM x g;n and the variety. The analogy extends to the Deligne–Mumford compactification M BB Baily–Borel compactification nD (though the former is a much nicer space). By construction, the boundary of nD BB is the union of lower-dimensional Hermitian locally symmetric spaces. For example, if D is equal to the Siegel upper half space n of degree n and nD is a Siegel modular variety, then the boundary @.nD BB / is the union of Siegel modular varieties of lower degrees. Similarly, the boundary of x DM is the union of moduli spaces of stable curves of smaller genus and with more M g;n punctures. Thus, the boundaries in both cases are hereditary in nature. DM x g;n Another analogy between these compactifications nD BB and M is the way BB they are constructed. The Baily–Borel compactification nD is the -quotient of a y of D that is obtained by adding rational boundary compopartial compactification D nents from the closure of D as a bounded symmetric domain, and giving it a suitable topology. The Teichmüller space Tg;n also admits a partial compactification called the augmented Teichmüller space Tyg;n in [1] that is also the completion with respect to the Weil–Petersson metric [84], and the quotient of Tyg;n by Modg;n is homeomorphic DM x g;n to the Deligne–Mumford compactification M . For any finite index subgroup of y Modg;n , the quotient nTg;n is also compact. For some special subgroups , such as those given in [59], the compactification nTyg;n is homeomorphic to a smooth DM x g;n . projective variety that admits a finite mapping onto M There are numerous natural complete Modg;n -invariant Riemannian and Finsler metrics on Tg;n . These include the Teichmüller (Finsler) metric, the Bergman metric, the Kobayashi metric (which was shown by Royden [70], [71] to be equal to the Teichmüller metric), the Carathéodory metric, the Kähler–Einstein metric, the McMullen metric, the Ricci metric, and the Liu–Sun–Yau metric (a perturbed Ricci metric). It was shown in [57] that all of these are, in fact, quasi-isometric.3 Looijenga [58] and Saper–Stern [76] proved that the L2 -cohomology of nD is canonically isomorphic to the (topological) intersection cohomology of nD BB for 2 We

shall assume this throughout the rest of this chapter. article [89] independently showed the quasi-isometry of some of these metrics.

3 The

Chapter 13. Lp -cohomology of moduli spaces

749

what is called the middle perversity.4 This result was conjectured by the second author [92] and is still often called the Zucker conjecture in the literature even though it is now a theorem. It is then a plausible guess that the L2 -cohomology of Mg;n , with respect to any of the above complete metrics, is canonically isomorphic to the intersection x DM is an orbifold, so has only x DM with coefficients in R. Since M cohomology of M g;n g;n x DM with Qquotient singularities, the latter is equal to the usual cohomology of M g;n coefficients. This is indeed true. With some motivation from a result of [93] saying that the Lp -cohomology of nD is equal to the ordinary cohomology of its reductive Borel–Serre compactification for sufficiently large and finite p, we prove, without so much effort, the following more general result. Theorem 1.1. For 1 < p < 1, the Lp -cohomology of Mg;n , with respect to the x DM . In above canonical complete metrics,5 is isomorphic to the cohomology of M g;n p 6 particular, the L -cohomology of Mg;n does not depend on the value of p. Since the reductive Borel–Serre compactification of nD is generally different from the Baily–Borel compactification when the rank of any irreducible factor of the bounded symmetric domain D is greater than one, or more generally when the covering symmetric space of some irreducible factor of nD has rank greater than 1,7 the Lp -cohomology of nD usually depends on the value of p. Thus, the above result says that the moduli space Mg;n behaves in this regard like rank-one locally symmetric spaces or their products. (See the paper [28] for results on rank-1 phenomena of the mapping class groups.) The Teichmüller space Tg;n also admits a canonical incomplete Kähler metric, the Weil–Petersson metric. In [75], it was proved that the L2 -cohomology group of Mg with respect to the Weil–Petersson metric is canonically isomorphic to the cohomology x DM . Though the theorem in [75] is stated only for the case of Mg;0 , the same of M g method works for general Mg;n ; the L2 -cohomology of Mg;n with respect to the x DM . In contrast with Weil–Petersson metric is isomorphic to the cohomology of M g;n Theorem 1.1, we have: Theorem 1.2. For 43 p < 1, the Lp -cohomology of Mg;n with respect to the Weil– x DM , whereas for 1 p < 4 , Petersson metric is isomorphic to the cohomology of M g;n 3 p the L -cohomology of Mg;n with respect to the Weil–Petersson metric is isomorphic the cohomology of Mg;n itself. 4 It seems to be accepted that the result for general can be deduced from the case where is neat. However, there is nothing in the literature that gives a proof of it. 5 Since it is complicated to define norms of differential forms and the volume form for a Finsler metric and since the Lp -cohomology groups of Riemannian manifolds only depend on the quasi-isometry class of the Riemannian metrics, when we say the Lp -cohomology of a Finsler metric, we meant the Lp -cohomology of a Riemannian metric that is quasi-isometric to it. 6 The case of p D 2 appeared already in [88]. 7 The Baily–Borel compactification of nD is always a quotient space of the reductive Borel–Serre compactification [96].

750


x DM , as described in Both Theorem 1.1 and Theorem 1.2 are obtained locally on M g;n Section 5. We wish to remind the reader that the mapping class groups, the topology of Mg;n , and its Weil–Petersson metric have been studied extensively by many people, too numerous to list here. On the other hand, the complete Riemannian metrics on Mg;n mentioned above have only recently been studied systematically in [57], where the aforementioned quasi-isometries were proved. Some related results had also been proved in [89]. The partial similarity also comes from the spectrum of the Laplace operators of the metrics. It is known that nD can have nonempty continuous spectrum.8 On the other hand, while it is proved in [47] that the Laplace operator of Mg;n with the Weil– Petersson metric has a discrete spectrum, the spectrum of the Laplace operator for any of the above complete Riemannian metrics is expected to contain rays of the form Œa; C1/, and their generalized eigenfunctions should be analogous to the Eisenstein series for locally symmetric spaces. Acknowledgments. We would like to thank Mark Goresky for many helpful correspondences and Athanase Papadopoulos for reading this chapter carefully. The first author was partially supported by NSF grant DMS 0905283. The second author was partially supported by NSF grant DMS 0600803.

2 Lp -cohomology L2 -cohomology is defined from square-integrable differential forms on a Riemannian manifold. Because L2 defines a Hilbert space, which is self-dual, it attracted special attention. Lp -cohomology is defined analogously for p ¤ 2. For compact manifolds, it only recovers the de Rham theorem. We intend to give here an account of the main thrust of the research in this direction of mathematics, we hope without any glaring omissions. We start with the case p D 2, for it was the one studied first. In essence, L2 -cohomology is the cohomological embodiment of two fundamental notions: L2 harmonic forms on a non-compact Riemannian manifold M , and cohomology classes in H .M; R/ that are represented by L2 differential forms (via the de Rham theorem). While both of these are pertinent considerations, the idea of defining .M; R/, as the hypercohomology of a complex of the L2 -cohomology, denoted H.2/ sheaves on suitable topological compactifications of M represents a “modern” point of view, which we take in this chapter. The two pioneering works, coming at the end of the 1970’s, were by Cheeger [15] and Zucker [91].9 8 The spectral decomposition of locally symmetric spaces has played a fundamental role in the Langlands program (see [55], [34]).


751

In terms of the latter, it is often the case that H.2/ .M; R/ is represented by a space 2 of L harmonic forms (e.g., for any metric when M is compact), and the reason why this can fail is easy enough to grasp. Also, the space of cohomology classes in H .M; R/ that are represented by L2 differential forms is just the image of the mapping H.2/ .M; R/ ! H .M; R/ that is induced by the inclusion of the L2 de Rham complex in the full de Rham complex. (As such, one might say L2 de Rham cohomology.) When M is (say) a complete complex Kähler manifold, the space of L2 C-valued harmonic forms has a decomposition according to complex bidegree (see (3.5) below). This gives rise to the very important Hodge decomposition of H .M; C/ in algebraic geometry when M is a smooth projective variety over C. We will treat these matters in greater detail below. Cohomology is defined for cochain complexes C (first, of vector spaces over R). More specifically, when the complex has terms only in non-negative degrees, we mean d0

d1

d2

0 ! C 0 ! C 1 ! C 2 ! ;

(2.1)

where the differentials d i satisfy d i B d i1 D 0, so im d i1 ker d i . The i -th cohomology of C is defined as the real vector space H i .C / ' .ker d i /=im d i1 : A morphism of complexes ‰ W B ! C , i.e., a set of linear mappings i W B i ! C i that are compatible with the differentials (dCi B i D iC1 B dBi ), induces mappings H i .B / ! H i .C /: ‰ is called a quasi-isomorphism (of complexes) when (2.2) is an isomorphism for all i . The R-valued C 1 forms on a smooth manifold M comprise a cochain complex A .M /, with Ai .M / being the i -forms and each d i given by the exterior derivative d . The the de Rham theorem identifies H i .A .M // ' H i .M; R/I

(2.2)

here the cohomology on the right-hand side is what is defined in algebraic topology, ˇ using, e.g., singular or Cech cochains with real coefficients. Moreover, cup product on cohomology is induced by the exterior product of differential forms. The de Rham theorem is best understood by introducing the sheaves of germs of be the complex (cf. (2.1)) of sheaves of vector differential forms on M [82]. Let AM spaces over R on M given by the data: for U open in M , U 7! A .U /. Sheaves allow i for a standard way to deduce global results from local ones. Because each sheaf AM admits partitions of unity, induced by arbitrarily fine partitions of unity on functions 9 An anecdote: There was an important conference “Analyse et topologie sur les espaces singuliers” at Luminy in July, 1981 (proceedings [80]). The second author recalls that at one point, Cheeger, being a differential geometer, said to MacPherson in his presence something like, “I was expecting/hoping there would be more L2 -cohomology.” MacPherson’s reply, “There are two experts in the world on L2 -cohomology, and both of them are here.”

752


0 (viz. AM ), so these are called fine sheaves. One deduces the de Rham theorem as the global consequence of the local calculation (Poincaré lemma) ´ R if i D 0; i (2.3) H .A .U // D 0 otherwise,

whenever U is a contractible open subset of M . The rather trivial complex of sheaves RŒ0, consisting of R in degree zero, and 0 in all other degrees, maps to A .M / with image the locally constant functions on M . The de Rham theorem follows from is a quasi-isomorphism in the category of the fact that the morphism RŒ0 ! AM complexes of sheaves on M , which is just (2.3). Next, we assume that M has a specified Riemannian metric g. This imparts a notion of length to tangent vectors to M , and thereby a length to all tensors on M . Thus, if ' 2 A .M /, one gets a non-negative function j'j, the (pointwise) length of ', on M . We also get from g a volume form d VM on M , and corresponding Lp seminorms: Z 1=p p j'j d VM (2.4) k'k.p/ D M

for any p with 1 p < 1. The definition of the L1 seminorm k'k.1/ goes in the usual way. We say that ' 2 A .M / is Lp when k'k.p/ < 1. The simplest way to proceed is: Definition 2.1. (i) The Lp de Rham complex of M is the largest subcomplex of A .M / contained in the Lp forms. Its elements are f' 2 A .M / W ' and d' are Lp g, and it is denoted by A.p/ .M /. (ii) The Lp -cohomology of M is the cohomology of the Lp de Rham complex of M , i.e., i .M / D H i .A.p/ .M //: H.p/ It is clear that the Lp -cohomology of M depends on g only to the extent that the Lp de Rham complex does; quasi-isometric metrics on M yield the same Lp de Rham complex, hence the same Lp -cohomology. (Metrics g1 and g2 are said to be quasi-isometric when there are positive real numbers c1 c2 such that g1 c1 g2 and g1 c2 g2 .) We return to the case p D 2. The starting point of the role of L2 -cohomology is the Hodge theorem: Theorem 2.2 ([43]). Let M be a compact Riemannian manifold. Then in every (de Rham) cohomology class in H .M; R/, there is one and only one harmonic differential form. One notes that there is no mention of L2 in the above statement since smooth differential forms on compact manifolds are automatically square integrable. The proof in [43] relies on the theory of integral equations. The reworking of the theorem


753

in terms of Hilbert spaces is due to Gaffney [32], [33], based on an earlier treatment by Kodaira [53]. Before we explain this, we point out direct consequences of the Hodge theorem. The basic reason is that cohomology groups can be studied using the unique canonical representatives instead of cohomology classes. (1) It gives an analytic proof that the de Rham cohomology groups of a compact Riemannian manifold are finite-dimensional. The reason is that the spectrum of the Laplacian (i.e., Hodge–Laplacian, or de Rham Laplace operator) on differential forms of a compact Riemannian manifold has a discrete spectrum (or more to the point that 0 is an isolated point in the spectrum with finite multiplicity) [81], p. 226. (2) The de Rham cohomology groups of a compact oriented Riemannian manifold satisfy Poincaré duality, since for each harmonic differential form, it is easy to find another harmonic form which pairs nontrivially with it (see (2.20) below or [81], p. 226). (3) On a compact orientable Riemannian manifold with nonnegative Ricci curvature, every harmonic 1-form is parallel. Furthermore, if the Ricci curvature is positive definite at some point, then every harmonic 1-form is zero and the first cohomology group is zero. In the proof, the Weitzenböck formula was applied to harmonic 1-forms, and the positivity of the Ricci curvature forces vanishing of the harmonic form. This result of Bochner has been developed into a powerful method called Bochner technique. One important result proved by this technique is the Kodaira vanishing theorem. See the book [87] and the survey [9] for many variants of the Bochner vanishing theorem on harmonic 1-forms. (4) The Hodge theorem on the existence and uniqueness of harmonic representatives of cohomology classes is a crucial step in the Hodge theory (or decomposition) which will be explained in the next section. See also the discussion in [16],§1.2, about three steps to obtain the Kähler package of a Kähler manifold. (5) When M is a compact locally symmetric space nD, the harmonic representatives of cohomology classes are automorphic forms, and the Hodge theorem establishes a connection between cohomology groups of arithmetic groups and automorphic forms. See the book [12] and references there. Let M be an oriented Riemannian manifold of dimension m, and let W Ai .M / W Ami .M / be the 0-th order operator for which we have at each point of M that h'; id VM D ' ^

whenever ';

2 Ai .M /:

The formal adjoint of d is the first-order operator ı W AiC1 .M / ! Ai .M / for which the formula hd'; i D h'; ı i

whenever ' 2 Ai .M / and 1

2 AiC1 .M /

(2.5)

holds locally on M . It is easy to check that ı D d D ˙ d ; the sign in the right-hand side is 1 for all i when m is even. If one of '; has compact support,

754


Stokes’ theorem implies that (2.5) holds for the global L2 inner product on A .M /; to distinguish the latter from the pointwise inner product, we introduce the notation h'; i.2/ . Unfortunately, the forms in Ai.p/ .M / (p ¤ 1 is temporarily arbitrary again) are merely dense in the Banach space of Lp i-forms, so one must pass to the completion to draw conclusions by functional analysis. Once that is done, d is only a denselydefined unbounded operator. One must then consider the various closures of d . At the extremes, we have the maximal, or weak, closure dNmax , whose domain is the set of Lp i -forms ' whose distributional exterior derivative d' is given by an Lp .i C 1/form, and the minimal closure dNmin determined by starting instead with C 1 forms of compact support. The extreme closures of ı are defined analogously. It is important to recognize the strict, as opposed to just formal, adjoint of a given closure of, say d , and we present it for dNmax . It is clear that (2.5) holds globally for an L2 differential form if and only if ' 7! hd'; i.2/ is a bounded functional of ' in the domain of dNmax , or equivalently, on smooth L2 forms '. One then writes .dNmax / . / for the L2 form representing that functional. It is clear that .dNmax / is a closure of ı, and indeed .dNmax / D ıNmin : In particular, if one knows that dNmax D dNmin , all closures of d coincide, likewise for ı, and also vice versa. Remark 2.3. One can find the following assertions in [94], §1, for instance. (i) One can replace A.p/ .M / by the larger complex L.p/ .M / given by the domain of dNmax . The inclusion A .M / ,! L .M / is a quasi-isomorphism. In other words, .p/

.p/

the two complexes define the same notion of Lp -cohomology.10 We thereby get the reformulation of (2.1) as i i1 i .M / D ker dNmax =im dNmax : H.p/

(2.6)

(ii) When M is the interior of a compact Riemannian manifold-with-corners (i.e., where the metric does not degenerate at the boundary), dNmax coincides with the closure of d on the complex of C 1 differential forms that are smooth at the boundary. The resulting complex is a quasi-isomorphic subcomplex of A .M /. Likewise, dNmax is the closure of ı on the subcomplex of such forms that vanish on the boundary. It is now necessary to be careful about what we mean by an L2 harmonic differential form. Harmonic forms are always smooth. At the formal level, they are given by the solutions of the equation ' D 0, where is the second-order Laplacian operator d ı C ıd that takes Ai .M / to itself. Already when i D 0, the solutions of this equation 10 Thus,

both dNmax and dNmin can be described in terms of closures of graphs.


755

can be infinite-dimensional. This operator has various closed extensions. One can x min as before, but the most useful closure is x str , the one given by x max and define N x str has as the strict term-by-term closure. Writing d from now on instead of dNmax , 2 N N N domain the space of L forms ' in the domain of d and d , such that d ' is in the domain of dN and dN ' is in the domain of dN . Under these conditions, one can expand x str '; 'i.2/ , making use of the global version of (2.5) (compare (2.8) below), h x str '; 'i.2/ D hdN '; dN 'i.2/ C hdN '; dN 'i.2/ : h x str ' D 0 if and only if dN ' D 0 and dN ' D 0. A differential From this we see that i-form ' will be said to be strictly harmonic when these equalities hold, and we write ' 2 H i .M /. In general ker dN i is a closed subspace of the L2 i -forms, but its subspace im dN i1 need not be. One has then that i .M / ker dN i D im dN i1 ˚ ker dN i \ .im ker dN i1 /? D ker dN i \ ker.dN i / D H.2/ From (2.6) with p D 2, we obtain

i i .M / D H.2/ .M / ˚ im dN i1 =im dN i1 H.2/

(2.7)

(algebraic direct sum). Some elementary functional analysis yields that the second summand is either 0 or of infinite dimension over R. The latter case occurs when M D R. Remark 2.4. (i) For a compact manifold M , (2.7) is basically what Gaffney proved in [33], where dN i1 is shown to have closed range. In [32], he proved that on a complete Riemannian manifold, dNmax D dNmin , so (2.5) holds in the L2 -complex, viz., hd'; i.2/ D h'; ı i.2/ ;

(2.8)

x max . x str D and then i i i (ii) If dim H.2/ .M / < 1, e.g., when M is compact, then H.2/ .M / ' H.2/ .M /. By (i), the right-hand side is just the space of all harmonic i -forms. This proves Theorem 2.2. i i (iii) It is possible that H.2/ .M / be infinite-dimensional because H.2/ .M / is infinitedimensional. (iv) For any p, the reduced Lp -cohomology can be defined as the following alteration of (2.6): ı i N i1 : x i .M / D ker dNmax H im d max .p/ x i .M / is always isomorphic to H i .M /, but the cohomological flavor With p D 2, H .2/ .2/ is gone. On M , there is a natural pairing m Ai.2/ .M; R/ Ami .2/ .M; R/ ! A.1/ .M; R/:

756


When M is compact, one can express the Poincaré duality (a perfect pairing) in terms of closed differential forms, Z '^ : P W H i .M; R/ H mi .M; R/ ! H m .M; R/ ' R; P .'; / D M

In particular, this can be given at the level of harmonic forms; indeed, as commutes with , one has an isomorphism ! H mi .M; R/; W H i .M; R/

P .'; / D h'; i.2/ :

(2.9)

2

The above carries over to complete manifolds M . The pairing on L harmonic forms is still a perfect pairing, so (2.9) induces a “Poincaré duality” i mi P W H.2/ .M; R/ H.2/ .M; R/ ! R:

(2.10)

With a little care, one defines and makes use of complexes of Lp sheaves (as promised). One notes first that the assignment, for U open in M , U 7! Ai.p/ .U / gives itself. To remember the rise only to the sheaf of i -forms that are locally Lp , i.e., AM p x of M as a Hausdorff global L condition, one must first specify a compactification M p x topological space. The L sheaf of i-forms on M is the assignment, for U open in x , U 7! Ai .U \ M /. For all M x , the resulting complex of sheaves A .M x / has M .p/ .p/ A.p/ .M / as its complex of global sections. For this fact to be useful for understanding x , so that A .M x/ the Lp -cohomology of M , one needs that the metric is fine on M .p/ x is a complex of fine sheaves (see [92]), §4; there must exist a partition of unity on M x have bounded differential with respect whose cut-off functions at the boundary of M to g, for multiplication only by such functions preserves the two Lp conditions in the definition of Ai.p/ .U \ M /. Whether such functions exist is something that depends x. on the choice of M Remark 2.5. The analogue of Remark 2.3 (i) holds at the level of sheaves. There is a x / for any compactification M x of M , containing weak Lp complex of sheaves L.p/ .M x / as a quasi-isomorphic subcomplex, for which L .M / is its complex of A.p/ .M .p/ global sections. There are useful variants of Lp -cohomology. Let M be a Riemannian manifold and w a positive, say smooth, function on M . The Lp seminorm with weight w is Z 1=p p j'j w d VM : (2.11) k'k.p/;w D M

When w D 1, this is just (2.4). Replacing (2.4) by (2.11) in Definition 2.1, one defines the weighted Lp -complex A.p/ .M I w/ and then the weighted Lp -cohomology of M : i .M I w/ D H i .A.p/ .M I w//: H.p/

x of M , denoted by The notion of weighted Lp sheaves on a compactification M x I w/, is correspondingly defined, and likewise the analogue of Remark 2.5. A.p/ .M


757

A second variant lies in differential forms with coefficients in a flat (complex) vector z denote the universal covering space of M . Then M D nM z , where bundle. Let M is the fundamental group of M . Assume that acts freely on a finite-dimensional z V vector space V =C, i.e., V is a representation space for . The projection of M z onto M induces, via the quotient by , with taken to be acting diagonally on the former, a vector bundle V over M . Any fixed v 2 V is taken to define a locally constant section of V; let V denote the sheaf of locally constant sections of V. The complex of sheaves A .M; R/ ˝ V , with differential given locally as d ˝ 1, whose complex of global sections is denoted A .M; V /, is quasi-isomorphic to V . If one puts a metric on V as well as M , one can define A.p/ .M; V / and Lp -cohomology x ; C/ ˝ V .11 groups H.p/ .M; V / parallel to Definition 2.1, and sheaves A.p/ .M The question that we are pursuing in this chapter is: Problem 2.6. If M is a given non-compact Riemannian manifold, is there a compactix of M for which the metric is fine on M x , and H .M / (is finite-dimensional fication M .p/ x? and) admits a topological interpretation on M When p D 2, the topological interpretation will have to satisfy Poincaré duality, because of (2.10). In the cases where Problem 2.6 has been solved, information x , are used. Indeed, drawn from the explicit asymptotics of the metric, as read on M DM x Theorems 1.1 and 1.2 are proved by examining A.p/ .Mg;n / for the respective metrics. Remark 2.7. One can generalize the above definition of Lp -cohomology to orbifolds. Differential forms, partitions of unity, Lp -sheaves, etc. can be defined on orbifolds. (See [77] and [23]). In this chapter, we concentrate on the case of smooth manifolds, with the sense that one can deduce the general case from that. Remark 2.8. We list here some interesting topics of a similar-sounding nature that do not pass through Problem 2.6, and we have elected to exclude. (i) The investigation of reduced L2 -cohomology, which is by Remark 2.4 (iv) focused on the L2 harmonic forms: [21], [22], [39]. (ii) The L2 -cohomology of a covering group (see [3], [20]). .M / ! H .M /, i.e., the (de Rham) cohomology classes (iii) The image of H.p/ p represented by an L differential form (with p D 2, [2], [11], [44]).

(iv) Bounded cohomology in the sense of Gromov [40]. (v) The cohomology of infinite chains with `p -coefficients (see [20]).

11Actually, weighted Lp -cohomology can be viewed as a special case of the preceding where the bundle V is trivial of rank one, metrized by w.

758


3 Hodge decomposition We retain the notation from the previous section. Suppose that our Riemannian manifold M is a complex manifold of (complex) dimension n (so m D 2n). We assume that the metric g is Hermitian, i.e., that the almost-complex structure tensor J of M is an isometry of g. The C-valued differential forms decompose according to bidegree (or type): M Ap;q .M / with p; q 0 and Ap;q .M / D Aq;p .M /; (3.1) Aj .M; C/ D pCqDj

with the bar denoting complex conjugation. An analogous decomposition holds for the sheaves Aj .M; C/. At issue is whether the decomposition (3.1) passes to H j .M; C/. One extends the Laplacian C-linearly to Aj .M; C/. For a j -form decomposed according to (3.1), X 'D ' p;q ; (3.2) P ' D ' p;q , of course. However, it is not true in general that ' p;q is a form of type .p; q/. When the Hermitian metric is Kählerian – one then says that M is a compact Kähler manifold – meaning that the alternating bilinear form !, given pointwise by !.X; Y / D g.JX; Y /; is a closed differential 2-form (of type .1; 1/),12 the following is true: Theorem 3.1 (Hodge decomposition). Let M be a compact Kähler manifold. Then: (i) .Ap;q .M // Ap;q .M /. Thus ' is harmonic if and only if each ' p;q (from (3.1)) is harmonic. L (ii) H j .M; C/ D pCqDj H p;q .M /, where H p;q .M / denotes the space of cohomology classes represented under (3.2) by harmonic forms of type .p; q/. Moreover, H p;q .M / D H q;p .M / (iii) The decomposition in (ii) is independent of the Kähler metric. Remark 3.2. We give some indication of the proof of Theorem 3.1. (i) The first assertion of Theorem 3.1 is usually proved by showing that when M is a Kähler manifold, can be expressed as an operator that, for obvious reasons, N with @ the type .1; 0/ preserves each Ap;q .M /. Using the decomposition d D @ C @, p;q pC1;q N component of d that takes A .M / to A .M /, and @ its complex conjugate, one has the analogues of (expressed at the formal level): @ D @@ C @ @;

N @N D @N @N C @N @:

Both of the above operators preserve Ap;q .M /. One says that a form ' of type .p; q/ is N @-harmonic (resp. @-harmonic) when @ ' D 0 (resp. @N D 0). Then one computes 12 It

is easy to see that ! n =nŠ is the volume form of M , and that ! D ! n1 =nŠ .


759

that D @ C @N C [cross-terms]: When the metric is Kählerian, the cross-terms vanish and @ D @N , so D 2 @N . Furthermore, (3.1) induces on the space of C-valued harmonic j -forms H j .M; C/, M H p;q .M /: (3.3) H j .M; C/ D pCqDj

Inserting this into Theorem 2.2, yields (ii). The abstract version of such a decomposition is called a (real) Hodge structure of weight j . Thus, one has constructed a real Hodge structure of weight j on H j .M; R/. (ii) The study of @N is fundamental in complex function theory in more than one variable, and it plays a role as asserted in item (i) above. Indeed, there are analogues of (2.2) and (2.3) for @N and OM .V /, the sheaf of holomorphic sections of a holomorphic p vector bundle V on M (the Dolbeault lemma), here with OM .V / D M , the sheaf of holomorphic p-forms on M . By the Dolbeault lemma (see [38]), the cohomology of N the @-complex, with differentials @N W Ap;q .M / ! Ap;qC1 .M /, can be identified with sheaf cohomology: p H q .M; M / ' H p;q .M / ' H p;q .M /:

(iii) Assertion (iii) of Theorem 3.1 may come as a surprise. It is proved by showing that H p;q .M / can be described by a sufficiently algebraic statement that is independent of the Kähler metric. Specifically, if one writes F p H j .M / for the sum of the cohomology classes of closed forms of types .r; s/, with r p, then H p;q .M / D F p H j .M / \ F q H j .M /I This filtration F is the restriction of one on A .M; C/, and that is induced from N ; the latter complex and its @the corresponding filtration (also denoted F ) of M (Dolbeault) resolution are quasi-isomorphic to CM . There are topological conditions imposed by the existence of a Kähler metric on M . For one, H 2 .M; R/ ¤ 0. Also, we see directly from Theorem 3.1(ii) that when j is odd, dim H j .M / must be even. One can go further. Let L denote the 0-th order operator (of type .1; 1/) given by wedge product with Œ!, viz., L W Aj .M; R/ ! Aj C2 .M; R/. Then L actually takes H j .M / to H j C2 .M / (as and L commute) as well as the canonical cup product with Œ! from H j .M / to H j .M /. One has Lnj W H j .M; R/ ! H 2nj .M; R/

(for j < n);

(3.4)

which is an isomorphism.13 There is additional structure on the cohomology of M , but we will not need that for our purposes. (See [83] for that, as well as for (3.4).) Examples. We present a good supply of Kähler manifolds. There are three standard ones: C (with the Euclidean metric), CPn (with the PGLnC1 -invariant Fubini–Study 13 There

is some relation between (3.4) and that makes use of the primitive decomposition.

760


metric), and a Hermitian symmetric space D of non-compact type (with a metric invariant under the isometry group of D). Two basic constructions provide many more. Any complex submanifold N of a Kähler manifold M is Kählerian (with the metric of M restricted to N ). In particular, every smooth algebraic subvariety of CPn is Kähler; in [54] Kodaira characterized smooth projective varieties as Kähler manifolds for which some Kähler metric satisfies Œ! 2 H 2 .M; Q/. The other construction is taking a quotient of M by a discrete group acting on M without fixed points. When is an arithmetically-defined subgroup of the isometries of D, the quotient nD is a locally symmetric variety. If M is a complete (non-compact) Kähler manifold with finite-dimensional L2 cohomology, one still has the decomposition of L2 harmonic forms ((3.3) in the compact case) M j p;q .M; C/ D H.2/ .M /; (3.5) H.2/ pCqDj

as the notions of formal and strict harmonic forms coincide by Remark 2.4 (i). This induces Hodge decompositions, via (2.7): M j p;q .M; C/ D H.2/ .M /: (3.6) H.2/ pCqDj

Problem 2.6 is asking for a topologically-defined cohomology group on some comx , to which (3.6) imparts Hodge structures of weight j in degree j . pactification M

4 Intersection (co)homology We will give a treatment of intersection cohomology with middle “perversity”, as it occurs for complex projective algebraic varieties. Leading to [35], Goresky and MacPherson were thinking: for a compact, orientable, stratified space X of dimension m (with singularities), one knows that Poincaré duality, i.e., the perfect pairing in the manifold case (even valid with Z-coefficients): H mi .X; Z/ H i .X; Z/ ! Z

(4.1)

(cf. (4.1), with coefficients in R, (2.10)) generally fails to hold. Is there a construction of something else for which Poincaré duality always holds, and which recovers the usual duality (4.1) when X is a manifold? They made a construction involving simplicial chains (as used by Lefschetz) with a triangulation of X , together with a parameter, the perversity, which specifies how much one relaxes the condition of general position for chains and their boundaries (compare Definition 2.1). For instance, if one insists on general position with respect to the strata of X (the 0 perversity), one gets the cohomology groups H mi .X/ from i -cycles, and with the vacuous condition one gets the homology groups Hmi .X/ from .m i /-cycles. The intersection pairing


761

(at the level of simplicial chains) induces the usual dual pairing H mi .X; Q/ Hmi .X; Q/ ! Q;

(4.2)

defined even with Z-coefficients. In general, there is a notion of a dual pair of perversities (see [35]). There is the analogue of (4.2) giving the duality between the corresponding notions of intersection cohomology groups. It is a fact of life that intersection cohomology, beyond usual homology and cohomology, is not a cohomology theory in the sense of algebraic topology, as it is not a homotopy invariant. For the purpose of this exposition, we assume that X is a compact algebraic subvariety of some CP` , with complex dimension n. One knows (see [37]) that X admits a stratification by subvarieties. By this, we mean that one has a descending chain of projective subvarieties X X 0 X 1 X n ; with codimX .X j / j I ı

then X j D X j X j C1 is the stratum of complex codimension j , a space without singularities, for which the Whitney conditions (see [37]) hold. This ensures that transverse slices along each stratum are locally constant, specifically are homeomorphic ı

to the (real) cone on the link ƒj of X j . Note that ƒj is necessarily of odd real dimension, namely 2j 1, and comes with an induced stratification with all strata of even real codimension. The middle perversity, m, for a space with all strata of ı

even codimension, is the specification that an i -chain is permitted to intersect X j in a chain of dimension at most i j . In particular, the intersection must be empty whenever j > i . The homology groups that one gets are IH .X /m ; we will drop the symbol m from the notation. Then intersection of chains defines a perfect pairing with coefficients in Q [35]: IH2ni .X; Q/ IHi .X; Q/ ! Q:

(4.3)

By using the dual numbering, we can rewrite (4.3) as cohomology: IH i .X; Q/ IH 2ni .X; Q/ ! Q: There is an analogue of (2.3) that characterizes intersection cohomology. Let U be a contractible open set about a point of X j of the form W Cone.ƒj /, with W a contractible open set in X j . ´ IH i .ƒj / if i < j; i IH .U / D (4.4) 0 otherwise: There is one instance in which intersection cohomology is expressible as ordinary cohomology, namely when X has only isolated singularities. In this case, X 1 D X n

762


and the global consequence of (4.4) is 8 ı i 0 ˆ ˆ if i < n; 1; indeed, all cj D 3. Thus, the Weil–Petersson metric is fine x DM . Saper invokes the L2 Künneth formula, applying it to (5.2) to yield on M g;n O .U / ' H.2/ .Uk / ˝ H.2/ .j W hj /: (5.1) H.2/ j

With Cheeger’s calculation [15] of the L2 -cohomology of a horn, one gets ´ R if i D 0; i . I hj / D H.2/ 0 otherwise; informally stated, the L2 -condition serves to fill in the origin of . The same holds for Uk , so he obtains that likewise ´ R if i D 0; i (5.2) H.2/ .U / D 0 otherwise; from which it follows that the sheaf of L2 -differential forms is quasi-isomorphic to the constant sheaf, and hence i x DM /: H.2/ .Mg;n I !wp / ' H i .M g;n One can carry out an analogous determination of H.p/ .Mg;n I !wp /. But first we place Theorem 1.2 in a broader context. By definition, the c-horn Hc .N / on a Riemannian manifold N is the product .0; 1 N with metric 2 ds 2 D dr 2 C r 2c dsN

for some c 1. It is incomplete. Cheeger’s general determination [15], Lemma 3.4, of the L2 -cohomology of a c-horn Hc .N / on a Riemannian manifold N (in terms of the L2 -cohomology of N ) has been generalized to Lp -cohomology by Youssin [90]. For our exposition, we extract a special case that is somewhat more general than what we need:


765

Proposition 5.1. Let Hc .N / be the c-horn on a compact Riemannian manifold N , whose dimension n is odd. Then ´ H j .N / if p < .n C c 1 /=j; j H.p/ .Hc .N // ' 0 otherwise. Corollary 5.2. The Lp -cohomology of Hc .S 1 / is given by 0 .Hc .S 1 // ' R for all p; (1) H.p/ ´ ; R if p < cC1 1 c .Hc .N // ' (2) H.p/ 0 otherwise; 2 .Hc .S 1 // D 0 for all p. (3) H.p/

Because S 1 is an abelian Lie group acting isometrically on the Lp de Rham complex of Hc .S 1 /, one can prove Corollary 5.2 directly, avoiding most of the detailed analysis needed for the proof of Proposition 5.1. We give here the argument. Lemma 5.3. Let Hc .S 1 / ' .0; 1 S 1 denote the c-horn on a circle. Then dr is an Lp 1-form for all p, and d is an Lp 1-form if and only if p < cC1 . c Proof. We note that the pointwise norms jdrjp D 1, jdjp D r pc and d V D r c dr d. Then Z 2 Z 1 p kdk.p/ D r cpc dr d: 0

0

p

It follows that d is an L 1-form on the horn if and only if c pc > 1, i.e., p < cC1 . c The preceding lemma suggests: Lemma 5.4. Let I D fr W 0 < r 1g be the half-open unit interval. Then the weighted Lp -cohomology of I (1 p < 1) with weights of the form r ˛ (˛ 2 R) is given by ´ R if ˛ > 1; 0 ˛ (1) H.p/ .I I r / D 0 if ˛ 1I 1 .I I r ˛ / D 0 for all p. (2) H.p/

Proof. The statement about H 0 is easy, for it is only a question of whether ˚ R1 1 2 Lp .I I r ˛ / WD f j 0 jf .r/jp r ˛ dr < 1 : As for H 1 , there is an evident operator B given by B.f .r/dr/ D F .r/: ´R r f .t /dt if ˛ > 1; F .r/ D R1r if ˛ 1; 0 f .t /dt

766


such that dF D f .r/dr in the weak sense. (Indeed, in the first case, B takes smooth 1forms to smooth functions, i.e., F 0 .r/ D f .r/ holds in the sense of smooth functions.) By Remark 2.3, it suffices to verify that B is a bounded operator on L.p/ .I I r ˛ /, which we show only in the second case, the first being similar. Using Fubini’s theorem, we estimate ˇ Z 1Z r Z 1 ˇZ r ˇ ˛ ˇ p ˇ ˇ f .t /dt ˇ r dr jf .t /jp dt r ˛ dr kB.f dr/k.p/ D ˇ 0 0 0 0 Z 1 Z 1Z 1 r ˛ dr jf .t /jp dt .t /t ˛ jf .t /jp dt D 0

t

kf kp.p/ D kf drkp.p/ ;

0

´

where .t/ D

t when ˛ < 1; t j log t j when ˛ D 1.

We can decompose A.p/ .Hc .S 1 / into the direct sum of the S 1 -invariant forms and the forms with integral zero over the S 1 -orbits; both are subcomplexes. The former is isomorphic to A.p/ .I I r c / ˚ A.p/ .I I r cpc /d; Applying Lemma 5.4 already produces the right-hand side of (2) in Corollary 5.2. It remains to show that the complementary factor is acyclic (i.e., has trivial cohomology). The main issue lies with H 1 . Let ' be a smooth 1-form and write ' D f .r; /dr C g.r; /d , with f 2 Lp .I S 1 I r c /, g 2 Lp .I S 1 I r cpc /, and the partial derivatives gr and f are equal. Let G.r; / denote the -antiderivative of g.r; / given by Z g.r; /d; G.r; / D 0 1

for any 0 2 S . This is well-defined whenever g has mean zero. Then G is a smooth function in Lp .I S 1 I r ccp /, by a calculation simpler than the one in (5.8)–(5.9); a fortiori it is in Lp .I S 1 I r c /. As gr D f , Gr .r; / D f .r; / f .r; 0 /: The preceding yields that ' D dG C f .r; 0 /dr. If we choose 0 so that f .r; 0 / 2 Lp .I S 1 I r c /, i.e., outside a set of measure zero in S 1 , we have by Lemma 5.4 that f .r; 0 /dr, hence also ', is exact in the sense of Lp -cohomology. This concludes our proof of Corollary 5.2. Remark 5.5. The way that Proposition 5.1 is proved, in effect, in [15] and [90] is by proving that the weak and minimal closures of d on A .Hc .N // coincide. Once one knows that, one can also adapt the criterion (2.29) from [92] (see also (2.41) of op.cit.), of which what we did above is a variant.


767

We note that when c D 3, the dividing point for the two cases in Corollary 5.2 occurs at p D 43 . We conclude, finishing the proof of Theorem 1.2: Proposition 5.6. a) For all p, the analogue of (5.1) holds: O H.p/ .U / ' H.p/ .Uk / ˝ H.p/ .j /: j

b) When 43 p < 1, the analogue of the identification in (5.2) holds for and x DM /: H .Mg I !wp / ' H .M

i H.p/ .U /,

g

.p/

c) When 1 p < .1/

4 , 3

´ i H.p/ . I hj /

D

R if i D 0 or i D 1 0 otherwise,

(so (5.2) must be correspondingly altered);17 .2/

H.p/ .Mg;n I !wp / ' H .Mg;n /:

Proof. Statement a) is an application of the Lp -Künneth theorem, as the factors have finite-dimensional Lp -cohomology. This yields the local and global assertions in b) and c). Complete metrics. Because the Weil–Petersson metric is incomplete, its geometry becomes more complicated. The statement of the Hodge theorem for !wp , which identifies the L2 -cohomology with a space of L2 harmonic forms, can impose (idealized) x str (compare the issue in Conjecture 4.4). For a complete boundary conditions for metric having finite-dimensional L2 -cohomology, the latter is isomorphic to the space of all L2 harmonic forms. There are several constructions and/or use of complete metrics in the literature. Foremost is the Teichmüller metric, which is defined as follows. For any two points in Tg;n (representing two marked Riemann surfaces of genus g with n punctures), S and S 0 , there is a unique quasi-conformal mapping S ! S 0 that has least “distortion” K 1 and is compatible with the markings. Then the Teichmüller distance d.S; S 0 / is defined to be log K. Other well-known examples of complete metrics include: the Bergman metric, Kähler–Einstein metric, McMullen metric [63], Ricci metric (the negative of the Ricci curvature of the Weil–Petersson), the perturbed Ricci metric of Liu, Sun and Yau [57], and any Poincaré metric adapted to the boundary of MgDM . Though there are infinitely many quasi-isometry classes of complete metrics available on a noncompact manifold, it was shown in [57] that all of the complete metrics just listed are quasi-isometric. In particular, they determine the same Lp -cohomology. Since the Teichmüller metric on Tg;n is the one studied and understood traditionally, 17 Informally,

the Lp -condition is mild enough that the puncture still counts.

768


one would naturally associate this Lp -cohomology to the Teichmüller metric. On the other hand, the Teichmüller metric is a non-Riemannian Finsler metric, and defining norms of differential forms and the volume form with respect to it is complicated. Since the Bergman metric is intrinsic for bounded domains in C n and is probably more familiar to the reader, we elect to use the Bergman metric !be in the following .Mg I !be /. computation and denote the Lp -cohomology group by H.p/ To proceed, we need the analogue of (5.2). It was shown in [57] and [89] that the Bergman metric is quasi-isometric to the Teichmüller metric, and we can use the asympototic bounds on the Teichmüller metric proved in [63] for the Bergman metric. Specifically, we use the formula from Proposition 4 in [56] which was proved in Theorem 1.7 of [63]; in the coordinates (5.1), we have up to quasi-isometry !be .U / !.Uk / C

X

!P .j /;

j

where !P is the Poincaré (or hyperbolic) metric. To be precise, McMullen ([63], Theorem 1.7) gives asymptotic formulas for the Teichmüller metric in Fenchel–Nielsen coordinates, showing the metric to be asymptotically hyperbolic in directions transverse to the boundary strata. (See also [56] for such a formula.) In [57], Theorem 4.3, the authors compare a Poincaré metric adapted to the Deligne–Mumford boundary to the Ricci metric. By the chain of equivalences, one sees that the Teichmüller metric and the Poincaré metric are quasi-isometric. In fact, a relation between the complex coordinates near the Deligne–Mumford boundary and the Fenchel–Nielsen coordinates is known; see Example 4.3 in [85] and also Remark 4.2 in [57]. Thus, the Bergman metric (and all of the aforementioned complete metrics) is also x DM . The Lp -cohomology of . ; !P / is known (see [93], §3) and is given fine on M g by the right-hand side of (5.4). By the same method as for Proposition 5.6, we obtain: Proposition 5.7. When 1 < p < 1, N (1) H.p/ .U / ' H.p/ .Uk / ˝ j H.p/;P .j /. ´ R if i D 0; i (2) Specifically, H.p/ .U / D 0 otherwise. x DM /. .Mg I !be / ' H i .M (3) H.p/ g

We observe that the above complete metrics yield the same Lp -cohomology as the incomplete Weil–Petersson metric when 43 p < 1. This is not so surprising, for the corresponding assertion was known to hold thirty years ago for the L2 -cohomology for the Euclidean (i.e., conical) and Poincaré metrics on [91]. Remark 5.8. When n D 0, the moduli space Mg D Mg;0 admits the so-called Satake compactification [5], obtained as follows. For each projective curve C of genus g, its Jacobian is an abelian variety of dimension g, with a canonical principal polarization.


769

Thus, it determines a point in the Siegel modular variety Sp.g; Z/ng . This gives an embedding i W Mg ,! Sp.g; Z/ng : BB

The closure of i.Mg / in the Baily–Borel compactification Sp.g; Z/ng turns out to be an algebraic variety, often called by algebraic geometers the Satake compactification of Mg . There is a (necessarily surjective) morphism of compactifications, which is x DM to the Satake compactification (see [42], for not an isomorphism when g 2, of M g;n instance). This compactification of Mg does not have many applications in algebraic geometry, probably because it does not have a good interpretation in terms of moduli. A natural question raised in [75] is whether the L2 -cohomology of Mg with respect to the restriction of the invariant metric of Sp.g; Z/ng to Mg is isomorphic to the intersection cohomology of the Satake compactification of Mg .

6 Some geometric questions concerning the complete metrics Looking ahead, we formulate several natural problems concerning the above metrics on Mg;n and Tg;n : (1) Compute the volume of Mg;n with respect to the complete metrics mentioned above. It can be shown easily that the metrics have finite volume. For arithmetic locally symmetric spaces given by congruent subgroups, the volumes can be expressed in terms of special values of zeta functions. The Weil–Petersson volume of the closely related moduli spaces of bordered Riemann surfaces can be computed inductively [64]. (2) Classify geodesics in Mg;n and prove the logarithmic law for geodesics with respect to the complete Riemannian metrics mentioned above. Geodesics that are eventually distance minimizing with respect to the Teichmüller metric, which is a Finsler metric, have been classified in [29] as in the case of locally symmetric spaces [46]. The logarithmic law for geodesics was first proved by Sullivan [79] for hyperbolic manifolds. It has been generalized for general locally symmetric spaces in [52]. For the Teichmüller metric, the logarithmic law for geodesics in Mg;n was proved by Masur in [61]. (3) Study the dynamics of the geodesic flow of Mg;n in any of the above complete Riemannian metrics. A lot of work has been done for the geodesic flow of locally symmetric spaces. See the book [24] and the survey [25] for references. See [26] for counting geodesics in the moduli space with respect to the Teichmüller metric.

770


The geodesic flow on Mg;n with respect to the Teichmüller metric is ergodic [62], and it was proved recently that the geodesic flow with respect to the Weil– Petersson metric is also ergodic [13]. A natural problem is to understand the geodesic flow with respect to other metrics mentioned above. (4) Determine the tangent cone of Tg;n with respect to any of the above complete Riemannian metrics. The tangent cones at infinity of symmetric spaces of noncompact type are Euclidean buildings [51], and the tangent cone of Modg;n has also been studied in [8], [7]. The tangent cone of Mg;n in any of the above complete Riemannian metrics is equal to the metric cone over a finite complex Modg;n nC .Sg;p /, where C .Sg;p / is the curve complex of the surface Sg;p [56], [30]. (5) Determine the quasi-isometry group of Tg;n with respect to any of the complete metrics. The quasi-isometry groups of symmetric spaces have been well-understood. See the survey [27] and references therein. (6) Obtain effective bounds on the first positive eigenvalue of Mg;n with respect to any of the above complete Riemannian metrics and also the Weil–Petersson metric. For arithmetic locally symmetric spaces, this is an important problem. For example, for quotients of the upper half plane by congruence subgroups of SL.2; Z/, there is the famous Selberg 14 -conjecture. (7) Find the explicit value of the bottom of the spectrum 0 .Tg;n / and the exponent in the exponential growth of the volume of balls in Tg;n with respect to any of the above complete metrics. For symmetric spaces, these two values can be computed explicitly, and they are closely related. For the Teichmüller metric on Tg , g 2, the exponent in the volume growth of balls with respect to a natural Modg -invariant measure has been computed in [4] and is equal to 6g 6. (8) Prove that the essential spectrum of the Laplacian of Mg;n with respect to any of the complete Riemannian metrics above is continuous and determine if it is of the form Œa; C1/ for some a 0. It is unlikely that one can get a detailed spectral resolution as in the case of locally symmetric spaces by Eisenstein series. On the other hand, it might be possible to prove asymptotic completeness as in the case of the many body problem in mathematical physics (see [78]). (9) Determine the Martin compactification of Tg;n with respect to any of the above complete Riemannian metrics. The Martin compactifications of symmetric spaces of noncompact type have been determined in [41]. By [63], the bottom of the spectrum 0 .Tg;n / of the


771

Laplacian of Tg with respect to any of the above complete Riemannian metrics, which are quasi-isometric to the Teichmüller metric, is positive. An easier problem is to show that the space of positive harmonic functions on Tg;n is infinite-dimensional. Another natural problem is to determine the Poisson boundary of Tg;n . Related problems on the Poisson boundaries of random walks on Teichmüller spaces Tg;n have been studied in [48], [49].

References [1]

W. Abikoff, The real analytic theory of Teichmüller space. Lecture Notes in Math. 820, Springer-Verlag, Berlin 1980. 748

[2]

A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds. Inst. Hautes Études Sci. Publ. Math. 25 (1965), 81–130. 757

[3]

M. Atiyah, Elliptic operators, discrete groups and von Neumann algebras. Astérisque 32–33 (1976), 43–72. 757

[4]

J. Athreya, A. Bufetov, A. Eskin, and M. Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space. Preprint, 2006; arXiv:math/0610715v5 [math.DS] 770

[5]

W. Baily, On the moduli of Jacobian varieties. Ann. of Math. (2) 71 (1960), 303–314. 768

[6]

W. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) 84 (1966), 442–528. 748

[7]

J. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space. Geom. Topol. 10 (2006), 1523–1578. 770

[8]

J. Behrstock and Y. Minsky, Dimension and rank for mapping class groups. Ann. of Math. (2) 167 (2008), no. 3, 1055–1077. 770

[9]

P. Bérard, From vanishing theorems to estimating theorems: the Bochner technique revisited. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 371–406. 753

[10] L. Bers, Finite-dimensional Teichmüller spaces and generalizations. Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 131–172. 747, 763 [11] A. Borel, Stable real cohomology of arithmetic groups. Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272. 757 [12] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition, Math. Surveys Monogr. 67, Amer. Math. Soc., Providence, R.I., 2000. 753 [13] K. Burns, H. Masur, and A. Wilkinson, The Weil–Petersson geodesic flow is ergodic. Ann. of Math., to appear; arXiv:1004.5343v3 [math.DS]. 770 [14] E. Cattani, A. Kaplan, and W. Schmid, L2 and intersection cohomologies for a polarizable variation of Hodge structure. Invent. Math. 87 (1987), 217–252. 762 [15] J. Cheeger, On the Hodge theory of Riemannian pseudomanifolds. In Geometry of the Laplace Operator, Proc. Sympos. Pure Math. 36, Amer. Math. Soc., Providence, R.I., 1980, 91–146. 750, 764, 766

772


[16] J. Cheeger, M. Goresky, and R. MacPherson, L2 -cohomology and intersection homology of singular algebraic varieties. In Seminar on differential geometry (S-T. Yau, ed.), Ann. of Math. Stud. 102, Princeton University Press, Princeton, N.J., 1982, 303–340. 753, 762, 763 [17] B. Chen, The Bergman metric on Teichmüller space. Internat. J. Math. 15 (2004), 1085–1091. [18] P. Deligne, Théorie de Hodge, II. Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–58. 762 [19] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109. 747, 748 [20] J. Dodziuk, L2 harmonic forms on complete manifolds. In Seminar on differential geometry (S-T. Yau, ed.), Ann. of Math. Stud. 102, Princeton University Press, Princeton, N.J., 1982, 291–302. 757 [21] H. Donnelly, L2 cohomology of the Bergman metric for weakly pseudoconvex domains. Illinois J. Math. 41 (1997), 151–160. 757 [22] H. Donnelly and C. Fefferman, L2 -cohomology and index theory for the Bergman metric. Ann. of Math. 118 (1983), 593–618. 757 [23] E. Dryden, C. Gordon, S. Greenwald, and D. Webb, Asymptotic expansion of the heat kernel for orbifolds. Michigan Math. J. 56 (2008), no. 1, 205–238. 757 [24] P. Eberlein, Geometry of nonpositively curved manifolds. Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1996. 769 [25] P. Eberlein, Geodesic flows in manifolds of nonpositive curvature. In Smooth ergodic theory and its applications, Proc. Sympos. Pure Math. 69, Amer. Math. Soc., Providence, R.I., 2001, 525–571. 769 [26] A. Eskin and M. Mirzakhani, Counting closed geodesics in moduli space. J. Mod. Dyn. 5 (2011), no. 1, 71–105. 769 [27] B. Farb, The quasi-isometry classification of lattices in semisimple Lie groups. Math. Res. Lett. 4 (1997), no. 5, 705–717. 770 [28] B. Farb, A. Lubotzky, and Y. Minsky, Rank-1 phenomena for mapping class groups. Duke Math. J. 106 (2001), no. 3, 581–597. 749 [29] B. Farb and H. Masur, Teichmüller geometry of moduli space, I: Distance minimizing rays and the Deligne–Mumford compactification. J. Differential Geom. 85 (2010), no. 2, 187–227. 769 [30] B. Farb and H. Masur, Teichmüller geometry of moduli space, II: M.S/ seen from far away. In In the tradition of Ahlfors-Bers, V (M. Bonk, J. Gilman, H. Masur, Y. Minsky, and M. Wolf, eds.), Contemp. Math. 510, Amer. Math. Soc., Providence, R.I., 2010, 71–79. 770 [31] B. Farb and S. Weinberger, The intrinsic asymmetry and inhomogeneity of Teichmüller space. Duke Math. J. 155 (2010), no. 1, 91–103. [32] M. Gaffney, A special Stokes’ theorem for complete Riemannian manifolds. Ann. of Math. (2) 60 (1954), 140–145. 753, 755 [33] M. Gaffney, Hilbert space methods in the theory of harmonic integrals. Trans. Amer. Math. Soc. 78 (1955), 426–444. 753, 755


773

[34] S. Gelbart and F. Shahidi, Analytic properties of automorphic L-functions. Perspect. Math. 6, Academic Press, Boston, Mass., 1988. 750 [35] M. Goresky and R. MacPherson, Intersection homology theory. Topology 19 (1980), 135–162. 760, 761 [36] M. Goresky and R. MacPherson, Intersection homology theory II. Invent. Math. 72 (1983), 77–129. 762 [37] M. Goresky and R. MacPherson, Stratified Morse theory. Ergeb. Math. Grenzgeb. (3) 14, Springer-Verlag, Berlin 1988. 761 [38] P. Griffiths and J. Harris, Principles of algebraic geometry. Wiley Classics Lib., John Wiley & Sons, New York 1994. 759 [39] M. Gromov, Kähler hyperbolicity and L2 -Hodge theory. J. Differential Geom. 33 (1991), 263–292. 757 [40] M. Gromov, Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–100. 757 [41] Y. Guivarc’h, L. Ji, and J. C. Taylor, Compactifications of symmetric spaces. Progr. Math. 156, Birkhäuser, Boston, Mass., 1998. 770 [42] J. Harris and I. Morrison, Moduli of curves. Grad. Texts in Math. 187, Springer-Verlag, New York 1998. 769 [43] W. V. D. Hodge, The theory and applications of harmonic integrals. Cambridge University Press, Cambridge 1941. 752 N Acta Math. 113 [44] L. Hörmander, L2 estimates and existence theorems for the @-operator. (1965), 89–152. 757 [45] W-C. Hsiang and V. Pati, L2 -cohomology of normal algebraic surfaces. Invent. Math. 81 (1985), 395–412. 763 [46] L. Ji and R. MacPherson, Geometry of compactifications of locally symmetric spaces. Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 457–559. 769 [47] L. Ji, R. Mazzeo, W. Müller, and A. Vasy, in preparation. 750 [48] V. Kaimanovich and H. Masur, The Poisson boundary of Teichmüller space. J. Funct. Anal. 156 (1998), 301–332. 771 [49] V. Kaimanovich and H. Masur, The Poisson boundary of the mapping class group. Invent. Math. 125 (1996), no. 2, 221–264. 771 [50] M. Kashiwara and T. Kawai, The Poincaré lemma for variations of polarized Hodge structure. Publ. Res. Inst. Math. Sci. 23 (1987), 345–407. 762 [51] B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math. 86 (1997), 115–197. 770 [52] D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces. Invent. Math. 138 (1999), 451–494. 769 [53] K. Kodaira, Harmonic fields in Riemannian manifolds (Generalized potential theory). Ann. of Math. 50 (1949), 587–665. 753 [54] K. Kodaira, On Kähler varieties of restricted type, Ann. of Math. 60 (1954), 28–48. 760 [55] R. Langlands, On the functional equations satisfied by Eisenstein series. Lecture Notes in Math. 544, Springer-Verlag, Berlin 1976. 750

774


[56] E. Leuzinger, Reduction theory for mapping class groups and applications to moduli spaces. J. Reine Angew. Math. 649 (2010), 11–31. 763, 768, 770 [57] K. Liu, X. Sun, and S. T.Yau, Canonical metrics on the moduli space of Riemann surfaces. I. J. Differential Geom. 68 (2004), no. 3, 571–637. 748, 750, 767, 768 [58] E. Looijenga, L2 -cohomology of locally symmetric varieties. Compositio Math. 67 (1988), no. 1, 3–20. 748, 762 [59] E. Looijenga, Smooth Deligne–Mumford compactifications by means of Prym level structures. J. Algebraic Geom. 3 (1994), no. 2, 283–293. 748 [60] H. Masur, Extension of the Weil–Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43 (1976), no. 3, 623–635. 764 [61] H. Masur, Logarithmic law for geodesics in moduli space. In Mapping class groups and moduli spaces of Riemann surfaces, Contemp. Math. 150, Amer. Math. Soc., Providence, R.I., 1993, 229–245. 769 [62] H. Masur, Interval exchange transformations and measured foliations. Ann. of Math. (2) 115 (1982), no. 1, 169–200. 770 [63] C. McMullen, The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. of Math. (2) 151 (2000), no. 1, 327–357. 767, 768, 770 [64] M. Mirzakhani, Weil–Petersson volumes and intersection theory on the moduli space of curves. J. Amer. Math. Soc. 20 (2007), no. 1, 1–23. 769 [65] T. Ohsawa, The Cheeger-Goresky-MacPherson conjecture for varieties with isolated singularities. Math. Z. 206 (1991), 219–224. 763 [66] P. Pansu, Lp -cohomology of symmetric spaces. Adv. Lect. Math. 6 (2008), 305–326. [67] P. Pansu, Lp -cohomologie et pincement. Comment. Math. Helv. 83 (2008), 327–357. [68] W. Pardon and M. Stern, The L2 -@N cohomology of complex projective varieties. J. Amer. Math. Soc. 4 (1991), 603–621. 763 [69] W. Pardon and M. Stern, Pure Hodge structure on the L2 -cohomology of varieties with isolated singularities. J. Reine Angew. Math. 533 (2001), 55–80. 763 [70] H. Royden, Intrinsic metrics on Teichmüller space. In Proceedings of the International Congress of Mathematicians, Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, 217–221. 748 [71] H. Royden, Invariant metrics on Teichmüller space. In Contributions to analysis,Academic Press, New York 1974, 393–399. 748 [72] M. Saito, Modules de Hodge polarisable. Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995. 763 [73] L. Saper, L2 -cohomology and intersection homology of certain algebraic varieties with isolated singularities. Invent. Math. 82 (1985), 207–255. [74] L. Saper, L2 -cohomology of Kähler varieties with isolated singularities. J. Differential Geom. 36 (1992), 89–161. 763 [75] L. Saper, L2 -cohomology of the Weil–Petersson metric. In Mapping class groups and moduli spaces of Riemann surfaces, Contemp. Math. 150, Amer. Math. Soc., 1993, 345–360. 749, 764, 769 [76] L. Saper and M. Stern, L2 -cohomology of arithmetic varieties. Ann. of Math. (2) 132 (1990), no. 1, 1–69. 748, 762


775

[77] I. Satake, The Gauss-Bonnet theorem for V -manifolds. J. Math. Soc. Japan 9 (1957), 464–492. 757 [78] I.M. Sigal and A. Soffer, Asymptotic completeness of N -particle long-range scattering. J. Amer. Math. Soc. 7 (1994), 307–334. 770 [79] D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149 (1982), 215–237. 769 [80] B. Teissier and J. L. Verdier (eds.), Analyse et topologie sur les espaces singuliers. I, Astérisque 100 (1982), II, III, Astérisque 101–102 (1983), Soc. Math. France, 1982. 751 [81] F. Warner, Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition, Grad. Texts in Math. 94, Springer-Verlag, New York 1983. 753 [82] A. Weil, Sur les théorèmes de de Rham. Comment. Math. Helv. 26 (1952), 119–145. 751 [83] A. Weil, Variétés Kähleriennes. Hermann, Paris 1971. 759 [84] S. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space. In Surveys in differential geometry, Vol. VIII, International Press, Somerville, Mass., 2003, 357–393. 748, 764 [85] S. Wolpert, The hyperbolic metric and the geometry of the universal curve. J. Differential Geom. 31 (1990), 417–472. 768 [86] S. Wolpert, The Weil–Petersson metric geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 47–64. 763 [87] H. Wu, The Bochner technique in differential geometry. Math. Rep. 3 (1988), no. 2, i–xii and 289–538. 753 [88] S.-K. Yeung, Bergman metric on Teichmüller spaces and moduli spaces of curves. In Recent progress on some problems in several complex variables and partial differential equations, Contemp. Math. 400, Amer. Math. Soc., Providence, R.I., 2006, 203–217. 749 [89] S.-K. Yeung, Quasi-isometry of metrics on Teichmüller spaces. Internat. Math. Res. Notices 4 (2005), 239–255. 748, 750, 768 [90] B.Youssin, Lp -cohomology of cones and horns. J. Differential Geom. 39 (1994), 559–603. 764, 766 [91] S. Zucker, Hodge theory with degenerating coefficients: L2 -cohomology in the Poincaré metric. Ann. of Math. 109 (1979), 415–476. 750, 762, 768 [92] S. Zucker, L2 -cohomology of warped products and arithmetic groups. Invent. Math. 70 (1982), 169–218. 749, 756, 766 [93] S. Zucker, On the reductive Borel–Serre compactification: Lp -cohomology of arithmetic groups (for large p). Amer. J. Math. 123 (2001), no. 5, 951–984. 749, 762, 768 [94] S. Zucker, The Hodge structures on the intersection homology of varieties with isolated singularities. Duke Math. J. 55 (1987), 603–616. 754 [95] S. Zucker, Lp -cohomology: Banach spaces and homological methods on Riemannian manifolds. Proc. Sympos. Pure Math. 54, Part 2, Amer. Math. Soc., Providence, R.I., 1993, 637–655. [96] S. Zucker, Satake compactifications. Comment. Math. Helv. 58 (1983), 312–343. 749 [97] S. Zucker, L2 -cohomology and intersection homology of locally symmetric varieties, III. In Théorie de Hodge, Astérisque 179–180 (1989), 245–278. 762

Part D

Teichmüller theory and mathematical physics

Chapter 14

The Weil–Petersson metric and the renormalized volume of hyperbolic 3-manifolds Kirill Krasnov and Jean-Marc Schlenker

Contents 1

2

3

4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Liouville theory . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Liouville theory and projective structures . . . . . . . . . . 1.3 The AdS-CFT correspondence and holography . . . . . . . 1.4 The definitions of the renormalized volume . . . . . . . . . 1.5 The first and second fundamental forms at infinity . . . . . . 1.6 Variation formulas . . . . . . . . . . . . . . . . . . . . . . 1.7 Maximization . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The renormalized volume and the Schwarzian derivative . . 1.9 The renormalized volume and Kleinian reciprocity . . . . . 1.10 The renormalized volume as a Kähler potential . . . . . . . 1.11 The relative volume of hyperbolic ends and the grafting map Two definitions of the renormalized volume . . . . . . . . . . . . 2.1 The renormalized volume . . . . . . . . . . . . . . . . . . . 2.2 The renormalized volume via equidistant foliations . . . . . 2.3 Renormalized volume as the W -volume . . . . . . . . . . . 2.4 Self-duality . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The W -volume and the Chern–Simons formulation . . . . . Description “from infinity” . . . . . . . . . . . . . . . . . . . . . 3.1 The metric at infinity . . . . . . . . . . . . . . . . . . . . . 3.2 Second fundamental form at infinity . . . . . . . . . . . . . 3.3 The Gauss and Codazzi equations at infinity . . . . . . . . . 3.4 Inverse transformations . . . . . . . . . . . . . . . . . . . . 3.5 Fundamental Theorem of surface theory “from infinity” . . . The Schläfli formula “from infinity” . . . . . . . . . . . . . . . . 4.1 The Schläfli formula for hyperbolic polyhedra . . . . . . . . 4.2 The Schläfli formula for hyperbolic manifolds with boundary 4.3 Parametrization by the data at infinity . . . . . . . . . . . . 4.4 Conformal variations of the metric at infinity . . . . . . . . 4.5 The renormalized volume as a function on Teichmüller space

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4.6 The second fundamental form at infinity as the real part of a holomorphic quadratic differential . . . . . . . . . . . . . . . . . . . . . . . 4.7 The second fundamental form as a Schwarzian derivative . . . . . . 4.8 The second fundamental form as the differential of WM . . . . . . . 5 Kleinian reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The first variation of the renormalized volume . . . . . . . . . . . . 5.3 The boundary of the convex core and the grafting map . . . . . . . 5.4 Deformations of representations and Poincaré duality . . . . . . . . 6 The renormalized volume as a Kähler potential . . . . . . . . . . . . . . 6.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Local deformations near the Fuchsian locus . . . . . . . . . . . . . 6.3 Kähler potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The relative volume of hyperbolic ends . . . . . . . . . . . . . . . . . . 7.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The first variation of the relative volume . . . . . . . . . . . . . . . 7.3 The grafting map is symplectic . . . . . . . . . . . . . . . . . . . . 8 Manifolds with particles and the Teichmüller theory of surfaces with cone singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Introduction We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichmüller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen’s quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a Kähler potential for the Weil– Petersson metric on Teichmüller space. Yet another is the fact that the grafting map is symplectic, which is proved using a variant of the renormalized volume defined for hyperbolic ends.

1.1 Liouville theory One of the early approaches to the problem of uniformization of Riemann surfaces was based on the so-called Liouville equation. Consider a closed Riemann surface S of genus g and fix an arbitrary reference metric h0 in the conformal class of S . Then, consider a conformally equivalent metric h D e 2 h0 . The condition that h has constant curvature minus one reads 0 K0 D e 2 ;

(1.1)

Chapter 14. Renormalized volume

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where 0 and K0 are the Laplacian and Gauss curvature of h0 respectively. One then tries to solve this equation for and thus find a hyperbolic metric on S. Historically, this approach to the uniformization turned out to be too difficult, and was abandoned in favor of the one based on Fuchsian groups. More recently, the set of ideas related to the Liouville equation came to the spotlight due to the central role it plays in Polyakov’s approach to string theory. In the so-called non-critical string theory a very important role is played by the Liouville functional, which (in one of its versions) can be written as Z 1 d vol0 jrj2 C e 2 2K0 : (1.2) SŒh0 ; D 8 When varied with respect to this functional gives the Liouville equation (1.1). As is implied by the uniformization theorem, there is indeed a unique solution to (1.1) on any given Riemann surface S . Let us denote this solution by hyp , where hyp stands for hyperbolic. One can evaluate the functional (1.2) on hyp and obtain a functional S Œh0 D S Œh0 ; hyp . When h0 is taken to be the hyperbolic metric, hyp D 0 and the value of the above functional is just 1=4 times the area of S evaluated using the hyperbolic metric, i.e. minus half the Euler characteristics of X . The values of S Œh0 at other points are not easy to find and one gets a rather non-trivial functional on the space of metrics h0 on S, which, in a certain sense, measures how far the metric h0 is from the hyperbolic one. This functional is not so interesting by itself, but serves as a prototype for the construction of a whole class of Liouville-type functionals that will play the central role in this chapter. The reason why S Œh0 is not so interesting is that one would rather have a functional on the space of conformal equivalence classes of metrics on S, i.e. the moduli space of Riemann surfaces, or at least on the Teichmüller space Tg , hoping that such a functional could be used to characterize the geometry of the moduli space in an interesting way. Such functionals are not easy to construct. Indeed, one could have obtained a functional on the moduli from S Œh0 by taking h0 to be some canonical metric in the given conformal class. However, as we have already seen, taking h0 to be the canonical metric of constant negative curvature gives a functional on the moduli space whose value at all points is a constant – the Euler characteristic. The reader can find another, discrete approach to Liouville theory in the survey by Kashaev in this volume [22].

1.2 Liouville theory and projective structures It turned out to be possible to use Liouville theory for a construction of interesting functions on the moduli space by first considering a certain function on the space of complex projective structures on Riemann surfaces. Then choosing the projective structure to be some canonical one, e.g. one related to the Schottky or quasifuchsian uniformization, one can get a functional on the (appropriate) moduli space itself. Since projective structures on Riemann surfaces are intimately connected to hyperbolic

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3-manifolds, such Liouville functionals of projective structure turn out to be related to the geometry of the corresponding 3-manifolds, and this is where the notion of the renormalized volume enters into the story. However, the first examples of non-trivial Liouville functionals on the Teichmüller space of Riemann surfaces had nothing to do with 3-manifolds, and worked entirely in the 2-dimensional setting. Thus, the first construction for the moduli space of surfaces of genus g was proposed in [39]. The idea was to base the construction on the so-called Schottky uniformization of the surface S . The latter is unique given a marking of S , which is a choice of g generators of 1 .S / corresponding to disjoint simple closed curves on S . The surface is then cut along the generators chosen, which results in a 2g-holed sphere. Thus, in the Schottky uniformization, the Riemann surface S is represented as a quotient of the complex plane under the action of the Schottky group , which is a group freely generated by g (loxodromic, i.e. non-elliptic) elements of PSL.2; C/. A fundamental region for the action of the Schottky group on C is the exterior of a set of 2g Jordan curves that get mapped into each other pairwise by the generators of . The flat metric jdzj2 of C can now be used as the reference structure for constructing a non-trivial Liouville functional, in place of h0 above. This functional is given by essentially the same quantity as in (1.2), with the flat metric Laplacian and K0 D 0. However, the Liouville field now has rather non-trivial transformation properties under the action of (so that the metric e jdzj2 can be pulled back to the quotient Riemann surface). As a result, the integral of the term jrj2 depends on a choice of the fundamental region. Thus, in order to define the functional in an unambiguous way one has to correct the “bulk” term (1.2) with a rather non-trivial set of boundary terms, see [39] for details. After this is done, one gets a well-defined functional of the conformal structure of S , of the marking of S that was used to get the Schottky uniformization as well as the Liouville field . One of the main properties of this functional is that the variation with respect to gives rise precisely to the Liouville equation D e , which can then be shown to have a unique solution with the required transformation properties. Evaluating the Liouville functional on this solution gives a non-trivial functional on the Schottky space, i.e. the moduli space of marked (by g curves) Riemann surfaces. One of the key properties of this functional is that its first variation (when the moduli are varied) gives rise to a certain holomorphic quadratic differential on the complex plane, which measures the difference between the Schottky and Fuchsian (used as reference) projective structures. From this one finds that the extremal Liouville functional is the Kähler potential for the so-called Weil–Petersson metric on the Schottky space, see [39] for details (and [46] for many key properties of the Weil–Petersson metrics). Even though this point of view is not at all developed in [39], it is convenient to think of the functional constructed as a special case of a more general Liouville functional on a surface equipped with a projective structure, specialized to the case where the projective structure in question is the Schottky one. This point of view suggests that there are other Liouville functionals out there, namely when one chooses the projective structure to be different. Indeed, a Liouville functional of the same


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type, but this time for the so-called quasifuchsian (or even more generally arbitrary Kleinian) projective structure was constructed in [41]. Since quasifuchsian projective structures are parametrized by the product of two copies of Teichmüller space Tg , one gets a functional on Tg Tg . Similar to the story in the Schottky case, this functional turns out to be a Kähler potential for the Weil–Petersson metric on Tg , as its .1; 1/ derivative with respect to moduli of the first copy of Tg turns out to be independent of the point in the second copy, see below for more details. In [40], a Liouville functional of the same type was constructed for the moduli .˛1 ;:::;˛n / space M.0;n/ of Riemann surfaces with n conical singularities of given angle

.˛1 ;:::;˛n / deficits 2 ˛i ; i D 1; : : : ; n. Moreover, a set of metrics on the space M.0;n/ was introduced, which for all ˛i D 1 (angle deficits 2) coincides with the usual Weil– Petersson metric on M.0;n/ . It was moreover shown that all these metrics are Kähler with the Kähler potential given by the Liouville functional. A choice of the projective structure is implicit in this case, as the natural projective structure on the complex plane with n marked points is used. Below we shall see how these Liouville functionals are intimately related to the geometry of the hyperbolic 3-manifolds which the corresponding projective structures define. We note however that, although hyperbolic surfaces with cone singularities fit nicely in the 3D geometric context considered here (see Section 8), the question of a geometrical (3-dimensional) interpretation of our last example – the functional on .˛1 ;:::;˛n / constructed in [40] – remains open. M.0;n/

1.3 The AdS-CFT correspondence and holography An initially unrelated development occurred in the context of the so-called AdS/CFT correspondence of string theory, see [44], in which asymptotically-hyperbolic manifolds play the key role. These are non-compact, infinite volume (typically Einstein, or Einstein with non-trivial “stress-energy tensor” on the right hand side of Einstein equations) Riemannian manifolds that have a conformal boundary, near which the manifold looks like the hyperbolic space of the corresponding dimension. Most interesting for physics is the case of 5-dimensional asymptotically-hyperbolic manifolds, for in this case the 5-dimensional gravity theory induces (or, as physicists say, is dual to) a certain non-trivial gauge theory on the boundary, see [44]. However, the simplest situation is that in three dimensions, where the Einstein condition implies constant curvature and one is led to consider simply a hyperbolic manifold M . Given such an asymptotically-hyperbolic manifold, physicists are interested in computing the Einstein–Hilbert functional (given by an integral of the scalar curvature plus a multiple of the volume form) on the metric of M . For the hyperbolic metric, the integrand reduces to a multiple of the volume form, so one is computing the volume of the hyperbolic manifold M , which is infinite. However, physicists are masters of extracting a finite answer from a divergent expression. And so it was found that in many situations there is a “canonical” way to extract a finite answer by

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“regularizing” the divergent volume, then subtracting the divergent contribution, and finally taking a limit to “remove the regulator”. This procedure, somewhat mysterious for a mathematician, was applied in the context of Schottky 3-dimensional hyperbolic manifolds (quotients of the hyperbolic space by the action of a Schottky group) in [24], with a somewhat unexpected result. Namely, it was shown that the “renormalized volume” of a Schottky manifold, as a function of the conformal factor for a metric in the conformal class of the hyperbolic boundary, is equal to the Schottky Liouville functional as defined in [39]. When one takes the conformal factor corresponding to the canonical metric of curvature 1 one gets the Liouville functional on the Schottky moduli space defined and studied in [39]. In view of the results of this reference, one finds that the renormalized volume, which is a purely 3-dimensional quantity in its definition, equals the Kähler potential for the Weil–Petersson metric on the Schottky space, which is a purely 2-dimensional quantity. One thus gets an example of what physicists like to refer to as a “holographic” correspondence (a relation between one quantity (or even theory) in n C 1 dimensions and another in n dimensions). The methods of [24], originally applicable only in the context of (classical) Schottky manifolds, were generalized and applied to arbitrary Schottky, quasifuchsian and even Kleinian manifolds in [41], with the end result being always the same: the renormalized volume turns out to be equal to the (appropriately defined) Liouville functional, and the latter is shown to be the Kähler potential on the corresponding moduli space. Even prior to the work [41], the set of ideas building upon the Kähler property of the Weil–Petersson metric on Teichmüller space (and closely related to the renormalized volume ideas, as was later shown in [41]) was used in [30] for a proof of the Kähler hyperbolicity property of the moduli space. The above story makes it clear that there is a deep relation between the geometry of the Teichmüller space of a Riemann surface S and the geometry of hyperbolic three-manifolds that realize S as its conformal boundary. This relation was recently demonstrated from a more geometrical perspective in [27], where it was shown that the key property of the Kähler potential, namely, that its first variation is given by a certain canonical quadratic differential on S, is a simple consequence of the Schläflilike formula of [35]. This geometrical perspective on the renormalized volume (and the Liouville functional) also made it clear that such a quantity may be defined not only for the Schottky, quasifuchsian and Kleinian projective structures considered in the literature, but, in fact, for an arbitrary projective structure. This idea leads to the notion of relative volume, defined and studied in [28]. It is on this more geometrical and, we believe, very simple, perspective on the renormalized volume that we would like to emphasize in this review.

1.4 The definitions of the renormalized volume Let M be a convex co-compact hyperbolic manifold, for instance a quasifuchsian manifold. The hyperbolic volume of M is infinite, but an interesting finite quantity can be extracted by a procedure that physicists refer to as renormalization. This


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proceeds by introducing a “regulator” that makes the quantity of interest finite, then removing the divergent contribution and then removing the regulator. For the case at hand an appropriate regulator is given by equidistant surfaces. Thus, the first step in the definition of the renormalized volume of M is to define a volume depending on an equidistant foliation F of M in the complement of a compact convex subset N . By an equidistant foliation we mean a foliation of M n N by closed, smooth, convex surfaces, so that, in each connected component of M n N , the surfaces are pairwise at constant distance. Since the foliation is equidistant, it is uniquely determined by N . Section 2 gives two different but equivalent definitions of a volume associated to N in M . The first definition is in terms of the asymptotic behavior of the volume of the set of points at distance at most from N as ! 1. This definition can be surprising but it extends to higher dimensions in the setting of conformally compact, Poincaré–Einstein metrics, see [16] for an original reference and [4] for a review. The other definition is simpler, but limited to 3 dimensions. It is in terms of the volume of N , corrected by a term involving the integral of the mean curvature of the boundary of N , as Z 1 H da: W .M; N / D V .N / 4 @N In 3 dimensions a convex co-compact hyperbolic manifold is completely specified by the conformal structure of its boundary @M . It then turns out that the dependence of W .N; M / on N is just the dependence on a metric in the conformal class of the boundary. Moreover, when this metric is varied the functional W .N; M / reaches an extremum on metrics of constant (negative) curvature. Thus, W .N; M / is nothing but the Liouville functional whose 2-dimensional realizations have been discussed above. Note that the hyperbolic 3-manifold M in which the volume is computed comes equipped with a projective structure on @M , and explains why the Liouville action in all cases needed a projective structure to be defined.

1.5 The first and second fundamental forms at infinity There is a natural description of the connected components of the complement of N in terms of the induced metric and second fundamental forms I , II of the corresponding boundary component of N . Section 3 contains an alternative description, in terms of naturally defined “induced metric” and “second fundamental form” I , II at infinity in the same end. There are simple transformation formulas from I , II to I , II and conversely. The conformal class of I is the conformal class at infinity of M . Moreover, I and II satisfy the Codazzi equation and an analog of the Gauss equation, which involves the trace of II instead of the determinant of II as in the usual Gauss equation for surfaces in H 3 . A subtle point, explained at more length in Section 3, is that, as we have already mentioned, not only N determines the metric at infinity I , but I also determines uniquely N . In general a metric I in the conformal class at infinity does not come

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from a choice of some convex subset N of M . However, for any such metric I , there is associated an equidistant foliation of a neighborhood of infinity in M . This is sufficient to define the renormalized volume W , and this makes W a function of M and I , rather than of M and N as defined above. The modified Gauss and Codazzi equations for I , II are described in Section 3. They have an interesting consequence. When K is constant, the trace H of II is also constant, so that the traceless part II0 of II also satisfies the Codazzi equation d r II0 D 0, where r is the Levi-Civita connection of I . It then follows (by an argument going back to Hopf [21]) that II0 is the real part of a holomorphic quadratic differential. In other words, II0 is canonically identified with a cotangent vector in TŒI T@M , where ŒI is the conformal class of I , a property that is going to play an important role below.

1.6 Variation formulas The function W .M; N / has a simple first-order variation formula in terms of the data on the boundary of N , recalled in Section 4. For any first-order deformation of M or of N in M , Z 1 ıW D ıH C hıI; II0 ida; 4 @N where H D trI II is the mean curvature of the boundary and II0 is the traceless part of the second fundamental form. A lengthy but direct computation then shows that the same formula (except for the sign) holds (when this function is interpreted as W .M; I /) in terms of the data at infinity: Z 1 ıW D ıH C hıI ; II0 ida; (1.3) 4 @N where H D trI II and II0 D II .H =2/I . A version of this formula that is valid in any dimension has been first obtained in [11] by a direct computation of the variation of the regularized volume, see also [4] for a more mathematical exposition. A geometrical viewpoint adopted in this review (that originates in [27]) interprets this variational formula as a version of the Schläfli formula.

1.7 Maximization A first consequence of Equation (1.3), together with a simple integration by parts argument which is explained in Subsection 4.3, is that the only critical point of W over the space of metrics of fixed area in a conformal class on @M is attained for the unique metric of constant curvature. Conversely, metrics of constant curvature are critical points of the restriction of W to metrics of fixed area in a conformal class, and those critical points happen to be always non-degenerate minima.


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This leads to the definition of the renormalized volume of M (no choice of I involved) as the maximum of W .M; I /, obtained precisely when the metric at infinity, I , has constant curvature K D 1. We call W .M / this function. By the Ahlfors– Bers theorem [1], M is uniquely determined by its conformal class at infinity, so W can also be considered as a function from the Teichmüller space T@M of the boundary to R. The second step is to vary the conformal class on @M while considering only constant curvature metrics. Then it follows directly from (1.3) that Z 1 hI 0 ; II0 ida ; d W .I 0 / D 4 @M which means that d W is identified (up to the factor 1=4) with II0 , considered as a 1-form on T@M .

1.8 The renormalized volume and the Schwarzian derivative There is another way to give a geometric meaning to II0 . Given a convex co-compact hyperbolic metric on M , it defines on the boundary at infinity @1 M a complex projective structure , see [12]. Let c be the underlying complex structure, so that c is the complex structure at infinity of M . There is another special complex projective structure on @M , obtained by applying the Riemann uniformization theorem to .@M; c/, we call it 0 . The image by the developing map of 0 of each connected component of @M is a disk, so 0 is called the Fuchsian complex projective structure associated to c. Let be the map between .@M; 0 / and .@M; /. By construction is holomorphic, so that we can consider its Schwarzian derivative ./, which is a holomorphic quadratic differential on .@M; c/. This holomorphic quadratic differential can also be considered as the difference between the projective structures and 0 . We have: Proposition 1.1. II0 D Re..//.

1.9 The renormalized volume and Kleinian reciprocity Suppose now that M is a quasifuchsian manifold, that is, it is convex co-compact and homeomorphic to the product of a closed oriented surface S of genus at least 2 with an interval. Let T be the Teichmüller space of S, and let Tx be the Teichmüller space of S with the opposite orientation. Given two complex structures cC 2 T , c 2 Tx , there is by Bers’ Double Uniformization Theorem a unique hyperbolic metric on M such that the complex structure at infinity on the upper boundary @C M of M is cC , while the complex structure at infinity on the lower boundary @ M of M is c . Given a quasifuchsian manifold M , we can also consider the corresponding complex projective structure C on @C M , and the Fuchsian complex projective structure 0;C on @C M obtained by applying the Riemann uniformization theorem to the

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complex structure cC . This defines, through the Schwarzian derivative construction recalled above, a holomorphic quadratic differential qC on .@C M; cC /, and the real part of qC defines a cotangent vector ˇC .c ; cC / 2 TcC T . The same construction for @ M yields a cotangent vector ˇ .c ; cC / 2 Tc Tx . McMullen’s quasifuchsian reciprocity [30] gives a subtle relation between the ways the complex projective structures on the two boundary components vary when the complex structure at infinity changes. In the following statement we consider a fixed conformal structure cC and vary c (resp. fix c and vary cC ), ˇC is then a map from Tx to TcC T (resp. ˇ is a map from T to Tc Tx ). We call DˇC (resp. Dˇ ) the differential of this map. Theorem 1.2 (McMullen [30]). The tangent maps and

DˇC .; cC / W Tc Tx ! TcC T Dˇ .c ; / W TcC T ! Tc Tx

are adjoint. This can be stated in different terms using the standard cotangent bundle symplectic structure on T T@M , which we will call ! here. Given .c ; cC / 2 Tx T , we call ˇ.c ; cC / D .ˇ .c ; cC /; ˇC .c ; cC // 2 Tc Tx TcC T , so that ˇ.c ; cC / 2 T.c ;cC / T@M . Thus ˇ defines a section of T T@M . Theorem 1.2 is a direct consequence of the following simpler statement. Theorem 1.3. The image ˇ.T@M / is Lagrangian in .T T@M ; ! /. In this form, the statement extends as it is to a much more general setting of convex co-compact (or geometrically finite) hyperbolic 3-manifolds. The proof of Theorem 1.2 from Theorem 1.3 is straightforward, as is the proof of Theorem 1.3 from the first-order variation of the renormalized volume as described above. Both are done in Subsection 5.2. It is proved there that Theorem 1.3 is actually equivalent to Theorem 1.2 along with the following proposition, which uses the same notation as Theorem 1.2 and is another direct consequence of Theorem 1.3. Corollary 1.4. For fixed c , ˇC .c ; / is a closed 1-form on T . Part 5.3 describes a second proof of Theorem 1.3, based on the geometry of the convex core and on the fact that the grafting map is symplectic (Theorem 1.5 below). The idea here is to prove first that the data on the boundary of the convex core defines, as the representation varies, a Lagrangian submanifold of the cotangent bundle of Teichmüller space, understood here in terms of hyperbolic metrics and measured laminations. But the natural map sending this data on the boundary of the convex core to the corresponding data at infinity is symplectic, and Theorem 1.3 follows.


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In Part 5.4 we describe a third argument, due to Kerckhoff, which works in the setting of deformations of the holonomy representation of the fundamental group of M . This uses topological arguments, more precisely a long exact sequence and Poincaré duality between cohomology spaces with value in an sl.2; R/-bundle over M . The equivalence with Theorem 1.3 is made through a result of Kawai [23] connecting the Goldman symplectic form on the space of complex projective structures on a surface to the cotangent symplectic structure on the cotangent bundle of Teichmüller space.

1.10 The renormalized volume as a Kähler potential A rather direct consequence of McMullen’s quasifuchsian reciprocity, explained in Section 6, is that although ˇC .c ; cC / clearly depends on c , its exterior (antiholomorphic, because ˇC .c ; cC / is a holomorphic one-form on T ) differential does not depend on c . So this exterior differential can be computed explicitly in the simplest possible case – when c D cC , that is, in the neighborhood of a “Fuchsian” hyperbolic manifold. This leads to a key result on the renormalized volume, namely that it is a Kähler potential for the Weil–Petersson metric on Teichmüller space.

1.11 The relative volume of hyperbolic ends and the grafting map So far we have considered the renormalized volume in the context of hyperbolic convex co-compact 3-manifolds. Such manifolds come equipped with a complex projective structure on each boundary component, and the renormalized volume we have discussed can be said to be the Liouville functional for the corresponding projective structure, as discussed in the beginning of this section. Section 7 is centered on a notion of renormalized volume, or Liouville functional, that is defined for an arbitrary projective structure on a Riemann surface S. This has been developed in [28]. The main idea is to use a variant of the renormalized volume, called the relative volume of a hyperbolic end, and then use an analog of Theorem 1.3 to obtain results on the grafting map. Thus, consider a closed surface S of genus at least 2. We denote by MLS the space of measured geodesic laminations on S, see e.g. [10], and by CP S the space of complex projective structures (or CP 1 -structures) on S , see e.g. [12]. The grafting map Gr W TS MLS ! CP S was defined by Thurston, see e.g. [13], [12]. It can be briefly described as follows. Let m 2 TS be a hyperbolic metric, and let l 2 MLS be a measured lamination with support a disjoint union of closed curves c1 ; : : : ; cn , each with a positive weight w1 ; : : : ; wn . The “grafted metric” on S is obtained by cutting open .S; m/ along each of the ci and gluing in a flat strip of width equal to wi . Then Gr.m; l/ is the complex projective structure naturally associated to this metric. The map Gr extends by continuity to a map defined on all measured laminations (not only those supported by multicurves).

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Thurston showed that the grafting map is a homeomorphism. His proof, although quite subtle, relies on a simple geometric idea, which is simpler to explain for a complex projective structure 2 CP S whose developing map dev W Sz ! CP 1 is injective. In this case, the boundary of the convex hull in H 3 of CP 1 n dev.Sz/ – here CP 1 is identified with the boundary at infinity of H 3 – is a convex pleated surface, on which 1 .S/ acts properly discontinuously. The quotient surface is endowed with a hyperbolic metric (its induced metric) and a measured lamination (its pleating lamination). This gives an element .m; l/ 2 TS MLS , which is the inverse image of under the grafting map. In this picture – which can be extended to the case where the developing map of is not injective – 1 .S/ acts properly discontinuously on the connected compoz of the complement in H 3 of the convex hull of nent bounded at infinity by dev.S/ 1 z CP n dev.S/, and the quotient is a hyperbolic end. In Section 7 we explain how to define the relative volume of such a hyperbolic end, and show that it satisfies a simple first-order variation formula, involving both a term “at infinity” similar to the one which we already mentioned for the renormalized volume, and a term on the “compact” boundary, involving the hyperbolic metric and the measured pleating lamination, which is very close to a Schläfli-type formula for the convex core of a quasifuchsian manifold, discovered by Bonahon [9]. There is a natural identification between TS MLS and the cotangent space T TS , obtained by considering the differential of the length of a measured lamination as a cotangent vector. Using this map, the grafting map can be considered as a map Gr W T TS ! CP S , and both sides are naturally symplectic manifolds (CP S is actually a complex symplectic manifold, but we consider here only the real part of its complex symplectic structure). It is proved in [28] – using mostly tools from Bonahon’s work [8], [9] – that this map is C 1 smooth (it is however not C 2 ). Theorem 1.5. .1=2/Gr is symplectic. The proof is a direct consequence of the first-order variation formula for the relative volume of hyperbolic ends, similar to the proof of Theorem 1.2 that follows from the first-order variations of the renormalized volume. Acknowledgements. We are grateful to Steve Kerckhoff for interesting conversations and for allowing us to present here the content of part 5.4. We would also like to thank Brice Loustau for pointing out an error in a previous version of this text, and for stimulating conversations on the questions considered here. The first author was supported by an EPSRC Advanced fellowship. The second author was partially supported by the ANR programs Repsurf (ANR-06-BLAN-0311) and ETTT (ANR09-BLAN-0116-01).

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2 Two definitions of the renormalized volume 2.1 The renormalized volume As mentioned in the introduction, the renormalized volume was motivated by physical considerations. This notion was first envisaged and used in the more general context of conformally compact (in particular Einstein) manifolds in an arbitrary number of dimensions, and only later specialized to the 3-dimensional setting considered here. In the more general case its definition uses a foliation of a particular type close to infinity (associated to a “defining function” of the boundary). The “canonical” renormalized volume independent of which foliation is used is then either the constant term (in even dimensions) or the logarithmic term (in odd dimensions) in the asymptotic expansion of the volume in terms of the parameter of the foliation. In this review, however, we are interested in the constant term in this asymptotic expansion for the odd-dimensional (3d) manifolds. This quantity depends on the choice of a metric in the conformal class at infinity. The higher-dimensional applications can be found e.g. in [16], [19], [4], see also [18], [11], while this review concentrates on a simple(r), but still extremely rich case of 3-dimensional manifolds.

2.2 The renormalized volume via equidistant foliations The limiting procedure via which the volume is defined can be somewhat de-mystified by considering for regularization a family of surfaces equidistant to a given one, following an idea already used by C. Epstein [33] (and more recently put to use in [26]). Thus, the main idea is to obtain the renormalized volume by taking a convex domain N M , and compute the renormalized volume of M with respect to N as X 2.gi 1/ (2.1) VR .M; N / D V .N / C lim V .@N; @N / .1=2/A.@N / !1

i

where V .@N; @N / is the volume between the boundary @N of the domain N and the surface @N located a distance from @N . The quantity A.@N / is the area of the surface @N , the sum in the last term is taken over all boundary components of M and gi are the genera of these boundary components. The convexity of the domain N ensures that the equidistant surfaces @N exist all the way to infinity. This ensures that the limit ! 1 can be taken. This limit exists and can be computed in terms of the volume of N , see below for the corresponding expressions. The limiting procedure used in [24], [41] is an example of the limiting procedure described above, for the Epstein surfaces [14] used in these references are equidistant.

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2.3 Renormalized volume as the W -volume The following formula for the renormalized volume (2.1) can be shown by an explicit (simple) computation: X VR .M; N / D W .N / .gi 1/; i

where the sum in the last term is taken over all the boundary components. Here the W -volume is defined as Z 1 W .N / WD V .N / H da: (2.2) 4 @N This formula for VR is a special case of a formula found by C. Epstein [33] for the renormalized volume of hyperbolic manifolds in any dimension. Thus, the renormalized volume of M with respect to N is, apart from an uninteresting term given by a multiple of the Euler characteristic of the boundary, just the W -volume of the domain N . It then makes sense to study this geometrical W -volume instead. Note already that W .N / is not equal to the Einstein–Hilbert functional of N with its usual boundary term, it differs from it in the coefficient of the boundary term.

2.4 Self-duality One of the interesting properties of the W -volume is that it is self-dual. Thus, we recall that the Einstein–Hilbert functional Z 1 IEH .N / WD V .N / H da (2.3) 2 @N for a compact domain N H 3 of hyperbolic space (note a different numerical factor in front of the second term) is nothing but the dual volume. Indeed, we recall that there is a duality between objects in H 3 and objects in dS3 , the .2 C 1/-dimensional de Sitter space. Under this duality geodesic planes in H 3 are dual to points in dS3 , etc. This duality between domains in the two spaces is easiest to visualize for convex polyhedra (see [34]), but the duality works for general domains as well. The fact that (2.3) is the volume of the dual domain is a simple consequence of the Schläfli formula, see below. Thus, we can write Z 1 V .N / D V .N / D V .N / H da 2 @N for the volume of the dual domain. This immediately shows that Z 1 V .N / C V .N / : W .N / D V .N / Hda D 4 @N 2


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Thus, the W -volume is self-dual in that this quantity for N is equal to this quantity for the dual domain N : W .N / D W .N /.

2.5 The W -volume and the Chern–Simons formulation An interesting remark is that there is a very simple expression for the W -volume in terms of the so-called Chern–Simons formulation of 2+1 gravity [45]. Let us briefly review this formulation. In the so-called first order formalism for gravity the independent variables are not components of the spacetime metric but instead the triads and the spin connection. Thus, for Riemannian signature gravity in 3 spacetime dimensions let us introduce a collection of 3 one-forms i , i D 1; 2; 3, such that P i 2 the spacetime interval can be written as ds D i ˝ i . We Pcan now construct from i an su.2/ Lie algebra-valued one-form by taking D i i i i , where the i are the 2 2 anti-symmetric Pauli matrices i j D ı ij Id C i ij k k . Using the Lie algebra-valued form one can write the metric as ds 2 D .1=2/Tr. ˝ /. The Einstein–Hilbert action as a functional of the metric g is given by Z Z 1 1 IEH Œg D dv .R C 2/ da H; (2.4) 4 M 2 @M where R is the Ricci scalar of g, H is the mean curvature of the boundary, and dv; da are the volume and area forms on M and @M respectively. When evaluated on a constant curvature metric with R D 6 the Einstein–Hilbert action reduces to (2.3). The functional (2.4) can be re-written in a very simple form in terms of by introducing a spin connection !, which is locally an su.2/-valued one-form. The action is then Z 1Z 1 1 IEH Œ ; ! D Tr ^ f .!/ ^ ^ C Tr. ^ !/: (2.5) 2 M 12 2 @M Here f .w/ D d! C ! ^ ! is the curvature of the spin connection !. When one varies this action with respect to ! one obtains the equation d! D 0, where d! is the covariant derivative with respect to the connection !. This equation can be solved for ! in terms of the derivatives of . Once one substitutes the solution back into the action one gets (2.4). In contrast, the linear combination of the two terms on the right-hand side of (2.2) that plays the role of the renormalized volume is obtained by evaluating on the constant curvature metric the following action: Z Z 1 1 IW Œg D dv .R C 2/ da H: (2.6) 4 M 4 @M This can be written in terms of the tetrad and the spin connection forms as follows: Z 1Z 1 1 IW Œ ; ! D Tr ^ f .!/ ^ ^ C Tr. ^ !/: (2.7) 2 M 12 4 @M

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One then notes that this is precisely the combination that appears in the Chern– Simons formulation. Indeed, let us introduce the Chern–Simons action of an suC .2/ connection A via Z 1 2 (2.8) Tr A ^ dA C A ^ A ^ A : ICS ŒA D 4i M 3 Now, defining the two suC .2/ connections i i A D ! C ; AN D ! 2 2 it is not hard to see that (2.7) is given by N IW D ICS ŒA ICS ŒA;

(2.9)

(2.10)

with precisely the right boundary term that comes from having to integrate by parts. In contrast, to obtain via the Chern–Simons formulation the combination (2.3) one N For more has to add a separate boundary term that is constructed from both A, A. details on the argument presented the reader is referred to [25], see formula (3.7) of this reference as well as the related discussion. It would be of interest to understand the relation, if any, between the self-duality of the W -volume and the fact that it has such a simple expression in the Chern–Simons formulation.

3 Description “from infinity” 3.1 The metric at infinity In this section we switch from a description of the renormalized volume from the boundary of a convex subset to the boundary at infinity of M . This description from infinity is remarkably similar to the previous one from the boundary of a convex subset. Lemma 3.1. Let M be a convex co-compact hyperbolic 3-manifold, and let N M be compact and “strongly” convex with smooth boundary. Let S be the equidistant surfaces from @N . The induced metric on S is asymptotic, as ! 1, to .1=2/e 2 I , where I D .1=2/.I C 2II C III / is defined on @N . Proof. Follows from the following lemma. Lemma 3.2. Let S be a surface in H 3 , with bounded principal curvatures, and let I , B be the first fundamental form and the shape operator of S correspondingly. Let S be the surface at distance from S . Then, for sufficiently small the induced metric on S is I .x; y/ D I .cosh./E C sinh./B/ x; .cosh./E C sinh./B/ y : (3.1) Here E is the identity operator.


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Note that this lemma also holds for a surface S in any hyperbolic 3-manifold M , not necessarily H 3 . We also note that when the surface S is convex, then the expression (3.1) gives the induced metric on any surface > 0, where increases in the convex direction. A proof of this lemma can be found in, e.g., [26]. It is the metric I that will play such a central role in what follows, so we would like to state some of its properties. Lemma 3.3. The curvature of I is K WD

2K : 1 C H C Ke

(3.2)

Proof. The Levi-Civita connection of I is given, in terms of the Levi-Civita connection r of I , by rx y D .E C B/1 rx ..E C B/y/: This follows from checking the three points in the definition of the Levi-Civita connection of a metric: • r is a connection, • r is compatible with I , • it is torsion-free (this follows from the fact that E C B verifies the Codazzi equation: .rx .E C B//y D .ry .E C B//x). Let .e1 ; e2 / be an orthonormal moving frame on S for I , and let ˇ be its connection 1-form, i.e., rx e1 D ˇ.x/e2 ; rx e2 D ˇ.x/e1 : Then the curvature of I p is defined as dˇ D Kda. Now let .e1 ; e2 / WD 2..E C B/1 e1 ; .E C B/1 e2 /; clearly it is an orthonormal moving frame for I . Moreover the above expression of r shows that its connection 1-form is also ˇ. It follows that Kda D dˇ D K da , so that K D K

2K da K D D : da .1=2/ det.E C B/ 1 C H C Ke

We note that the metric I is defined for any surface S M . However, it might have singularities (even when the surface S is smooth) unless S is strictly horospherically convex, i.e., its principal curvatures are less than 1 (which implies that it remains on the concave side of the tangent horosphere at each point). If S is a strictly horospherically convex surface S embedded in a hyperbolic end of M then the metric I is guaranteed to be in the conformal class of the (conformal) boundary at infinity of M . For a general surface S the “asymptotic” metric is not directly related to the conformal infinity, and in particular, it does not have to be in the conformal class of the boundary.

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3.2 Second fundamental form at infinity We have already defined the metric “at infinity”. Let us now add to this a definition of what can be called the second fundamental form at infinity. Definition 3.1. Given a surface S with first, second and third fundamental forms I; II and III , we define the first and second fundamental forms “at infinity” as follows: 1 1 .I C 2II C III / D .I C II /I 1 .I C II / 2 2 1 D I..E C B/; .E C B//; 2 1 1 1 II D .I III / D .I C II /I 1 .I II / D I..E C B/; .E B//: 2 2 2 I D

(3.3)

It is then natural to define B WD .I /1 II D .E C B/1 .E B/;

(3.4)

and III WD I .B ; B / D I..E B/; .E B//: An interesting point is that I ; II and III determine the full asymptotic development of the metric close to infinity: the induced metrics I on the surfaces S are given by Equation (3.6) below. This extends Lemma 3.1, and can be considered as an analog of Equation (3.1). Note that, for a surface which has principal curvatures strictly bounded between 1 and 1, III is also a smooth metric and its conformal class corresponds to that on the other component of the boundary at infinity. This is a simple consequence of Lemma 3.2 and the fact that when the principal curvatures are strictly bounded between 1; 1 the foliation by surfaces equidistant to S extends all the way through the manifold M . Such manifolds were called almost Fuchsian in our work [26]. As before, these definitions make sense for any surface, but it is only for a convex surface (or more generally for a horospherically convex surface) that the fundamental forms so introduced are guaranteed to have something to do with the actual conformal infinity of the space.

3.3 The Gauss and Codazzi equations at infinity We also define H WD tr.B /. The Gauss equation for “usual” surfaces in H 3 is replaced by a slightly twisted version. Lemma 3.4. H D K : the mean curvature at infinity is equal to minus the curvature of I .


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Proof. By definition, H D tr..E C B/1 .E B//. An elementary computation (for instance based on the eigenvalues of B) shows that H D

2 2 det.B/ : 1 C tr.B/ C det.B/

But we have seen (as Equation (3.2)) that K D 2K=.1 C H C Ke /. The result follows because, by the Gauss equation, K D 1 C det.B/. However, the “usual” Codazzi equation holds at infinity.

Lemma 3.5. d r B D 0. Proof. Let u, v be vector fields on @1 M . Then it follows from the expression of r found above that

.d r B /.x; y/ D rx .B y/ ry .B x/ B Œx; y D .E C B/1 rx ..E C B/B y/ .E C B/1 ry ..E C B/B x/ B Œx; y D .E C B/1 rx ..E B/y/ .E C B/1 ry ..E B/x/ .E C B/1 .E B/Œx; y D .E C B/1 .d r .E B//.x; y/ D 0:

3.4 Inverse transformations The transformation I; II ! I ; II is invertible. The inverse is given explicitly, and the inversion formula exhibits a remarkable symmetry. Lemma 3.6. Given I , II , the fundamental forms I , II such that (3.3) holds are obtained as 1 1 I D .I C II /.I /1 .I C II / D I ..E C B /; .E C B //; 2 2 (3.5) 1 1 1 II D .I C II /.I / .I II / D I ..E C B /; .E B //: 2 2 Moreover, B D .E C B /1 .E B /: Having an expression for the fundamental forms of a surface in terms of the ones at infinity, one can re-write the metric of Lemma 3.2 induced on surfaces equidistant to S in terms of I , II .

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Lemma 3.7. The metric (3.1) induced on the surfaces equidistant to S can be rewritten in terms of the fundamental forms “at infinity” as I D

1 1 2 e I C II C e 2 III : 2 2

(3.6)

This lemma shows the significance of II as being the constant term of the metric. This lemma also shows clearly that when the equidistant foliation extends all the way through M (i.e. when the principal curvatures on S are in .1; 1/), the conformal structure at the second boundary component of M is that of III D II .I /1 II . Thus, in this particular case of almost Fuchsian manifolds, the knowledge of I on both boundary components of M is equivalent to the knowledge of I ; II near either component. In other words, II is determined by I . This statement is, of course, more general and works for manifolds other than almost Fuchsian ones.

3.5 Fundamental Theorem of surface theory “from infinity” Let us now recall that the Fundamental Theorem of surface theory states that given I , II on S satisfying the Gauss and Codazzi equations, there is a unique embedding of S into the hyperbolic space with induced metric and second fundamental form equal to I and to II . Then (3.1) gives an expression for the metric on equidistant surfaces to S, and thus describes a hyperbolic manifold M in which S is embedded, in some neighborhood of S. It would be possible to state a similar result for hyperbolic ends, uniquely determined by I and II at infinity. But there is also an analogous theorem, based on a classical result of Bers [5], in which the first (and only the first) form at infinity is used. This can be compared with arguments used in [36], where a corresponding second fundamental form and the Gauss and Codazzi equations “at infinity” were introduced. Theorem 3.8. Given a convex co-compact 3-manifold M , and a metric I (on all the boundary components of M ) in the conformal class of the boundary, there is a unique foliation of each end of M by convex equidistant surfaces S M such that .1=2/.I C 2II C III / D e 2 I , where I , II , III are the fundamental forms of S . Remark 3.2. Note that one does not need to specify II . The first fundamental form I (but on all the boundary components) is sufficient. Proof. The surfaces in question can be given explicitly as an embedding of the universal cover SQ of S into the hyperbolic space. Thus, let .; y/; > 0; y 2 C be the usual upper half-space model coordinates of H 3 . Let us write the metric at infinity as I D e jdzj2 ;

(3.7)


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where is the Liouville field covariant under the action of the Kleinian group giving M on S 2 . The surfaces are given by the following set of maps: Eps W S 2 ! H 3 ; z 7! .; y/ (here Eps stands for Epstein, who described these surfaces in [14]): p =2 2e e D ; (3.8) 1 C .1=2/e 2 e jz j2 e 2 e y D z C zN : 1 C .1=2/e 2 e jz j2 As is shown by an explicit computation, the metric induced on the surfaces S is given by (3.6) with II D

1 . dz 2 C N d zN 2 / C z zN dzd z; N 2 1 D zz .z /2 : 2

(3.9) (3.10)

Thus, we see that II is determined by the conformal factor in (3.7). Remark 3.3. This theorem implies that the renormalized volume only depends on I . Indeed, the foliation .S / of the ends does depend only on I , and this foliation can be used for regularization and subtraction procedure. Then the fact that the W -volume is essentially the renormalized volume implies that the W -volume is a functional of I only. In the next section we will find a formula for the first variation of this functional. Corollary 3.9. If the principal curvatures at infinity (eigenvalues of B ) are positive the map Eps is a homeomorphism onto its image for any . Proof. We first note that the map Eps is not always a homeomorphism, and the surfaces S are not necessarily convex, but for sufficiently large both things are true. A condition that guarantees that Eps is a homeomorphism for any is stated above. This condition can be obtained from the requirement that the principal curvatures of surfaces S are in Œ1; 1. Let us consider the surface S WD SD0 the first and second fundamental forms of which are given by (3.5) (this immediately follows from (3.6)). The shape operator of this surface is then given by B D .E C B /1 .E B /. It is then clear that the principal curvatures of S are given by ki D .1 ki /=.1 C ki /, where the ki are the “principal curvatures” (eigenvalues) of B . The latter are easily shown to be given by p D e z zN ˙ N : (3.11) k1;2 It is now easy to see that the condition k1;2 2 .1; 1/ is equivalent to the condition > 0. This is a necessary and sufficient condition for the foliation by surfaces S to k1;2 extend throughout M . If this condition is satisfied the map Eps is a homeomorphism for any .

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Interestingly, this condition makes sense not only in the quasifuchsian situation but is more general. Thus, for example, it applies to Schottky manifolds. But for Schottky manifolds, with their single boundary component, the foliation by equidistant surfaces S cannot be smooth for arbitrary . It is clear that surfaces must develop singularities for some value of . We therefore get an interesting corollary: Corollary 3.10. There is no Liouville field on C invariant under a Schottky group such that z zN is greater than jzz .1=2/z2 j everywhere on C. Proof. Indeed, if such a Liouville field existed, we could have used it to construct a smooth equidistant foliation for arbitrary values of , but this is impossible. A similar statement holds for a Kleinian group with more than two components of the domain of discontinuity.

4 The Schläfli formula “from infinity” In this section we obtain a formula for the variation of the renormalized volume.

4.1 The Schläfli formula for hyperbolic polyhedra Recall first the classical Schläfli formula (see e.g. [31]), which is a good motivation for what follows. Consider a hyperbolic polyhedron P . Under a first-order deformation of P , the first-order variation of the volume of P is given by 1X Le d e : (4.1) dV D 2 e Here the sum is over the edges of P , Le is the length of the edge e, and e is its exterior dihedral angle. There is also an interesting “dual” Schläfli formula. Let 1X Le e V DV 2 e be the dual volume of P ; then, still under a first-order deformation of P , 1X e dLe : dV D 2 e This follows from the Schläfli formula (4.1) by an elementary computation.

(4.2)

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4.2 The Schläfli formula for hyperbolic manifolds with boundary As we have seen in the previous sections, the renormalized volume of a convex cocompact hyperbolic 3-manifold M can be expressed as the W -volume of any convex domain N M . The W -volume is equal to the volume of N minus the quarter of the integral of the mean curvature over the boundary of N . Let us consider what happens if one changes the metric in M . As was shown in [35], the following formula for the variation of the volume holds Z 1 (4.3) ıH C hıI; II i da: 2ıV .N / D 2 @N Here H is the trace of the shape operator B D I 1 II , and the expression hA; Bi stands for tr.I 1 AI 1 B/. We can use this to get the following expression for the variation of the W -volume: Z Z Z 1 1 1 1 ıW .N / D ıHda H ı.da/; ıH C hıI; II i da 2 @N 2 4 @N 4 @N so that 1 ıW .N / D 4

Z

D H E ıH C ıI; II I da: 2

@N

(4.4)

To get the last equality we have used the obvious equality 1 1 tr.I 1 ıI / D hıI; I ida: 2 2 The formula (4.4) can be further modified using ıda D

ıH D ı.tr.I 1 II // D tr.I 1 .ıI /I 1 II / C tr.I 1 ıII / D hıI; II i C hI; ıII i: We get ıW .N / D

1 4

Z

D

ıII

@N

E H ıI; I da: 2

(4.5)

(4.6)

(4.7)

It is this formula that will be our starting point for transformations to express the variation in terms of the data at infinity.

4.3 Parametrization by the data at infinity Let us now recall that given the data I , II on the boundary of N one can introduce the first and second fundamental forms “at infinity” via (3.3). Conversely, knowing the fundamental forms I ; II “at infinity” one can recover the fundamental forms on @N via (3.5). Our aim is to rewrite the variation (4.7) of the W -volume in terms of the variations of the forms I ; II .

802


Lemma 4.1. The first-order variation of W can be expressed as Z D H E 1 ıW .N / D ıI ; I da : ıII 4 @N 2

(4.8)

A proof can be found in [27]; it follows from a direct computation based on the formulas expressing I , II in terms of I , II . This formula could be compared to a more general formula applicable in any dimension given in [11] and reviewed in [4]. A similar formula for the case of 4dimensional manifolds is given by Anderson in [3], and more recently by Albin [2] in higher dimensions. Our derivation here is different from that in the cited references, for we interpret the variational formula as a version of the Schläfli formula. We also note that the situation is simpler for even-dimensional manifolds, since the renormalized volume is then canonically defined, while in odd dimension it depends on the choice of a metric in the conformal class at infinity. In odd dimension there is another, canonically defined “renormalized volume”, namely the logarithmic term in the asymptotic expansion of the volume as a function of the parameter of an equidistant foliation. This quantity is different from the one used here (defined in (2.1)) which is the constant term in the same asymptotic expansion. Formula (4.8) looks very much like the original formula (4.7), except for the minus sign and the fact that the quantities at infinity are used. The fact that we have got the same variational formula as in terms of the data on @N is not too surprising. Indeed, the variational formula (4.8) was obtained from (4.7) by applying the transformation (3.5). As it is clear from (3.3), this transformation applied twice gives the identity map. In view of this, it is hard to think of any other possibility for the variational formula in terms of ıI , ıII except being given by the same expression (4.7), maybe with a different sign. This is exactly what we see in (4.8). There is another expression of the first-order variation of W , dual to (4.4), which will be useful below. Corollary 4.2. The first-order variation of W can also be expressed as Z 1 ıH C hıI ; II0 ida ; ıW D 4 @N where II0 is the traceless part (for I ) of II .

4.4 Conformal variations of the metric at infinity We can now use Corollary 4.2 to show that, when varying the W -volume with the area of the boundary defined by the I metric kept fixed, the variational principle forces the metric I to have constant negative curvature. The variations we consider here do not change the conformal structure of the metric I , and thus do not change the manifold M . Geometrically they correspond to small movements of the surface @N inside the fixed manifold M . Thus, let us consider a conformal deformation of the


803

metric I of the form ıI D 2uI , where u is some function on @N . Clearly for such variations hıI ; II0 i D 0, precisely because II0 is traceless. Let us consider the following functional Z F .N / D W .N / da (4.9) 4 @N appropriate for finding an extremum of the W -volume with the area computed using the metric I kept fixed. The first variation of this functional gives, using Corollary 4.2: Z Z Z Z 1 1 ıF D .ıH /da 4uda D .ıK /da 4uda : 4 @N 4 @N 4 @N 4 @N Z

But

Z

K da D

ı @N

.ıK / C 4uK da D 0

@N

by the Gauss-Bonnet formula, so that Z ıF D .uK u /da : @N

It follows that critical points of F are characterized by the fact that K D . It is not hard to compute the second variation and show that the critical points of F are local maxima, see [27] for details. Remark 4.1. As we have already discussed, the renormalized volume W .N / is actually a functional of metrics I on all the boundary components of M . We have just established that this functional has an extremum, for variations that keep the total area of the boundary components fixed, at the constant curvature metric I . However, this is precisely the defining property of the Liouville functional we have discussed in the Introduction. This establishes the renormalized volume – Liouville action functional relation. Moreover, one can turn the argument around and use the renormalized volume W .N / (as a functional of the metrics I on all the boundary components) as a definition of the Liouville functional. This point of view explains why there is not one, but a whole set of Liouville functionals – depending on which hyperbolic three-manifold is used – and it also explains why it is so hard to define the Liouville functional in intrinsically 2-dimensional terms – because it is in fact a 3-dimensional quantity.

4.5 The renormalized volume as a function on Teichmüller space Let us now consider the renormalized volume as a function over the Teichmüller space of @N ; in other terms, for each conformal class on @N , we consider the extremum of W over metrics of given area within this conformal class. We have just seen that this extremum is obtained at the (unique) constant curvature metric. The main goal here is to recover by simple differential geometric methods important results of Takhtajan and

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Zograf [39], Takhtajan and Teo [41] – showing that the renormalized volume provides a Kähler potential for the Weil–Petersson metric. So the “volume” that we consider here is now defined as follows. Definition 4.2. Let g be a convex co-compact hyperbolic metric on M , and let c 2 T@M be the conformal structure induced on @1 M . We call WM .c/ the value of W on the equidistant foliation of M near infinity for which I has constant curvature 1. In other terms, by the results obtained above, WM .c/ is the maximum of W over the metrics at infinity which have the same area as a hyperbolic metric, for each boundary component of M . Throughout this section the metric at infinity I that we consider is the hyperbolic metric, while the second fundamental form at infinity, II , is uniquely determined by the choice that I is hyperbolic. Its traceless part is denoted by II0 .

4.6 The second fundamental form at infinity as the real part of a holomorphic quadratic differential It is interesting to remark that, in the context considered here – when I has constant curvature – the second fundamental form at infinity has a complex interpretation. This can be compared with the same phenomenon, discovered by Hopf [21], for the second fundamental form of constant mean curvature surfaces in 3-dimensional constant curvature spaces. Lemma 4.3. When K is constant, II0 is the real part of a holomorphic quadratic differential (for the complex structure associated to I ) on @1 M . This holomorphic quadratic differential is given explicitly by (3.10). Proof. By definition II0 is traceless, which means that it is at each point the real part of a quadratic differential: II0 D Re.h/. Moreover, we have seen in Remark 3.5 that B satisfies the Codazzi equation, d r B D 0. It follows as for constant mean curvature surfaces (see e.g. [26]) that h is holomorphic relative to the complex structure of I .

4.7 The second fundamental form as a Schwarzian derivative The next step is to show that, for each boundary component @i M of M , II0i is just the real part of the Schwarzian derivative of a natural equivariant map from the hyperbolic plane (with its canonical complex projective structure) to @i M with its complex projective structure induced by the hyperbolic metric on M . In the terminology used is the difference between the complex projective structure at by McMullen [30], II0i infinity on @i M and the Fuchsian projective structure on @i M . To state this result, let us call F the “Fuchsian” complex projective structure on @i M , obtained by applying the Poincaré uniformization theorem to the conformal


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metric at infinity on @i M . The universal cover of @i M , with the complex projective structure lifted from F , is projectively equivalent to a disk in CP 1 . We also call QF the projective structure induced on @i M by the hyperbolic metric on M . Here “QF ” stands for quasifuchsian (while M is only supposed to be convex co-compact). This notation is used to keep close to the notation in [30]. The map W .@i M; F / ! .@i M; QF / is conformal but not projective between .@i M; F / and .@i M; QF /, so we can consider its Schwarzian derivative ./. Lemma 4.4. II0 D Re..//. A simple way to prove this assertion is to use the formula (3.10) for the holomorphic quadratic differential whose real part gives the traceless part of II . The Liouville field that enters into this formula can be simply expressed in terms of the conformal map from @i M to the hyperbolic plane. It is then a standard and simple computation to verify that is equal to the Schwarzian derivative of this map, see e.g. [39]. It is possible to reformulate the statement (4.4) slightly, setting i to be the restriction to @i M of ./. This notation is analogous to the notation used in [30], where the index i is useful to recall that this quantity is related to @i M . Then i is a quadratic holomorphic differential (QHD) on @i M , and, still using the notation in [30], the definition of i can be rephrased as i D QF F . The lemma D Re. i /. A geometric proof of this lemma is given in the can then be written as II0i appendix of [27]. Remark 4.3. Note that i can also be considered as a complex-valued 1-form on the Teichmüller space of @i M . Indeed, it is well known that the cotangent vectors to TS , where S is a Riemann surface, can be described as holomorphic quadratic differentials q on S. The pairing with a tangent vector (Beltrami differential ) is given by the integral of q over S . The complex structure on TS can then be described as follows: the image of the cotangent vector q under the action of the complex structure J is simply J.q/ D iq. Another, more geometric way to state the action of J is to note that it exchanges the horizontal and vertical trajectories of q. Thus, holomorphic quadratic differentials q on S are actually holomorphic 1-forms on TS .

4.8 The second fundamental form as the differential of WM There is another simple interpretation of the traceless part of the second fundamental form at infinity. Lemma 4.5. The differential d WM of the renormalized volume WM , as a 1-form over the Teichmüller space of @M , is equal to .1=4/II0 .

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Proof. This is another direct consequence of Corollary 4.2 because, as one varies I among hyperbolic metrics, H (which is equal to K ) remains equal to 1, so that ıH D 0. Corollary 4.6. i D 4@WM . Proof. This follows directly from the lemma, since we already know that i is a holomorphic differential. Remark 4.4. We would like to emphasize how much simpler is the proof given above than that given in [39], [41]. Unlike in these references, which obtain the above result on the gradient of WM via an involved computation using a reasonably complicated cohomology machinery, Corollary 4.2 implies this result in one line. This demonstrates the strength of the geometric method used here. Our proof can be immediately extended even to situations where the methods of [41] are inapplicable, such as manifolds with cone singularities. See more remarks on this case below. Lemma 4.5 and, in particular, its corollary above, is the key fact needed to demonstrate that the renormalized volume plays the role of the Kähler potential on Teichmüller space. The remainder of the proof is in part 6.3 below, after some considerations on quasifuchsian reciprocity, which are partly based on the results of this section and are needed in the proof.

5 Kleinian reciprocity 5.1 Statement Kleinian reciprocity, as defined by McMullen [30], is the extension of Theorem 1.3 to the more general setting of a geometrically finite hyperbolic 3-manifold M . Let GF .M / be the space of complete geometrically finite hyperbolic metrics on M . Each metric g 2 GF induces a complex projective structure .g/ on @M . Theorem 5.1. .GF / is Lagrangian in CP @M . As explained in the introduction and proved in Subsection 5.2.1 below, Theorem 1.2 is a direct corollary of this statement. The proof of Theorem 1.2 given in [30] can be described as analytic, as it takes place in the universal cover of M and uses the symmetry of a certain kernel related to the Beltrami problem. By contrast, the arguments considered here are mostly geometric and take place in M . We will describe here three (other) proofs of Theorem 5.1, corresponding to different ways to consider the space of complex projective structures CP .


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• When CP is considered in complex terms, and identified with T TC , the cotangent bundle of the space of complex structures on @M , Theorem 5.1 follows from the first variation of the renormalized volume, as explained in the introduction. This is the argument described in the introduction. • When CP is considered in hyperbolic terms, and identified with T TH , Theorem 5.1 follows from the dual Bonahon–Schläfli formula for the first variation of the dual volume of the convex core. The equivalence with the previous viewpoint is clear through Theorem 1.5. • When CP is considered as (a connected component of) a space of equivalence classes of representations of 1 .@M / into PSL.2; C/, Theorem 5.1 can be proved by a completely different argument, based on exact sequences and Poincaré duality, which was discovered (previously) by S. Kerckhoff. The equivalence with the complex or the hyperbolic viewpoint follows from the fact that the Goldman form on the space of representations of 1 .@M / into PSL.2; C/ is equal (up to multiplication by a constant) to the symplectic form obtained from the cotangent symplectic form on T T , as proved by Kawai [23]. We briefly describe these three arguments in the next subsections. We consider here for simplicity the case of quasifuchsian manifolds; however all arguments can be extended without difficulty to the more general context of geometrically finite hyperbolic 3-manifolds.

5.2 The first variation of the renormalized volume We give here the proofs announced in the introduction of quasifuchsian reciprocity from the first-order variation formula for the renormalized volume. 5.2.1 From Theorem 1.3 to Theorem 1.2. Suppose that Theorem 1.3 holds, and 0 0 ; 0/; .0; cC / of tangent vectors in Tc Tx TcC T . In addition to the consider a pair .c notation from Section 1.8, we introduce the notation B W T@M ! T T@M for the map to the total space of the bundle T T@M , corresponding to the section ˇ of T T@M . We use the notation D for the differential as in the introduction, and call r the Levi-Civita connection of the Weil–Petersson metric on the cotangent bundle of both T and Tx (we could use another connection). Thus r determines a connection on T@M D T Tx , and this connection defines an identification T .T T@M / ' T T@M T T@M . With this notation we have 0 0 0 0 hDˇC .c ; 0/; cC i hDˇ .0; cC /; c i 0 0 D h.rc0 ˇ ; DˇC .c ; 0//; .0; cC /i 0 0 0 ˇC /; .c ; 0/i h.Dˇ.0; cC /; rcC 0 0 D ! .B .c ; 0/; B .0; cC // D 0;

where the last equality follows from Theorem 1.3. This proves Theorem 1.2.

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Corollary 1.4 is also a direct consequence of Theorem 1.3. Indeed, for fixed c , ˇC .c ; T / is a Lagrangian submanifold in T T by Theorem 1.3. It is also the graph of a 1-form by definition, and it is well known (and can be checked by a direct computation) that a 1-form is closed if and only if its graph is Lagrangian. Conversely, Theorem 1.3 follows quite directly from Theorem 1.2 together with 0 2 TcC T . Then Corollary 1.4. To see this, let c 2 Tx ; cC 2 T ; v 2 Tc Tx ; vC ; vC the computation above, done backwards, shows that ! .B .v ; 0/; B .0; vC // D 0: In addition, 0 0 ! .B .0; vC /; B .0; vC // D h.DvC ˇ ; rvC ˇC /; .0; vC /i 0 ˇ ; rv 0 ˇC /; .0; vC /i h.DvC C 0 0 ˇC ; vC i D hrvC ˇC ; vC i hrvC 0 D dˇC .vC ; vC / D 0;

where the last equality comes from Corollary 1.4. Theorem 1.3 follows by linearity. 5.2.2 Proof of Theorem 1.3. Theorem 1.3 follows very directly from the basic properties of the renormalized volume WM as they are described above. Indeed we have seen that 1 ˇ D II0 D d WM : 4 So ˇ is closed. This argument extends as it is to the more general setting of Theorem 5.1.

5.3 The boundary of the convex core and the grafting map The second argument leading to quasifuchsian reciprocity is also based on hyperbolic geometry, and more specifically on the geometry of the convex core of quasifuchsian 3-manifolds. It rests on an extension of the classical Schläfli identity for convex cores of quasifuchsian manifolds, found by Bonahon [9], which leads to an analog of Theorem 5.1 where the renormalized volume is replaced by the volume of the convex core, and the cotangent bundle of Teichmüller space is considered in “hyperbolic” terms. 5.3.1 The convex core of quasifuchsian manifolds. Let M be a quasifuchsian hyperbolic 3-manifold. M contains a smallest non-empty geodesically convex subset, its convex core C.M /, which is compact and homeomorphic to M . The boundary of C.M / is therefore the disjoint union of two copies of a surface S of genus at least 2, which we call the “upper” and “lower” boundary components of C.M /. Since


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C.M / is a minimal convex subset, it has no extreme points, so @C.M / is locally convex and ruled (there is a geodesic segment of the ambient manifold, contained in C.M /, containing each point). It follows that the induced metric m on @C.M / is hyperbolic (it has constant curvature 1), but @C.M / is pleated along a measured geodesic lamination l. More details can be found in [42]. 5.3.2 Measured laminations as cotangent vectors. When thinking of Teichmüller space in terms of hyperbolic metrics on surfaces, it is natural to associate its cotangent bundle with measured laminations, rather than with holomorphic quadratic differentials. The identification goes as follows. Let l 2 ML be a measured lamination, and let m 2 T be a hyperbolic metric, both on S. It is then possible (see [10]) to define the length of l for m, Lm .l/. This defines a smooth function L .l/ W T ! R0 : The differential dL .l/ at m is an element of Tm T , and the map ML ! Tm T is a homeomorphism (see e.g. [28]). This construction defines an identification ı between T ML and T T . But T T has a cotangent symplectic structure, which we call !H here. It can be pulled back to T ML, where we still call it !H . It can be defined quite explicitly in terms of the intersection form on ML, see [38]. 5.3.3 A Lagrangian submanifold. Given a quasifuchsian metric g 2 QF , we can consider the induced metrics on the upper and lower boundary components of the convex core, mC ; m 2 T , and the corresponding measured bending laminations, lC ; l 2 ML. So we have two points .mC ; lC /; .m ; l / 2 T ML. This defines a map H W QF ! T T@M . Theorem 5.2. H.QF / is a Lagrangian submanifold of .T T@M ; !H /. The main ideas of the proof are explained in the next subsection. Theorem 1.3 directly follows from this result and from Theorem 1.5, according to which the grafting map is symplectic (up to a constant). 5.3.4 The Bonahon–Schläfli formula for the volume of the convex core. The convex core of a quasifuchsian manifold is, in some ways, reminiscent of a convex polyhedron. The main differences are: it has no vertices and edges are replaced by a measured lamination. This gives, in a sense, a much richer structure. Bonahon [8] has extended the Schläfli formula recalled in Subsection 4.1 to this setting as follows. Let M be a convex co-compact hyperbolic manifold (for instance, a quasifuchsian manifold), let be the induced metric on the boundary of the convex core, and let be its measured bending lamination. By a “first-order variation” of M we mean a first-order variation of the representation of the fundamental group of M . Bonahon shows that the first-order variation of under a first-order variation of M is

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described by a transverse Hölder distribution 0 , and there is a well-defined notion of length of such transverse Hölder distributions. This leads to a version of the Schläfli formula. Lemma 5.3 (The Bonahon–Schläfli formula [8]). The first-order variation of the volume VC of the convex core of M , under a first-order variation of M , is given by d VC D

1 L . 0 /: 2

Here 0 is the first-order variation of the measured bending lamination, which is a Hölder cocycle so that its length for can be defined, see [6], [7], [8], [9]. 5.3.5 The dual volume. Just as for polyhedra above, we define the dual volume of the convex core of M as 1 VC D VC L . /: 2 Lemma 5.4 (The dual Bonahon–Schläfli formula). The first-order variation of V under a first-order variation of M is given by 1 d VC D L0 . /: 2 This formula has a very simple interpretation in terms of the geometry of Teichmüller space: up to the factor 1=2, d V is equal to the pull-back by ı of the Liouville form of the cotangent bundle T T . Note also that this formula can be understood in an elementary way, without reference to a transverse Hölder distribution: the measured lamination is fixed, and only the hyperbolic metric varies. The proof can be found in [28], it is based on Lemma 5.3. Theorem 5.2 is a direct consequence of Lemma 5.4: since d VC coincides with the Liouville form of T T@M on H.QF /, it follows immediately that H.QF / is Lagrangian for the symplectic form !H on T T@M .

5.4 Deformations of representations and Poincaré duality Steve Kerckhoff found another (unpublished) proof of Theorem 5.1 based on topological ideas and in particular on his earlier work with Craig Hodgson [20]. This proof works in the context of deformations of PSL.2; C/ representations, the symplectic form on CP is here the Goldman symplectic form !G on CP . Recall (from [15]) that given a complex projective structure on @M , its holonomy representation is a morphism from 1 .@M / to PSL.2; C/. The tangent space to CP at is then naturally identified with the cohomology space H 1 .@M I E/, where E is an sl.2; C/ bundle over @M naturally associated to – it is the bundle of local projective vector fields for on @M . Given two cohomology classes u; v 2 H 1 .@M I E/, one can

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consider their cup product u Y v 2 H 2 .@M; C/, and integrate it over @M . This defines the Goldman symplectic form !G W H 1 .@M I E/ H 1 .@M I E/ ! C: Kawai [23] proved that this symplectic form is equal, up to a constant, to the canonical symplectic form on CP @M , obtained for instance by identification of CP @M with T T@M through the Schwarzian derivative, see [12]. Kerckhoff’s proof is based on the long exact sequence in cohomology for the pair .M; @M /: ˇ

˛

! H 1 .M; @M I E/ ! H 1 .M I E/ ! H 1 .@M I E/ ! H 2 .M; @M I E/ ! : Here ˛ is restriction of the deformation from M to @M . Note that the map H 1 .M; @M I E/ ! H 1 .M I E/ is zero, since any non-trivial deformation of the hyperbolic structure on M induces a non-zero deformation of the complex projective structure on the boundary (this follows for instance from the Ahlfors–Bers theorem). As a consequence, ˛ is injective. Part of the long exact sequence above can be extended as the commutative diagram below, taken from [20], p. 42: H 1 .M I E/ H 2 .M; @M I E/

˛

ˇ

/ H 1 .@M I E/ / H 1 .@M I E/

ˇ

˛

/ H 2 .M; @M I E/ / H 1 .M I E/ .

Here the vertical arrows are the Poincaré duality maps. Recall that Poincaré duality can be defined using the cup product as above. In particular, the Poincaré dual u of a form u, for instance in H 1 .@M I E/, is such that, for all v 2 H 1 .@M I E/, !G .u; v/ D hu ; vi. Let u; v 2 H 1 .M I E/. It follows from the above commutative diagram that !G .˛.u/; ˛.v// D h˛.u/; ˛.v/ i D h˛.u/; ˇ .v /i D hˇ B ˛.u/; v i D 0: It also follows from the above exact sequence (or from the upper part of the commutative diagram and the fact that ˛ is injective) that dim H 1 .@M I E/ D 2 dim H 1 .M I E/ (see [20] for the details). So ˛.H 1 .M I E// is Lagrangian in H 1 .@M I E/, and this, along with Kawai’s result [23], provides yet another proof of Theorem 5.1.

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6 The renormalized volume as a Kähler potential In this section we again consider the setting of quasifuchsian manifolds and recover the result of Takhtajan and Teo [41]: the renormalized volume WM with c fixed is a Kähler potential for the Weil–Petersson metric on T@C M .

6.1 Notation To simplify notation a little, we set c WD .c ; /, so that the real part of c is ˇC .c ; /. Since we already know that c D 4@WM , we only need to prove that N c / D 2!WP , where !WP is the Kähler form of the Weil–Petersson metric on @.i T@C M . An important part of the argument is that d c , as a 2-form on T@C M , does not depend on c . This appears as Theorem 7.2 in McMullen’s paper [30]. We include a proof for completeness, following the proof given in [30]. Proposition 6.1. The differential d c , considered as a complex-valued 2-form on T@C M , does not depend on c . Proof. Let v 2 Tc T@ M , we want to show that the corresponding first-order variation Dv .d c / of d c vanishes. This will follow from the fact that the first-order variation of c corresponding to v , Dv c , is the differential of a function defined on T@C M , namely the function fv defined by fv .cC / D h c .cC /; v i; where h; i is the duality pairing. The fact that Dv c D dfv can be proved by evaluating both sides on a vector vC 2 TcC T@C M and using the quasifuchsian reciprocity. Since c is a complex 1-form with real part equal to ˇC .c ; /, Theorem 1.2 indicates that hDv c ; vC i D hDˇC .c ; cC /.v ; 0/; vC ; i D hDˇ .c ; cC /.0; vC /; v i D dfv .vC /: It clearly follows that d c , as a 2-form on T@C M , does not depend on c .

6.2 Local deformations near the Fuchsian locus That WM is a Kähler potential is now reduced to a simple computation in the Fuchsian situation. Proposition 6.2. Suppose that M is a Fuchsian manifold, with cC D c . Let I be the hyperbolic metric in the conformal class cC . Under a first-order deformation which does not change c , the variation of I and II0 on @C M are related by ıII0 D ıI :


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The proof is quite elementary, it is based on the fact that, for a quasifuchsian manifold which is “close to Fuchsian”, the metrics at infinity on the upper and lower components of the boundary are I and III respectively (where III is the third fundamental form at infinity on the upper boundary component). Moreover, if I has constant curvature, then III also has constant curvature (see [27]).

6.3 Kähler potential We can reformulate this statement by setting R WD Re. c /, so that, by Lemma 4.4, R .X / D hX; II0 i. Using the previous proposition, this can then be stated as .DX R /.Y / D hX; Y iW P ; where D is the Levi-Civita connection of the Weil–Petersson metric on T@C M . N c , denoting by J the complex We can now compute explicitly an expression of @ structure on T@ M . N c .X; Y / D .DX c /.Y / C i.DJX c /.Y / @ D .DX R /.Y / i.DX R /.J Y / C i..DJX R /.Y / i.DJX R /.J Y // D hX; Y i ihX; J Y i C i hJX; Y i C hJX; J Y i D 2.hX; Y i ihX; J Y i/: N c .X; JX/ D 2i kXk2 , and we recover the result of This means precisely that @ WP Takhtajan and Teo [41] that WM is a Kähler potential for the Weil–Petersson metric. Note that this statement could be compared to the fact (proved recently in [17]) that, on the Fuchsian locus, the Hessian of the area of the (unique) closed minimal surface, as a function on Tx T , is compatible with the hypothesis that this area is also a Kähler potential for the Weil–Petersson metric.

7 The relative volume of hyperbolic ends So far we have considered a version of the renormalized volume defined for a hyperbolic 3-manifold M as a whole. This means that only certain very special projective structures can arise at boundary components @M . It is interesting, however, to extend the notion of renormalized volume (and thus of Liouville action) to arbitrary projective structures at the boundary. This is achieved by the notion of the relative volume that we consider in this section. When the projective structure in question is such that a non-singular hyperbolic 3-manifold M realizing it exists, then the renormalized volume of M is just the sum of relative volumes of its hyperbolic ends and the (dual) volume of the convex core C.M /.

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We first recall some results due to Bonahon concerning the first variation of the volume of the convex core of a quasifuchsian manifold. We then introduce the relative volume of a hyperbolic end, and give a first variation formula for it. This establishes an analog of Kleinian reciprocity in the relative volume context, and proves that the grafting map is symplectic.

7.1 Definition The relative volume is defined for (geometrically finite) hyperbolic ends rather than for hyperbolic manifolds. Thus, consider a hyperbolic end M . The procedure used in the definition of the renormalized volume can be used in this setting, leading to the relative volume of the end. We will say that a geodesically convex subset K M is a collar if it is relatively compact and contains the metric boundary @0 M of M (possibly all geodesically convex relatively compact subsets of M are collars, but it is not necessary to consider this question here). Then @K \ M is a locally convex surface in M . The relative volume of M is related both to the (dual) volume of the convex core and to the renormalized volume; it is defined as the renormalized volume, but starting from the metric boundary of the hyperbolic end. We follow the same path as for the renormalized volume and start from a collar K M . We set Z 1 1 H da C L . /; W .K/ D V .K/ 4 @K 2 where H is the mean curvature of the boundary of K, is the induced metric on the metric boundary @0 M of M , and is its measured bending lamination. As for the renormalized volume we define the metric at infinity as I WD lim 2e 2 I ; !1

where I is the set of points at distance from K. The conformal structure of I is equal to the canonical conformal structure c1 at infinity of M . Here again, W only depends on I (and on M ). Not all metrics in c1 can be obtained from a compact subset of M , however all metrics do define an equidistant foliation close to infinity in M , and it is still possible to define W .I / even when I is not obtained from a convex subset of M . So W defines a function, still called W , from metrics in the conformal class c1 to R. Lemma 7.1. For fixed area of I , W is maximal exactly when I has constant curvature. The proof follows directly from the arguments used in [27] (and reviewed in Section 7) so we do not repeat it here. It takes place entirely on the boundary at infinity so it makes no difference whether one considers a hyperbolic end or a geometrically finite hyperbolic manifold.


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Definition 7.1. The relative volume VR of M is defined to be W .I / when I is the hyperbolic metric hyperbolic metric in the conformal class at infinity on M .

7.2 The first variation of the relative volume Proposition 7.2. Under a first-order variation of the hyperbolic end, the first-order variation of the relative volume is given by Z 1 0 1 0 hI 0 ; II0 ida : (7.1) VR D L . / 2 4 @1 E The proof is based on the arguments described in the previous sections, both for the first variation of the renormalized volume and for the first variation of the volume of the convex core.

7.3 The grafting map is symplectic Since hyperbolic ends are in one-to-one correspondence with CP 1 -structures, we can consider the relative volume VR as a function on the space of projective structures CP . This space is canonically identified with the (complex) cotangent bundle T TC , where the subscript C indicates that one is talking about the complex Teichmüller space. Let ˇC be the Liouville form on T TC . Consider now the space T ML associated with the metric boundary of our hyperbolic end. In Section 5.3.2 we have discussed how this space can be naturally identified by the map ı with T TH . Let H be the Liouville form on T TH . We can now consider the grafting map Gr W T ML ! CP 1 , and the composition ı B Gr 1 W CP ! T TH . The latter map turns out to be C 1 (see [28]), a fact which is somewhat surprising since there is no C 1 structure on ML. It pulls back H as .ı B Gr 1 / H D L0 . /: Under the identification of CP with T TC through the Schwarzian derivative, the expression of C is Z 0 C D hI ; II0 ida : @1 M

So Proposition 7.2 can be formulated as d VR D

1 1 .ı B Gr 1 / H C ; 2 4

and it follows that 2.ı B Gr 1 / !H D !C . This means that the grafting map preserves (up to a constant) the symplectic form and is thus symplectic. This statement can also be rephrased in a way analogous to (5.1) by saying that the subspace of the space .T ML/ CP that can be realized on the two boundaries of a hyperbolic end is a Lagrangian submanifold in .T ML/ CP .

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8 Manifolds with particles and the Teichmüller theory of surfaces with cone singularities One key feature of the arguments presented in this work is that they are always local, in the sense that they depend on local quantities defined on the boundaries of compact subsets of quasifuchsian manifolds. Thus, we make only a very limited use of the fact that the quasifuchsian manifolds are actually quotients of hyperbolic 3-space by a group of isometries. One place where this is used is in the proof of the fact that II is determined by I (actually a direct consequence of the Bers double uniformization theorem). We expect that all the results should extend from quasifuchsian (more generally geometrically finite) manifolds to the “quasifuchsian manifolds with particles” which were studied e.g. in [26], [32]. Those are actually cone-manifolds, with cone singularities along infinite lines running from one connected component of the boundary at infinity to the other, along which the cone angle is less than . In the (non-singular) quasifuchsian setting the Bers double uniformization theorem shows that everything is determined by the conformal structure at infinity. The corresponding statement holds for “quasifuchsian manifolds with particles”; a first step towards it is made in [32], while the second step is made in [29]. The corresponding statement for more general, convex co-compact manifolds, remains however elusive. Those results can be used to obtain results on the Teichmüller-type space of hyperbolic metrics with n cone singularities of prescribed angles on a closed surface of genus g. Note that this space, which can be denoted by Tg;n; (with D . 1 ; : : : ; n / 2 .0; /n ) is topologically the same as the “usual” Teichmüller space Tg;n of hyperbolic metrics with n cusps (with a one-to-one correspondence from [43]) but it has a natural “Weil–Petersson” metric which is different. It follows from the considerations made here, extended to quasifuchsian manifolds with particles, that this “Weil–Petersson” metric is still Kähler, with the renormalized volume playing the role of a Kähler potential – a result also obtained by different arguments by Schumacher and Trapani [37]. We leave the detailed investigation of this extension to quasifuchsian cone manifolds for future work.

References [1]

L. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics. Ann. of Math. (2) 72 (1960), 385–404. 787

[2]

P. Albin, Renormalizing curvature integrals on Poincaré–Einstein manifolds. Adv. Math. 221 (2009), no. 1, 140–169. 802

[3]

M. T.Anderson, L2 curvature and volume renormalization ofAHE metrics on 4-manifolds. Math. Res. Lett. 8 (2001), no. 1–2, 171–188. 802

[4]

M. T. Anderson, Geometric aspects of the AdS/CFT correspondence. In AdS/CFT correspondence: Einstein metrics and their conformal boundaries, IRMA Lect. Math. Theor. Phys. 8, Eur. Math. Soc., Zurich, 2005, 1–31. 785, 786, 791, 802


817

[5]

L. Bers, Simultaneous uniformization. Bull. Amer. Math. Soc. 66 (1960), 94–97. 798

[6]

F. Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston’s symplectic form. Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), no. 2, 233–297. 810

[7]

F. Bonahon, Geodesic laminations with transverse Hölder distributions. Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 2, 205–240. 810

[8]

F. Bonahon, A Schläfli-type formula for convex cores of hyperbolic 3-manifolds. J. Differential Geom. 50 (1998), no. 1, 25–58. 790, 809, 810

[9]

F. Bonahon, Variations of the boundary geometry of 3-dimensional hyperbolic convex cores. J. Differential Geom. 50 (1998), no. 1, 1–24. 790, 808, 810

[10] F. Bonahon, Geodesic laminations on surfaces. In Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), Contemp. Math. 269, Amer. Math. Soc., Providence, R.I., 2001, 1–37. 789, 809 [11] S. de Haro, K. Skenderis, and S. N. Solodukhin, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence. Comm. Math. Phys. 217 (2001), no. 3, 595–622. 786, 791, 802 [12] D. Dumas, Complex projective structures. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 455–508. 787, 789, 811 [13] D. Dumas and M. Wolf, Projective structures, grafting and measured laminations. Geom. Topol. 12 (2008), no. 1, 351–386. 789 [14] C. L. Epstein, Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. Preprint, 1984; http://www.math.upenn.edu/~cle/papers/WeingartenSurfaces.pdf 791, 799 [15] W. M. Goldman, The symplectic nature of fundamental groups of surfaces. Adv. in Math. 54 (1984), no. 2, 200–225. 810 [16] C. R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence. Nuclear Phys. B 546 (1999), no. 1–2, 52–64. 785, 791 [17] R. Guo, Z. Huang, and B. Wang, Quasi-Fuchsian 3-manifolds and metrics on Teichmüller space. Asian J. Math. 14 (2010), no. 2, 243–256 813 [18] M. Henningson and K. Skenderis, The holographic Weyl anomaly. J. High Energy Phys. (1998), no. 7, Paper 23. 791 [19] M. Herzlich, A remark on renormalized volume and Euler characteristic for ACHE 4manifolds. Differential Geom. Appl. 25 (2007), no. 1, 78–91. 791 [20] C. D. Hodgson and S. P. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. J. Differential Geom. 48 (1998), 1–60. 810, 811 [21] H. Hopf, Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4 (1951), 232–249. 786, 804 [22] R. Kashaev, Discrete Liouville equation and Teichmüller theory. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume III, EMS Publishing House, Zurich 2012, 821–851. 781 [23] S. Kawai, The symplectic nature of the space of projective connections on Riemann surfaces. Math. Ann. 305 (1996), no. 1, 161–182. 789, 807, 811

818


[24] K. Krasnov, Holography and Riemann surfaces. Adv. Theor. Math. Phys. 4 (2000), no. 4, 929–979. 784, 791 [25] K. Krasnov, On holomorphic factorization in asymptotically AdS 3D gravity. Classical Quantum Gravity 20 (2003), no. 18, 4015–4042. 794 [26] K. Krasnov and J.-M. Schlenker, Minimal surfaces and particles in 3-manifolds. Geom. Dedicata 126 (2007), 187–254. 791, 795, 796, 804, 816 [27] K. Krasnov and J.-M. Schlenker, On the renormalized volume of hyperbolic 3-manifolds. Comm. Math. Phys. 279 (2008), no. 3, 637–668. 784, 786, 802, 803, 805, 813, 814 [28] K. Krasnov and J.-M. Schlenker, A symplectic map between hyperbolic and complex Teichmüller theory. Duke Math. J. 150 (2009), no. 2, 331–356. 784, 789, 790, 809, 810, 815 [29] C. Lecuire and J.-M. Schlenker, The convex core of quasifuchsian manifolds with particles. Preprint, 2009; arXiv:0909.4182v1 [math.GT]. 816 [30] C. T. McMullen, The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. of Math. (2) 151 (2000), no. 1, 327–357. 784, 788, 804, 805, 806, 812 [31] J. Milnor, The Schläfli differential equality. In Collected papers, vol. 1, Publish or Perish, Houston, TX, 1994, 281–295. 800 [32] S. Moroianu and J.-M. Schlenker, Quasi-Fuchsian manifolds with particles. J. Differential Geom. 83 (2009), no. 1, 75–129. 816 [33] S. J. Patterson and P. A. Perry, The divisor of Selberg’s zeta function for Kleinian groups. Appendix A by Charles Epstein. Duke Math. J. 106 (2001), no. 2, 321–390. 791, 792 [34] I. Rivin and C. D. Hodgson, A characterization of compact convex polyhedra in hyperbolic 3-space. Invent. Math. 111 (1993), 77–111. 792 [35] I. Rivin and J.-M. Schlenker, The Schläfli formula in Einstein manifolds with boundary. Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 18–23. 784, 801 [36] J.-M. Schlenker, Hypersurfaces in H n and the space of its horospheres. Geom. Funct. Anal. 12 (2002), no. 2, 395–435. 798 [37] G. Schumacher and S. Trapani, Weil–Petersson geometry for families of hyperbolic conical Riemann surfaces. Michigan J. Math. 60 (2011), no. 1, 3–33. 816 [38] Y. Sözen and F. Bonahon, The Weil-Petersson and Thurston symplectic forms. Duke Math. J. 108 (2001), no. 3, 581–597. 809 [39] L. Takhtajan and P. Zograf, On uniformization of Riemann surfaces and the Weil–Petersson metric on the Teichmüller and Schottky spaces. Mat. Sb. 132 (1987), 303–320; English transl. Math. USSR Sb. 60 (1988), 297–313. 782, 784, 804, 805, 806 [40] L. Takhtajan and P. Zograf, Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on M0;n . Trans. Amer. Math. Soc. 355 (2003), no. 5, 1857–1867 (electronic). 783 [41] L. A. Takhtajan and L.-P. Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography. Comm. Math. Phys. 239 (2003), no. 12, 183–240. 783, 784, 791, 804, 806, 812, 813 [42] W. P. Thurston, Three-dimensional geometry and topology. Originally notes of lectures at Princeton University, Princeton 1979; recent version available on http://www.msri.org/publications/books/gt3m/. 809


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[43] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821. 816 [44] E. Witten, Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2 (1998), 253–291. 783 [45] E. Witten, 2 C 1-dimensional gravity as an exactly soluble system. Nuclear Phys. B 311 (1988/89), no. 1, 46–78. 793 [46] S. A. Wopert, The Weil–Petersson metric geometry. In Handbook of Teichmüller theory (A. Papadopoulos, ed.), Volume II, EMS Publishing House, Zurich 2009, 47–64. 782

Chapter 15

Discrete Liouville equation and Teichmüller theory Rinat M. Kashaev

Contents 1 2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical discrete Liouville equation . . . . . . . . . . . . . . . . . . . . 2.1 Discretization from the Liouville formula . . . . . . . . . . . . . . 2.2 Discrete Liouville equation and Teichmüller space . . . . . . . . . 3 Quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Faddeev’s quantum dilogarithm . . . . . . . . . . . . . . . . . . . 3.2 Quantum discrete Liouville equation . . . . . . . . . . . . . . . . . 3.3 Algebra of observables and the evolution operator . . . . . . . . . . 3.4 Integrable structure of the quantum discrete Liouville equation . . . 3.5 The case N D 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Relation to quantum Teichmüller theory . . . . . . . . . . . . . . . . . . 4.1 Highlights of quantum Teichmüller theory . . . . . . . . . . . . . . 4.2 The quantum discrete Liouville equation and quantum Teichmüller theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

821 823 823 825 828 828 833 833 834 836 837 837

. 843 . 849

1 Introduction The Liouville equation [32] is a partial differential equation of the form 1 @2 D e ; @z@zN 2 which has a number of applications both in mathematics and mathematical physics. For example, it describes surfaces of constant negative curvature, thus playing an indispensable role in uniformization theory of Riemannian surfaces of negative Euler characteristic [33]. Indeed, let p W H ! † be a universal covering map for a hyperbolic surface †, where H is the upper half plane with the standard Poincaré metric ds 2 , and W U ! H, U †, a local section of p. Then, the pull-back metric ds 2 in conformal form e jdzj2 , z being a local complex coordinate on U , gives a solution of the Liouville equation on U .

822

Rinat M. Kashaev

In theoretical physics, the Liouville equation is often considered in analytically continued form with z D x C t and zN D x t as real independent coordinates. In this case, it takes the form of a classical equation of motion for a relativistic 1 C 1dimensional field theoretical system: @2 @2 2 D 2e : (1.1) @t 2 @x The invariance with respect to (holomorphic) re-parameterizations, associated with the diffeomorphism group of the circle, makes the Liouville equation relevant to twodimensional gravity [26] and Conformal Field Theory (CFT) [9]. It is also a basic ingredient in the theory of noncritical strings [34]. These are the reasons for the recent interest in the Liouville equation, especially its quantum theory [13], [14], [23], [24], and its (quantum) integrability properties [4], [5], [6]. There are few, seemingly different aspects of the relationship between the Liouville equation and Teichmüller theory. One such aspect is purely classical, which comes through the above mentioned uniformization theory of surfaces of negative Euler characteristic. It exhibits even more profound features when one considers the perturbative approach to quantum Liouville theory [35], [36], [37]. Another aspect is purely quantum, and it originates in a conjecture of H. Verlinde [39], which states that there is a mapping class group equivariant isomorphism between the space of quantum states of quantum Teichmüller theory on a given surface and the space of conformal blocks of the quantum Liouville theory on the same surface, see [38] for the up-to-date situation with the Verlinde conjecture. One more aspect of the connection of the Liouville equation to Teichmüller theory has been considered recently in the works [20], [19], [22] through the consideration of a specific discretized version of the Liouville equation both on classical and quantum levels. The discrete Liouville equation has the form m;n1 m;nC1 D .1 C m1;n /.1 C mC1;n /;

(1.2)

where the “discrete space-time” is represented by the integer lattice Z2 and the dynamical field m;n is a strictly positive real function on this lattice. To see in what sense this is a discretized version of the Liouville equation, let us take a small positive parameter as the lattice spacing of the discretized space-time, and consider the combination .x; t/ D log. 2 m;n / in the limit, where ! 0, m; n ! 1 in such a way that the products x D m, and t D n are kept fixed. If a solution m;n of the discrete Liouville equation is such that such a limit exists, then the limiting value 0 .x; t / solves the dynamical version (1.1) of the Liouville equation. It is worth repeating here the remark made in the paper [22] that the discrete Liouville equation is among the simplest examples of Y -systems [42]. There are also other interesting connections of (quantum) discrete integrable systems with (hyperbolic) geometry, see for example, [10], [7], [8], [11]. Finally, let us

Chapter 15. Discrete Liouville equation and Teichmüller theory

823

note that the Liouville equation is also treated, in relation with the theory of renormalized volume, by K. Krasnov and J.-M. Schlenker in another chapter of this volume [31]. The purpose of this exposition is, following the works [20], [19], [22], [29], to review the relationship between the discrete Liouville equation and Teichmüller theory both classically and quantum mechanically. Acknowledgements. It is a pleasure to thank L. D. Faddeev and A. Yu. Volkov in collaboration with whom some of the results described in this survey were obtained. I would also like to thank V. V. Bazhanov and V. V. Mangazeev and the anonymous referee for valuable comments on the initial version of this chapter. This work was partially supported by FNS Grant No. 200020-121675.

2 Classical discrete Liouville equation This section is devoted to the classical discrete Liouville equation. We first describe the discretization of Faddeev and Volkov, and then give an interpretation in terms of a mapping class dynamics in the Teichmüller space of an annulus with marked points on the boundary. The key role in this interpretation is played by the shear coordinates of Thurston–Bonahon–Fock in Teichmüller space.

2.1 Discretization from the Liouville formula The interpretation of solutions of the Liouville equation in terms of pull-backs of the Poincaré metric leads to the Liouville formula for a general solution of the dynamical version (1.1) of the Liouville equation e .x;t/ D 4

f 0 .u /g 0 .uC / ; .f .u / g.uC //2

u˙ D x ˙ t;

(2.1)

where f .x/, g.x/ are two arbitrary smooth functions on the real line. From the structure of the Liouville formula it follows that it makes sense on the entire real plane provided the functions f .x/, g.x/ satisfy the conditions f 0 .x/g 0 .y/ < 0;

f .x/ ¤ g.y/

for all .x; y/ 2 R2 :

(2.2)

Despite the fact that in the dynamical version (1.1) of the Liouville equation the complex analytical aspects of the uniformization of hyperbolic surfaces are somehow lost, there is still an action of the group PSL.2; R/ on the functions f .x/ and g.x/ given by the transformations f .x/ 7!

af .x/ C b ; cf .x/ C d

g.x/ 7!

ag.x/ C b ; cg.x/ C d

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Rinat M. Kashaev

which leave unchanged the Liouville solution (2.1). One can show that the set of PSL.2; R/-orbits of pairs .f; g/ is in bijection with the set of solutions of the Liouville equation. For example, in the physically interesting case of periodic solutions .x C L; t / D .x; t / which correspond to a space-time of the topological type of the cylinder S 1 R, the functions f .x/ and g.x/ are quasi-periodic: f .x C L/ D where the PSL.2; R/-matrix

af .x/ C b ; cf .x/ C d

g.x C L/ D

ag.x/ C b cg.x/ C d

(2.3)

a b T Œ D c d

is called the monodromy matrix associated with a periodic solution . The basic idea of A.Yu. Volkov behind the discrete Liouville equation is to write first a PSL.2; R/-invariant finite difference analogue of the Liouville formula (2.1) by replacing its right-hand side by a cross-ratio of four shifted quantities f .x t ˙ /, g.x C t ˙ /. Namely, if we define h .x; t / D

.f .u C / g.uC C //.f .u / g.uC // ; .f .u C / f .u //.g.uC C / g.uC //

u˙ D x ˙ t; (2.4)

where we assume Conditions (2.2), then it is easily seen that the limit `.x; t/ D lim 2 h .x; t / !0

exists and the formula e .x;t/ D

1 `.x; t /

coincides with the Liouville formula (2.1). On the other hand, one observes that the four shifted cross-ratios h .x ˙ ; t /, h .x; t ˙ / depend on six values f .x t C k/, g.x C t C k/, with k 2 f0; ˙2g. Taking into account the PSL.2; R/-invariance of the cross-ratios, this means that among those four cross-ratios only three are independent, and one identifies the following relation h .x; t C /h .x; t / D .1 C h .x C ; t //.1 C h .x ; t // which, after the substitutions x D m and t D n takes the form of the discrete Liouville equation (1.2) for the function m;n D h .m; n/;

.m; n/ 2 Z2 :

(2.5)

Notice that one and the same pair of functions .f; g/ is used for constructing solutions of both the Liouville equation and its discrete counterpart. When the quasi-periodicity conditions (2.3) are satisfied, Function (2.4) is periodic: h .x C L; t / D h .x; t /;


825

so that we have periodic solutions of the discrete Liouville equation as soon as the lattice spacing is chosen to be a rational multiple of the period L. Namely, for D LM=N with positive mutually prime integers M and N , the function (2.5) satisfies the equation mCN;n D m;n : In particular, when N D 1, m;n is independent of the first argument, and the discrete Liouville equation becomes a one-dimensional discrete equation of the form n1 nC1 D .1 C n /2 : This latter equation first appeared in this context in [22] where it has been interpreted as the evolution of the “zero-modes” of the continuous Liouville equation.

2.2 Discrete Liouville equation and Teichmüller space The purpose of this subsection is to show that the discrete Liouville equation can be realized as a mapping class group dynamics on Teichmüller space. We consider an annulus with N marked points on each of its boundary components (2N points in total), labeled A1 ; : : : ; AN for one boundary component, and B1 ; : : : ; BN , for another. Additionally, choose the ideal triangulation shown in Figure 1, where the variables f1 ; : : : ; f2N not only serve to identify the interior edges, but also denote the shear coordinates of Thurston–Bonahon–Fock associated to the triangulation in the Teichmüller space of the annulus. Notice that here we consider an unusual situation Br1

Q

Q

f1

r

A1

Br3 p

Br2

Q

Q

f3

Qr

A2

BrN C1

Q

Q

Qf 2 Q

p pBrN Q

Qf

Q f

Q

Q

4

2N

Q

Qr p p p r A3

AN

f2N C1

Q Qr

AN C1

Figure 1. An ideally triangulated annulus with N marked points on each boundary component. The leftmost and the rightmost vertical edges are identified with equalities f2N C1 D f1 , AN C1 D A1 , BN C1 D B1 .

where all marked points are located on the boundary of the annulus, and for the parametrization of the Teichmüller space one does not need to associate coordinates on the boundary ideal arcs. This can be seen by looking at a fundamental domain in the Poincaré upper half plane which is an ideal polygon with 2N C 2 ideal vertices, see Figure 2 for the case N D 2. Clearly, the isometry class of such a polygon is determined by 2N 1 real parameters corresponding to the positions of the 2N C 2 vertices modulo the three-dimensional isometry group PSL.2; R/. This is less than the number of shear coordinates fi , 1 i 2N , but one more degree of freedom comes from the gluing condition: one chooses a PSL.2; R/-matrix restricted by the condition that it should map a given ordered pair of boundary points (the endpoints .A1 ; B1 / of

826

Rinat M. Kashaev H

f1 f2 f3 f4 f5

r

A1

r

A2

r

A3

r

B3

r

B2

r

B1

Figure 2. A fundamental domain in the Poincaré upper half plane for an ideally triangulated annulus with N D 2 marked points on each boundary component. Under the projection map p W H ! H=Z the images of the biggest circle with label f1 and the middle smallest circle with label f5 coincide so that p.A1 / D p.A3 /, p.B1 / D p.B3 /.

the half circle labeled f1 ) to another ordered pair (the endpoints .AN C1 ; BN C1 / of the half circle labeled f2N C1 ), and it is known that there is a one parameter family of such matrices. The mapping class group of our annulus is given by all homeomorphisms preserving the set of marked points, not necessarily point-wise. We are interested in the unique mapping class, denoted D 1=N , which fixes the set fA1 ; : : : ; AN g point-wise and cyclically permutes the set fB1 ; : : : ; BN g: B1 7! B2 7! B3 7! 7! BN 7! B1 : As the notation suggests, the N -th power of this class is nothing else than the unique Dehn twist of the annulus which fixes the boundary point-wise. Theorem 1 ([19]). The discrete dynamical system on the Teichmüller space of an annulus with N marked points on each boundary component, corresponding to the mapping class D 1=N , is described by the discrete Liouville equation (1.2) on the sublattice m C n D 1 .mod 2/ with the 2N -periodic boundary condition mC2N;n D m;n ; the evolution step being identified with the translation along the “light-cone”: m;n 7! 0m;n D m1;nC1 : Proof. Recall that under a flip the shear coordinates transform according to the formulae: a0 D a=.1 C 1=e/; d 0 D d=.1 C 1=e/; (2.6) b 0 D b.1 C e/; c 0 D c.1 C e/; e 0 D 1=e;

827


where the variables are shown in Figure 3 and all other variables stay unchanged. We remark that this transformation law still applies even if some of the sides of the r @

a r @

e

r @

@ b @ @

a0 @r

@ c@

!

0 c@ @

d

@

e0

r @ @

@ b0 @ @

@r

@r

d0 @r

Figure 3. A flip transformation corresponding to equations (2.6).

quadrilateral are a part of the boundary. The only modification is that there is no coordinate associated to a boundary edge, and thus there is nothing to be transformed on this edge. Now, from Figure 4 and the transformation law (2.6) it follows that the mapping class D 1=N acts in the Teichmüller space according to the following formulas f2j 7! f2j0 D 1=f2j 1 ;

f2j C1 7! f2j0 C1 D f2j .1 C f2j 1 /.1 C f2j C1 /: (2.7)

If we identify the variables f1 ; : : : ; f2N with the initial data for the 2N -periodic p p Bp rj Q

Bj C1

Q

Q f 2j

Q

ppp r

Aj

Bj C2

r Q Q

f2j C1Q f2j C2

Q Qr

Aj C1

p p Bp rj 1

rp p p

Q

Q

Qr p p p

Aj C2

Brj

D 1=N

!

Brj pC1 pp

f2j 1

f2j

p p p r

f2j C1

r

Aj

rp p p

Aj C1

Aj C2

# p p Bp rj 1 Q Q

Bj

Q

Aj

rp p p

Q

0 fQ 2j

ppp r

Bj C1

r Q

0 Q f2j0 C2 f2j C1

Q Qr

Aj C1

Q

Q

Qr p p p

Aj C2

Figure 4. The action of the mapping class D 1=N on the triangulated annulus: it is the identity on the bottom boundary and a cyclic shift to the right by one spacing on the top boundary.

discrete Liouville equation (1.2) on the sublattice m C n D 1 .mod 2/ along the zig-zag line n 2 f1; 0g according to the formulae ´ if m D 1 .mod 2/I m;0 fm D 1=m;1 otherwise;

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Rinat M. Kashaev

then, the transformation formulas (2.7) exactly correspond to the light-cone evolution m;n 7! 0m;n D m1;nC1 for the time instants n 2 f1; 0g. Remark 1. One can consider more general dynamical systems of this type on arbitrary surfaces with boundary. Let † D †g;s;N1 ;N2 ;:::;Nn be an oriented surface of genus g with s interior marked points and n boundary components, where the i -th boundary component carries Ni marked points. Then, one can consider a dynamical system in the Teichmüller space of †, where the discrete time evolution is given by the twist 1=N Di i on the i -th boundary component. Moreover, one can show that there exists a coordinate system in the corresponding Teichmüller space such that for large Ni the evolution is still described by the discrete Liouville equation but with some specific boundary conditions. In the simplest case of an N -gon † D †0;0;N the corresponding dynamics is evidently N -periodic, and one can show that it corresponds to the simplest Zamolodchikov Y -system associated with the pair of Dynkin diagrams .AN 3 ; A1 /.

3 Quantum theory In what follows we describe the quantum discrete Liouville equation and its connection with quantum Teichmüller theory. The quantum theories of Teichmüller space and of the discrete Liouville equation are built up on the base of a particular special function, called Faddeev’s quantum dilogarithm. It is a building block for the operators realizing elements of the mapping class group in quantum Teichmüller theory, while in the case of quantum Liouville equation it is used for the construction of the evolution operator and the Baxter’s Q-operator (the operator reflecting the integrable structure of the discrete Liouville equation). This is why we start by describing some of the properties of this function.

3.1 Faddeev’s quantum dilogarithm Let b be a complex number with non-zero real part 0. We can perform the integration in (3.1) by the residue method. The result can be written in the form 'b .z/ D .e 2.zcb /b where

1

2

q ´ e ib ; .xI y/1 ´

1 Y

I qN 2 /1 =.e 2.zCcb /b I q 2 /1 ; qN ´ e ib

.1 xy j /;

j D0

2

;

x; y 2 C; jyj < 1:

(3.8)

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Rinat M. Kashaev

Formula (3.8) defines a meromorphic function on the entire complex plane, satisfying the functional equations (3.3) and (3.4), with essential singularity at infinity. So, it is the desired extension of Definition (3.1). It is easy to read off the location of its poles and zeroes: zeroes of .'b .z//˙1 D f˙.cb C mib C nib 1 / W m; n 2 Z0 g: The behavior at infinity depends on the direction along which the limit is taken: 8 1; j arg.z/j > 2 C arg.b/I ˆ ˆ ˆ 2 < i z ˇ inv ; j arg.z/j < 2 arg.b/I e 'b .z/ˇjzj!1 (3.9) ˆ .ib 1 zI b 2 /=.qN 2 I qN 2 /1 ; j arg z =2j < arg bI ˆ ˆ : 2 2 j arg z C =2j < arg b; .q I q /1 = .ibzI b 2 /; where the standard notation for the -function is used: X 2 e i n C2 inz ; = > 0: .zI / ´ n2Z

Thus, for complex b, double quasi-periodic -functions, generators of the field of meromorphic functions on complex tori, describe the asymptotic behavior of Faddeev’s quantum dilogarithm. 3.1.3 Integral Ramanujan identity. Consider the following Fourier integral: Z 'b .x C u/ 2 iwx e dx; (3.10) ‰.u; v; w/ ´ R 'b .x C v/ where =.u C cb / > 0;

=.v C cb / > 0;

=.u v/ < =w < 0:

(3.11)

Restrictions (3.11) actually can be considerably relaxed by deforming the integration path in the complex x-plane, keeping the asymptotic directions of the two ends within the sectors ˙.j arg xj =2/ > arg b. So, the domain enlarged in this way for the variables u; v; w has the form: j arg.iz/j < arg b;

z 2 fw; u v w; v u 2cb g:

(3.12)

Integral (3.10) can be evaluated explicitly by the residue method, the result being 'b .u v C cb /'b .w cb / 2 iw.vcb / 1 e o 'b .u v w C cb / 'b .v C w u cb / D e 2 iw.uCcb / o ; 'b .v u cb /'b .w C cb /

‰.u; v; w/ D

(3.13) (3.14)

where the two expressions in the right-hand side are related to each other through the inversion relation (3.3). In [20] this identity has been demonstrated to be an integral counterpart of the Ramanujan 1 1 summation formula.

831


3.1.4 Fourier transforms. Particular values of ‰.u; v; w/ lead to the following Fourier transform formulas: Z C .w/ ´ 'b .x/e 2 iwx dx D ‰.0; v; w/jv!1 R (3.15) 2 iwcb iw 2 1 De o ='b .w C cb / D e o 'b .w cb /; and

Z

.w/ ´ D

.'b .x//1 e 2 iwx dx D ‰.u; 0; w/ju!1 R 2 e 2 iwcb o1 'b .w cb / D e iw o ='b .w C cb /:

The corresponding inverse transforms read Z .'b .x//˙1 D ˙ .y/e 2 ixy dy;

(3.16)

(3.17)

R

where the pole at y D 0 is surrounded from below. 3.1.5 Other integral identities. Faddeev’s quantum dilogarithm satisfies also integral analogs of other basic hypergeometric identities, see for example [25]. For any n 1 define Z n Y 'b .x C bj cb / i2x.wcb / ; dx e ‰n .a1 ; : : : ; an I b1 ; : : : ; bn1 I w/ ´ lim !0 R 'b .x C aj / j D1

(3.18) where bn D i for some > 0, =.bj / > 0;

=.cb aj / > 0;

n X

=.bj aj cb / < =.w cb / < 0:

j D1

The integral analog of the 1 the form

1 -summation formula of Ramanujan in this notation takes

‰1 .aI w/ D o

'b .a C w cb / : 'b .a/'b .w/

Equivalently, we can rewrite it as follows: Z 'b .x C a/ x ‰1 .aI w/ ´ e i2x.wCcb / dx R 'b .x C cb i0/ De

i2.aCcb /.wCcb /

‰1 .aI w/ D

o1

'b .a/'b .w/ : 'b .a C w C cb /

(3.19)

(3.20)

By using the integral Ramanujan formula, one can obtain an integral analog of the Heine transformation formula of the 1 2 basic hypergeometric series: ‰2 .a; bI cI w/ D

'b .c b/ ‰2 .c b; wI a C wI b/: 'b .a/

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By using the evident symmetry ‰2 .a; bI cI w/ D ‰2 .b; aI cI w/; we come to an integral analog of the Euler–Heine transformation formula: ‰2 .a; bI cI w/ 'b .c b/'b .c a/'b .a C b C w c/ ‰2 .c a; c bI cI a C b C w c/: D 'b .a/'b .b/'b .w/ (3.21) Performing the Fourier transform on the variable w and using the equation (3.19), we obtain an integral analog of the summation formula of Saalschütz: ‰3 .a; b; cI d; a C b C c d cb I cb / 'b .a C b d cb /'b .b C c d cb /'b .c C a d cb / : D o3 e id.2cb d / 'b .a/'b .b/'b .c/'b .a d /'b .b d /'b .c d / (3.22) One special case of this formula is obtained by taking the limit as c ! 1: ‰2 .a; bI d I cb / D o3 e id.2cb d /

'b .a C b d cb / : 'b .a/'b .b/'b .a d /'b .b d /

(3.23)

From equation (3.22) one can derive an integral or non-compact analog of the Bailey lemma, which is equivalent to the following: if some operators p and q satisfy the Heisenberg commutation relation (3.6), then the operator valued function Q.u; v/ D Q.u; vI p; q/ 2

´ e i q 'b .u q/'b .v q/

'b .p C u C v/ 2 'b .u C q/'b .v C q/e i q ' b . p/ (3.24)

gives a commuting operator family in the variables u and v, and it acts diagonally on a one-parameter family of vectors 2

Q.u; v/j˛s i D j˛s i'b .u C s/'b .v C s/'b .u s/'b .v s/e i2s ;

(3.25)

where the vectors j˛s i are defined by their matrix elements hxj˛s i D

'b .s x cb C i 0/ i2.xCcb /s e ; 'b .s C x C cb i 0/

s 2 R0 ;

(3.26)

with respect to the “position” basis hxj, x 2 R, where the operator q is diagonal, and p acts as differentiation: hxjq D xhxj;

hxjp D

1 @ hxj: 2 i @x


833

Some of these and other interesting properties of Faddeev’s quantum dilogarithm as well as their interpretation in the context of integrable systems are also described in [40], [8].

3.2 Quantum discrete Liouville equation The quantum version of Equation (1.2) (with 2N -periodic boundary conditions) reads as wm;tC1 wm;t1 D .1 C q wmC1;t /.1 C q 1C2ıN;1 wm1;t /; (3.27) where the field variables wm;t are elements of the observable algebra (see below), satisfying the periodicity condition wmC2N ;t D wm;t ; 2

and q D exp.ib /, b being the coupling constant (or its square root). The latter is expected to be related with the Virasoro central charge through the formula cVir D 1 C 6.b C b 1 /2 : Remark 3. Notice that the case N D 1 is special, where the two terms in the righthand side of Equation (3.27) are given in terms of one and the same value of the field variable wmC1;t D wm1;t . This is similar to the affine Cartan matrix of type A.1/ N for N D 1. A modification shows up also in the defining commutation relations of the observable algebra, see Relations (3.29) below.

3.3 Algebra of observables and the evolution operator The algebra of observables is generated by a finite set of self-adjoint operators fr1 ; : : : ; r2N g. We shall think of them as an operator family parameterized by integers frj gj 2Z satisfying the periodicity condition rj C2N D rj :

The defining commutation relations are as follows: ´ .1/m .1 C ıN;1 /.2 i/1 if n D m ˙ 1 .mod 2N /I Œ rm ; rn D 0 otherwise:

(3.28)

(3.29)

Taking into account the interpretation in terms of the Teichmüller space of an annulus (with N marked points on each boundary component), the operators rj can be considered as quantized logarithmic shear coordinates, the commutation relations (3.29) exactly corresponding to the Poisson structure of Teichmüller space. The initial data for the field variables in (3.27) are exponentials of the generating elements: w2j C1;0 D e 2b r2j C1 ; w2j;1 D e 2b r2j :

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Proposition 1. Let the operator Ulc be defined by the formula Ulc ´ G

N Y

'Nb .r2j /;

'Nb .x/ ´ 1='b .x/;

(3.30)

j D1

where the operator G is defined through the system of linear equations Grj D .1/j rj 1 G;

j 2 Z:

(3.31)

Then, the field variables defined by the formula wj;t ´ Utlc e 2b rj Ct Ut lc ;

j C t D 1 .mod 2/;

(3.32)

satisfy the 2N -periodic quantum discrete Liouville equation (3.27). Proof. By using Formula (3.32), Equation (3.27) is equivalent to the equality 1C2ıN;1 Ulc f2kC1 U2 f2k1 /; lc f2k1 Ulc D .1 C q f2kC1 /.1 C q

(3.33)

where fi D e 2b ri and 2k D j C t . On the other hand, Equation (3.31), the commutation relations (3.29), and the functional relations (3.4) imply that U1 lc f2k1 Ulc D f2k ; Ulc f2kC1 U1 lc

D .1 C q f2kC1 /.1 C q

1C2ıN;1

(3.34) 1 f2k1 /f2k :

(3.35)

Identities (3.34), (3.35) straightforwardly imply Identity (3.33). Remark 4. Note that the operator Ulc is identified with the “light-cone” evolution operator in the sense that it realizes the appropriate translation in the “space-time” lattice through the formula Ulc wj;t U1 lc D wj 1;t C1 :

Remark 5. Because of the duality symmetry b $ b 1 in the theory, there actually exist two types of exponential fields exp.2b ˙1 rm / which satisfy two dual quantum discrete Liouville equations.

3.4 Integrable structure of the quantum discrete Liouville equation The quantum discrete Liouville equation is integrable in the sense of the quantum inverse scattering method [17]. This means that it admits a set of commuting operators with the evolution operator being one of them, and there is a system of linear difference equations, called Baxter equations, relating these operators. In what follows, the order in products with non-commuting entries will be indicated as follows: Y Y ai ´ an an1 amC1 am ; ai ´ am amC1 an1 an : ni m

min

835


Consider the algebra AN of operators with a generalized linear basis of the form Y e 2 iri xi ; 1i2N

where the self-adjoint operators ri satisfy the commutation relations (3.29), and the variables xi take real or complex values. The term “generalized” here means that a generic element of the algebra AN is an integral of the form Z Y f .x1 ; : : : ; x2N / e 2 iri xi dx1 dx2N ; X 2N

1i 2N

where f .x1 ; : : : ; x2N / is a complex valued distribution (generalized function), and X 2N C 2N is a 2N -dimensional (over R) sub-manifold. The ascending cyclic product is a set of linear mappings, oCj W AN ! AN ;

j 2 Z; oCj D oCj C2N ;

acting diagonally on the basis monomials: Y Y oC 1 e 2 iri xi ´ e 2 ix2N x1 1i 2N

e 2 iri xi μ oCj

1i2N

Y

e 2 iri xi :

j ij C2N 1

We define the “transfer-matrices”

Y

t˙ ./ D oC 1 Tr

Lj˙ ;

(3.36)

1j 2N

where Lj˙

j

˙1

e .1/ b .rj / D .1/j b ˙1 .r / j e

! j ˙1 e .1/ b .Crj / Œj C 12 ; j ˙1 e .1/ b .Crj /

(3.37)

Œj 2 D .1 .1/j /=2; and the trace is that of two-by-two matrices. We also define a “Q-operator” Y Q./ D oC 1 wŒj 2 .; rj / G; 2 C;

(3.38)

1j 2N

where

´ e 2 ix 'b .x /wi .; x/ ´

'b .x C / 1

if i D 0I if i D 1;

(3.39)

and G is defined by equation (3.31). Proposition 2. The transfer-matrices (3.36) and the Q-operator (3.38) commute among themselves, Œt ./; t˙ . / D ŒQ./; Q. / D Œt˙ ./; Q. / D 0;

D ˙;

(3.40)

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Rinat M. Kashaev

solve the following Baxter equations, t˙ ./Q./ D Q. C ib ˙1 =2/ C .1 e 4b

˙1

/N Q. ib ˙1 =2/;

(3.41)

and the evolution operator Ulc of the quantum discrete Liouville equation is given by the formula Ulc D Q.0/: (3.42) Formula (3.42) is verified straightforwardly, the commutativity part of the proposition is the standard argument by using the Yang–Baxter equations, while the proof of the Baxter equations given in [29] uses a less standard argument. C C L2iC1 is equivalent to the Remark 6. The product of two neighboring L-operators L2i spectral parameter dependent L-operator introduced in [21] for the description of the (continuous) Liouville equation in the framework of the inverse scattering method.

3.5 The case N D 1 When N D 1, the algebra AN is generated by a single Heisenberg pair of position and momentum operators p and q: r1 D p q;

r2 D p C q;

Œp; q D .2 i/1 :

Calculation of the operators (3.36) and (3.38) at N D 1 gives the following result: t.z/ D LC .p; q/ ´ e 2b p C 2 cosh.2b q/;

o3 Q

z c b

2

Z D R

e i2cb .xCcb / Q.x cb C i0; 1I p; q/e i2xz dx;

where o is defined in Equation (3.2) and 2

2

Q.u; 1I p; q/ D e i q 'b .u C q/'Nb .p/'b .u q/e i q ; see also Equation (3.24). Calculation of the integral gives the formula 2

2

Q.0/ D 'b .p C q/e i2.cb q / :

By acting on the vectors j˛s i, and using equation (3.25), we obtain t.z/j˛s i D LC .p; q/j˛s i D j˛s i2 cosh.2bs/

and Q

z c b

2

j˛s i D j˛s io1 e i.s

where the vector

Z Œzj ´ R

2 C2c 2 / b

2

dxe i2xzix hxj

Œzj˛s i;

(3.43) (3.44)


837

is such that Œzjp D pz Œzj; Œzjq D qz Œzj; (3.45) 1 @ 1 @ z; qz ´ : (3.46) pz ´ 2 i @z 2 i @z On the other hand, the Baxter equation (3.41) at N D 1 can formally be written in the form z c b D0 (3.47) .LC .pz ; qz / LC .p; q//Q 2 which, when applied to the vector j˛s i, is reduced to an identity by using Equations (3.43)–(3.45).

4 Relation to quantum Teichmüller theory Quantum Teichmüller theory can be thought of as a particular family of projective unitary representations of the mapping class group of a given punctured surface in a Hilbert space. It is more convenient to enlarge the mapping class group to the groupoid of (decorated) ideal triangulations of the surface. The reason is that, unlike the mapping class group, this groupoid admits a rather simple and uniform presentation. The principal building block of the theory is an operator T which is an example of a general notion of (projective) semi-symmetric T -matrix. We give a brief description of the quantum Teichmüller theory following the line of the papers [28], [30]. Then, following the paper [19], we explain the connection between the quantum Liouville equation and the quantum Teichmüller theory of an annulus. In the last part of the section, we consider a special decorated ideal triangulation of the annulus, where the Dehn twist is represented by a single T-operator. It should be remarked that quantum Teichmüller theory is not an example of TQFT in the sense of Atiyah [1]. It can be considered as a restricted TQFT, defined on only those cobordisms which are given by mapping cylinders of homeomorphisms of punctured surfaces. Nonetheless, a true TQFT in the sense of Atiyah can be constructed for complexified Teichmüller theory related with the bending pleated surfaces of Bonahon in three-dimensional hyperbolic space [12], where only the phases of shear coordinates are quantized while their absolute values are kept classical. When considering compact oriented three-manifolds within such a TQFT one needs to introduce also a link realized as a Hamiltonian sub-complex of a triangulation [27], [3].

4.1 Highlights of quantum Teichmüller theory 4.1.1 Groupoid of decorated ideal triangulations. Let † D †g;s be an oriented surface of genus g with s punctures. Denote M D 2g2Cs and assume that M s > 0. The surface † admits ideal triangulations with 2M triangles.

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Rinat M. Kashaev

Definition 1. A decorated ideal triangulation of † is an ideal triangulation , where all triangles are provided with a marked corner, and a bijective ordering map

N W f1; : : : ; 2M g 3 j 7! Nj 2 T . / is fixed. Here T . / is the set of all triangles of . Graphically, the marked corner of a triangle Ni is indicated by an asterisk and the index i is put inside the triangle. The set of all decorated ideal triangulations of † is denoted by † . Recall that if a group G acts freely on a set X then there is an associated groupoid defined as follows. The objects are the G-orbits in X , while morphisms are G-orbits in X X with respect to the diagonal action. Denote by Œx the object represented by the element x 2 X and Œx; y the morphism represented by the pair of elements .x; y/ 2 X X . Two morphisms Œx; y and Œu; v, are composable if and only if Œy D Œu and their composition is Œx; yŒu; v D Œx; gv, where g 2 G is the unique element sending u to y. The inverse and the identity morphisms are given respectively by Œx; y1 D Œy; x and idŒx D Œx; x. In what follows, products of the form Œx1 ; x2 Œx2 ; x3 : : : Œxn1 ; xn will be written as Œx1 ; x2 ; x3 ; : : : ; xn1 ; xn . Remarking that the mapping class group M† of † freely acts on † , denote by G† the corresponding groupoid, called the groupoid of decorated ideal triangulations. It admits a presentation with three types of generators and four types of relations. The generators are of the form Œ ; , Œ ; i , and Œ ; !ij , where is obtained from by replacing the ordering map N by the map N B , where 2 S2M is a permutation of the set f1; : : : ; 2M g, i is obtained from by changing the marked corner of the triangle Ni as in Figure 5, and !ij is obtained from by applying the flip transformation in the quadrilateral composed of the triangles Ni and Nj as in Figure 6. s @

@ @ s

i

s

s

@

i

!

s

i

@ @s

Figure 5. Transformation i .

s i @ @

s @ @ j

@s

@s

!ij

!

s @ i @

s @ @ j @s

Figure 6. Transformation !ij .

@s


839

There are two sets of relations satisfied by these generators. The first set is as follows: Œ ; ˛ ; . ˛ /ˇ D Œ ; ˛ˇ ; ˛; ˇ 2 S2M ; Œ ; i ; i i ; i i i D idŒ ; Œ ; !ij ; !ik !ij ; !j k !ik !ij D Œ ; !j k ; !ij !j k ; Œ ; !ij ; i !ij ; !j i i !ij D Œ ; .ij / ; j .ij / ; i j .ij / :

(4.1) (4.2) (4.3) (4.4)

The first two relations are evident, while the other two are shown graphically in Figures 7 and 8. r #c # c

# r # B i B B Br

c

j

cr !ij k ! r

&!j k

r #cc # # c

r ##c c

# i c # r cr Z BZ k B Z B j ZZ Br Zr

# r cr !i k k Z BZ i ! Z B B j ZZ B r Zr

r #c # c # c

r ##c c

# r cr !ij j B i ! B B k r Br

.

!j k

# i c # r cr B j B B k r Br

Figure 7. Pentagon relation (4.3).

s @ @

s i @ @

j

@s

!ij

!

@s

s j @ @

i s @

@ s

s @ @ j @s

@s

!

!

.ij / B i j

s @ @

s @ i @

i

!ij

s @ i @

s @ @ j @s

Figure 8. Inversion relation (4.4).

@s

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Rinat M. Kashaev

The following commutation relations fulfill the remaining second set of relations: Œ ; i ; .i / D Œ ; ; 1 .i / ;

(4.5)

Œ ; !ij ; .!ij / D Œ ; ; ! 1 .i / 1 .i / ;

(4.6)

Œ ; j ; i j D Œ ; i ; j i ; Œ ; i ; !j k i D Œ ; !j k ; i !j k ; i 62 fj; kg; Œ ; !ij ; !kl !ij D Œ ; !kl ; !ij !kl ; fi; j g \ fk; lg D ;:

(4.7) (4.8) (4.9)

4.1.2 Hilbert spaces of square integrable functions. In what follows, we work with the Hilbert spaces H ´ L2 .R/; H ˝n ´ L2 .Rn /: Any two self-adjoint operators p and q, acting on H and satisfying the Heisenberg commutation relation (3.6), can be realized as differentiation and multiplication operators. Such a “coordinate” realization in Dirac’s bra-ket notation has the form 1 @ (4.10) hxj; hxjq D xhxj: hxjp D 2 i @x Formally, the set of “vectors” fjxigx2R forms a generalized basis of H with the following orthogonality and completeness properties: Z jxidxhxj D 1: hxjyi D ı.x y/; R

For any 1 i m we shall use the following notation: i W End H 3 a 7! ai D 1 ˝ ˝ 1 ˝ a ˝ 1 ˝ ˝ 1 2 End H ˝m : „ ƒ‚ … i1 times

Besides that, if u 2 End H then we shall write

˝k

for some 1 k m and fi1 ; i2 ; : : : ; ik g f1; 2; : : : ; mg,

ui1 i2 :::ik ´ i1 ˝ i2 ˝ ˝ ik .u/:

The permutation group Sm naturally acts in H ˝m : P .x1 ˝ ˝ xi ˝ / D x 1 .1/ ˝ ˝ x 1 .i / ˝ ;

2 Sm :

(4.11)

4.1.3 Semi-symmetric T -matrix. Fix some self-conjugate operators p; q satisfying the Heisenberg commutation relation (3.6). Choose a parameter b satisfying the condition .1 jbj/=b D 0; and define two unitary operators 2

2

A ´ e i=3 e i3 q e i.pCq/ 2 End H ; Té

i2 p1 q2

'b .q1 C p2 q2 / 2 End H

(4.12) ˝2

:

(4.13)

841


They satisfy the following relations characterizing a projective semi-symmetric T matrix: A3 D 1; T12 T13 T23 D T23 T12 ; T12 A1 T21 D A1 A2 P.12/ ;

(4.14) (4.15) (4.16)

where 2

´ e icb =3 ;

i

.b C b 1 /; 2 and the operator P.12/ is defined by Equation (4.11) in the case where cb ´

(4.17)

S2 3 D .12/ W 1 7! 2 7! 1: The operator A is characterized (up to a normalization factor) by the equations AqA1 D p q;

ApA1 D q:

Note that Equations (4.14)–(4.16) correspond to relations (4.2)–(4.4). This fact is the base of using the former to realize the latter. 4.1.4 Useful notation. For any operator a 2 End H we shall denote akO ´ Ak ak Ak1 ;

akL ´ Ak1 ak Ak :

(4.18)

It is evident that a LO D a OL D ak ; k

k

a OO D akL ; k

a LL D akO ; k

where the last two equations follow from Equation (4.14). In particular, we have pkO D qk ;

qkO D pk qk ;

(4.19)

pkL D qk pk ;

qkL D pk :

(4.20)

Besides that, it will be also useful to use the notation P.kl:::mk/ O ´ Ak P.kl:::m/ ;

1 P.kl:::mk/ L ´ Ak P.kl:::m/ ;

(4.21)

where .kl : : : m/ is the cyclic permutation .kl : : : m/ W k 7! l 7! 7! m 7! k: Equation (4.16) in this notation takes a rather compact form: T12 T21O D P.121/ O :

Remark 7. One can derive the following symmetry property of the T -matrix: 1 1 D P.121/ T12 D T12 T21O T21 O O T21O D T1O 2O P.121/ 1O 1 D T1O1 P.1O 2O 1/ L D T1O 2O T1O 2O T2O 1L D T2O 1L : 2O

(4.22)

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Rinat M. Kashaev

4.1.5 Quantum functor. As it was remarked in the beginning of the section, quantum Teichmüller theory is not a TQFT in the sense of Atiyah, but a restricted version of it defined on only the cobordisms given by mapping cylinders of orientation preserving homeomorphisms of oriented punctured surfaces. Technically, the restriction is reflected in constructing a functor from the groupoid of (decorated) ideal triangulations to the category of vector spaces. Notice that the groupoid of decorated ideal triangulations can be naturally identified with a sub-category of three-dimensional cobordisms between punctured surfaces. Specifically, quantum Teichmüller theory is defined by a quantum functor, F W G† ! End H ˝2M ;

which means that we have an operator-valued function F W † † ! End H ˝2M ;

satisfying the equations F. ; / D 1; F. ; 0 /F. 0 ; 00 /F. 00 ; / 2 C n f0g F.f . /; f . 0 // D F. ; 0 /

for all ; 0 ; 00 2 † ; (4.23)

for all f 2 M† ;

(4.24)

F. ; i / D Ai ;

(4.25)

F. ; !ij / D Tij ;

(4.26)

F. ; / D P

for all 2 S2M ;

(4.27)

where the operator P is defined by Equation (4.11). The consistency of these equations is ensured by the consistency of Equations (4.14)–(4.16) with Relations (4.2)– (4.4). In other words, the functor F defines a projective representation of the groupoid of decorated ideal triangulations G† in the automorphism group of the Hilbert space H ˝2M . A particular case of Equation (4.23) corresponds to 00 D : F. ; 0 /F. 0 ; / 2 C n f0g:

(4.28)

As an example, we can calculate the operator F. ; !ij1 . //. Denoting 0 D !ij1 . / and using Equation (4.28), as well as Definition (4.26), we obtain 1 F. ; !ij . // D F.!ij . 0 /; 0 / ' .F. 0 ; !ij . 0 ///1 D Tij1 ;

(4.29)

where ' means equality up to a numerical multiplicative factor. A projective unitary representation of the mapping class group M† is obtained if we choose a decorated ideal triangulation and define M† 3 f 7! .f / D F. ; f . // 2 End H ˝2M : Indeed, .f / .h/ D F. ; f . //F. ; h. // D F. ; f . //F.f . /; f .h. /// ' F. ; f h. // D .f h/:

843


4.2 The quantum discrete Liouville equation and quantum Teichmüller theory Again, as in the classical case, we consider an annulus with N marked points on each of its boundary components and choose the decorated ideal triangulation N shown in Figure 9. Equivalently, N can be thought of as an infinite triangulated strip, where the triangles are numbered according to Figure 9 with the periodicity condition

NN .n C 2N / D NN .n/ r Q

Q

r

r Q

Q

2

Q Q

r Q

3

rp p p r Q

r Q

4

Q

Q

1

for all n 2 Z:

2N

Q

Q

Q

r p p p r Q

2N 1 QQ

r Q

Figure 9. A decorated ideal triangulation of an annulus with N marked points on each of the boundary components. The leftmost and the rightmost vertical edges are identified.

Recall that D n=N , n 2 Z, is the mapping class twisting the top boundary component with respect to the bottom one by the angle 2 n=N so that the marked points on the top component are cyclically translated by n spacings. When n D N we get a pure Dehn twist D N=N D D. Clearly, D m=N B D n=N D D .mCn/=N : From Figure 10 it follows that the quantum realization of the transformation D 1=N ppp r Q

Q

2j

Q

ppp r

2j C 2

Q Qr

2j C 1Q

r

D 1=N

Q

2j 1 Q

ppp r

rp p p

r Q Q

2j 2

!

2j 1

p p p r

r p p p Q

#

" .: : : ; j; j 1; : : :/ ppp r Q

Q

2j 1

Q

ppp r

2j 2 Q

rp p p

r Q Q

Q Qr

2j C 1

Q

2j

Q

Q

Q

j

r p p p Q

ppp r Q

r

Q j

2j 1

Q

Q

p p p r

2j 2 Q

rp p p

1 1 2j !2j C1;2j 2j

r

Q

j1

2j C 1

2j

Q

rp p p

Q Qr

Q

2j C 1

rp p p

Q

2j

Q

Q

Qr p p p

Figure 10. Transformation D 1=N as a morphism in the groupoid of decorated ideal triangulations.

has the form N .D 1=N / ' D1=N ´ N 6=N P.:::j;j C1:::/

2N Y kD1

Ak

N Y lD1

T2lC1;2l };

(4.30)

844

Rinat M. Kashaev

where the normalization factor corresponds to a certain choice of normalization of Dehn twists in quantum Teichmüller theory. Notice that here we use notation (4.18): 1 T2lC1;2l } D A2l T2lC1;2l A2l : We define Dn=N ´ .D1=N /n

for all n 2 Z:

Consider the following faithful reducible realization of the observable algebra AN in L2 .R2N /: ´ pj C pj 1 if j D 0 .mod 2/I .rj / D qj C qj 1 otherwise; .G/ D N C6=N e i2

PN

j D1 p2j p2j C1

2N Y

Ak1 P.:::;l;l1;:::/ :

kD1

Theorem 2 ([19]). One has the following equality: .Ulc / D D1=N :

(4.31)

4.2.1 Working in another triangulation. Here we consider a special decorated ideal triangulation, where the Dehn twist D is represented by a single T-operator. It happens that with respect to this triangulation all operators Dn=N , 0 < n < N , are represented in terms of products of only N C 1 T-operators. Consider a decorated ideal triangulations of the form

nWN ´ nC1 B nC2 B B N . N /;

1 n < N;

where the transformations n are defined in Figure 11. The transformation n acts non-trivially only in the minimal annular part of nWN , which by itself is nothing else but the triangulated annulus n . r rp p p ppp r r rp p p ppp r 1 ! 2n 2n;2n1 2n 2n 2 1 2n 2n 2 2n 1 1 2n 1 2n r rp p p p p p r r rp p p p p p r 1 1 1 2n2 1 !2n;1 11 # 2n2 !2n1;2n2 n ppp r r rp p p 2n 1 2n 2 1 2n p p p r r rp p p Figure 11. The transformation n reduces n to n1 and two triangles attached to the boundary.

Quantum realizations of any element ' of the (extended) mapping class group with respect to decorated ideal triangulations and 0 are conjugate to each other by the


845

operator F . ; 0 /:

F 0 ; '. 0 / ' F 0 ; '. / F '. /; '. 0 /

' F 0 ; F . ; '. // F ; 0 D Ad F 0 ; F . ; '. // :

In particular, from Figure 11 it follows that Y 1 F. N ; nWN / ' W.n/ ´ .T2j ;2j 1 T

z

T 1 /: 2j 1;2j 2 2j;1L

N j >n

(4.32)

We would like to find the realization of the transformations D n=N with respect to the decorated ideal triangulation 1WN : z n=N ´ W.1/1 Dn=N W.1/: F 1WN ; D n=N . 1WN / ' D Proposition 3. The formula W.n/1 D1=N W.n/

D n1C6=N

n1 Y j D1

Y

T 1

} 2j C1;2j

(4.33)

T2N 1;2k1

N >k>n

b

1 T2N 1;2n } T1;2N 1 T2n1;2n

2n2 Y

Al1 PQN

lD1

mDn .2m1;2m/

P.:::;s;s1;:::/

holds for 1 n < N . In particular, when n D 1, Y 1 z 1=N D 6=N D T2N 1;2j 1 T2N 1;2L T1;2N O P.:::;2kC1;2k1;:::/ 1 T1;2

(4.34)

N >j >1

Proof. The proof is by decreasing induction in n. First, let us check that Equation (4.33) holds true at n D N 1. We write N 6=N W.N 1/1 D1=N W.N 1/

Q1

D T2N;1L T2N 1;2N 2 T 1

2N 1;2N

N 2 Y j D1

T 1

Q1 1

1 }T

2j C1;2j

1 T2N; 1L

T 1 }

2N 2;2N 3 2N ;2N 1

2N Y

Ak1 P.:::;l;l1;:::/

kD1

where we cancelled one pair of T-operators. Applying now the Pentagon relation to the underlined fragment and slightly reshuffling the commuting terms, we create

846

Rinat M. Kashaev

another pair of T-operators: D

N 2 Y

T 1

} 2j C1;2j

j D1

T

Q1 1

T 1 T 2N 1;2N 2 2N 1;1L 2N 3;2N 2

Q1 1

1

1

T1 T 1 T 1 T 1 } ;2N 1 2N 3;2N 2 2N 2;2N 3 2N 1;2N 2N

2N Y

Ak1 P.:::;l;l1;:::/ :

kD1

Applying twice the inversion relation to eliminate four T’s in the second line, we arrive at the desired result. Now, assuming that Formula (4.33) holds for 1 < n C 1 < N , we prove that it also holds for n: n6=N Ad W.n/1 D1=N 1 D n6=N Ad.T2nC2;1L T2nC1;2n / B Ad W.n C 1/1 D1=N }T

b

2nC2;2nC1

D

b

1 T2nC2;1L T2nC1;2n T1 } }T 2nC2;2nC1 2nC1;2n

Y

j D1

T1

} 2j C1;2j

z

b

1 T2N 1;2k1 T2N 1;2nC2 T1;2N 1 T2nC1;2nC2

N >k>nC1

b

n1 Y

1 T2nC2;2n }T}

b

T1

Q

2n;2n1 2nC2;2N 1

2n Y

Al1 PQN

mDnC1 .2m1;2m/

lD1

P.:::;s;s1;:::/ :

Applying the Pentagon relation to the underlined fragment, n1 Y Y T1 } T2N 1;2k1 D 2j C1;2j j D1

N >k>nC1

b

Q

b

T2nC2;1L T1

b

T1 T T1 T } 2nC1;2nC2 2nC2;2N 1 1;2N 1 2nC1;2nC2 2nC2;2n

b

b

Q

1 T2nC2;2n T1 } T2n; } 2n1 2nC2;2N 1

2n Y lD1

Al1 PQN

mDnC1 .2m1;2m/

P.:::;s;s1;:::/ :

Applying again the Pentagon relation and hiding one T into the product over k, n1 Y Y 1 T1 } T2N 1;2k1 T2nC2;1L T2n;2nC2 T T1 D 2nC2;2N 1 1;2N 1 2j C1;2j j D1

Q

N >k>n

T2n;2nC2 T1 }

b

T1

z

2n;2n1 2N 1;2nC2

2n Y lD1

Al1 PQN

mDnC1 .2m1;2m/

P.:::;s;s1;:::/


847

and two more Pentagon relations with subsequent cancellation of two pairs of T’s yield D

n1 Y

Y

T1

} 2j C1;2j

j D1

2n Y

N >k>n

1 1 T2N 1;2k1 T2N 1;2n } T1;2N 1 T }

Al1 PQN

mDnC1 .2m1;2m/

lD1

b

2n;2n1

P.:::;s;s1;:::/ :

Finally, an application of the inversion relation to the last T gives Equation (4.33). Formula (4.34) is a particular case of (4.33) corresponding to n D 1. Proposition 4. The formula z .nN /=N D 6.N n/=N D

Y nj >1

Y

1 T2nC1;2j 1 T2nC1;2L T1;2nC1 T1;2 O

1 T1;2kC1 P.:::;2l1;2lC1;:::/

n

(4.35)

n1

Y

1 T2nC1;2l1 T2nC1;1 T2nC1;2L T2N 1;2nC1

nl>1

1

Y

T2N 1;2

1 T2N 1;2k1 P.:::;2m1;2mC1;:::/

n

:

nC1nC1

nl>1

1

1 1 T2N 1;2L T2nC1;2L T1;2N O T2N 1;2 1 T2N 1;2nC1 T1;2 Y n 1 T2N : 1;2k1 P.:::;2m1;2mC1;:::/ nC1nC1

nl>1

1 T1;2N O T2;2N O 1 T2;1 1

Y

1 T2N 1;2k1 P.:::;2m1;2mC1;:::/

n

;

nC1l>nC1

1 T2N 1;2k1 P.:::;2m1;2mC1;:::/

n

:

nC11

PN 2 N n 'Nb .g2kC1;2N C1 / e i lD1 r2l G

n

Handbook of Teichmuller Theory, Volume III

Handbook of Teichmuller theory,

Handbook of Teichmuller Theory