An introduction to Teichmüller spaces

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With 43 Illustrations

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to Teichmiiller Spaces An Introduction

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Y. Imayoshi . M. Taniguchi

Yorcut IMAYOSHI lvayosHr YOICHI Departmentof of Mathematics, Mathematics,College college of of General GeneralEducation, Department Education,Osaka osaka University, University,Toyonaka, Toyonaka, Osaka560. 560.Japan Japan Osaka Ivfrrs.lHrxo Tesrcucnr M-\SAHIKO T-\:"IGliCHI Departmentof of Mathematics, Mathematics,Faculty Facultyof of Science, Science,Kyoto Kyoto University, Department University,Sakyo-ku, Sakyo-ku,Kyoto Kyoto 606, 606, Japan Japan

ISBN 4-431-70088-9 4-431-70088-9 Springer-Verlag Tokyo Berlin ISBN Springer-Verlag Tokyo Berlin Heidelberg HeidelbergNew New York York ISBN 3-540-70088-9 3-540-70088-9 Springer-Verlag ISBN Springer-Verlag Berlin Berlin Heidelberg HeidelbergNew New York Tokyo Tokyo ISBN 0-387-70088-9 0-387-70088-9 Springer-VerlagNew ISBN Springer-Verlag New York Berlin Berlin Heidelberg Heidelberg Tokyo Tokyo Springer-Verlag Tokyo1992 1992 © Tokyo @ Springer-Verlag Printed in Hong Hong Kong Kong This work work is subject All rights are reserved, reserved, whether the whole or part of the material subject to copyright. All i s concerned, c o n c e r n e d , specifically s p e c i f i c a l l y tthe h e rrights i g h t s of is o f ttranslation, r a n s l a t i o n , rreprinting, e p r i n t i n g , rreuse e u s e of o f illustrations, i l l u s t r a t i o n s , rrecitation, ecitation, broadcasting, ways, and storage banks. broadcasting, reproduction on microfilms or in other ways, storage in data banks. etc. in this publication does The use use of registered registered names, names, trademarks, etc. does not imply, imply, even in the absence absence of names protective o f a specific s p e c i f i c statement, s t a t e m e n t , tthat h a t such a m e s are r o t e c t i v e laws such n a r e exempt e x e m p t from f r o m tthe h e rrelevant elevant p l a w s and and regulations general use. regulations and therefore free for general use. Printing and binding: Best-set Best-set Typesetter, Ltd., Ltd., Hong Hong Kong Kong

August, 1991

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This is the English translation of our introductory book on Teichmiiller space written in Japanese. We have taken advantage of the opportunity afforded by this translation to correct some minor errors in the original text, and to include several new related topics as additional sections or subsections. Among other things, we add the construction of Thurston's compactification of the Teichmiiller space in §4 of Chapter 3, and the Thurston and Bers theory on the classification of Teichmiiller modular transformations in §5 of Chapter 6. We also include a sketch of the proof of celebrated theorems of Royden (§4 of Chapter 6), connection between Teichmiiller theory and deformation theory of the complex structures of Riemann surfaces due to Kodaira and Spencer (§2.4 of Chapter 7), and a derivation of negativity of curvatures of the Teichmiiller space with respect to the Weil-Petersson metric (§3.4 of Chapter 7). Further, we indicate how to verify that the compactified moduli space of a compact Riemann surface, constructed in Appendix B, is actually compact. We include several new references chiefly related to added parts. Other than these, the main body of the text is unchanged. Professors Tadashi Kuroda and Kotaro Oikawa kindly read the manuscript of the English edition and gave much valuable advice. We also record our gratitude to all friends and colleagues, especially Sadayoshi Kojima, Makoto Masumoto, Toshiyuki Sugawa, and Harumi Tanigawa, who rendered us much help by reading various parts of the manuscript. Finally, we would like to express our hearty thanks to Professor Kenji Ueno of Kyoto University, and to Springer-Verlag Tokyo for their kind and helpful support in achieving this English edition of our book.

Preface to the English Edition

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A Riemann surface is a connected one-dimensional complex manifold. Two Riemann surfaces R 1 and R2 are biholomorphically equivalent if there exists a biholomorphic mapping from R 1 onto R 2 . It is said that R 1 and R 2 have the same complex structure if they are biholomorphically equivalent. A Riemann surface can be also regarded as a real two-dimensional oriented ~ifferentiable manifold. Even ifthere is an orientation-preserving diffeomorphism between two Riemann surfaces, they are not necessarily biholomorphically equivalent. The question naturally arises as to how many distinct complex structures could be assigned on a given oriented two-dimensional differentiable manifold. This is called Riemann's moduli problem. Tracing its evolution, let us examine this problem more closely for closed Riemann surfaces. Let M g be Riemann's moduli space of genus g, that is, the set of all biholomorphic equivalence classes of closed Riemann surfaces of genus g. Since every closed Riemann surface of genus zero is biholomorphically equivalent to the Riemann sphere, obviously M o consists of one point. As is well known classically, the theory of elliptic functions and elliptic curves shows that M 1 is identified with the complex plane. In 1857, Riemann asserted that M g (g ~ 2) is parametrized by 3g - 3 complex parameters. He represented closed Riemann surfaces of genus

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In this book, we present the theory of Teichmiiller spaces which give a parametrization of all the complex structures on a given surface. This subject lies in the intersection of many important areas of mathematics. These include complex manifolds, Fuchsian groups, Kleinian groups, Lie groups, automorphic forms, complex analysis, algebraic geometry, differential geometry, topology in two and three dimensions, differential equations, complex dynamics, and ergodic theory. Recently, the theory of Teichmiiller spaces has begun playing important roles in the string theory. We have attempted to make the book as self-contained as possible. We begin from the most elementary aspects and primitive motivations. We also present subjects through typical examples and heuristic methods. We hope that these plans help the reader grasp the substance of Teichmiiller spaces.

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The problem of how to parametrize the variations of complex structures on a fixed base surface originated with G. F. Bernhard Riemann. This problem has spurred extensive investigations, and progress has been considerable in the areas of the theory of Riemann surfaces, algebraic geometry, and differential geometry.

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as finite finite branched branched covering covering surfaces surfacesof of the the Riemann gg as Riemann sphere, sphere, and and determined determined parameters of the number number of of parameters of M Mo by the the by the number of degrees of freedom of number of degrees of freedom of the the g points. branch points. branch In this this book, book, we we treat treat moduli moduli spaces In spaces through through Teichmiiller Teichmiiller spaces spaces and and groups as Teichmiiller modular modular groups as follows. follows. Teichmiiller Let R R be be aa closed closed Riemann genus g, g, and Riemann surface surface of Let of genus and let let E X be be aa marking marking on on i.e., aa canonical canonical system ft, i.e., system of generatorsof of generators R, of aa fundamental fundamental group group of pairs of R. .R.Two Two pairs (R,D) and (R', (B', E') D') are arc defined defined to to be be equivalent equivalent if (R, E) and if there there exists exists aa biholomorphic biholomorphic mapping f: -R'such that f. is equivalent equivalentto mapping -+ R' such that (E) is E] the to E'. Dt. Denote Denote by by [R, the f : RR--/.(X) [E,X] equivalenceclass classof of (R, (R,E). Such an an equivalence equivalenceclass equivalence E). Such E] is called a marked class [R, I] is called a ma"rked [R, closed Riemann surfaceof genusg. g. The Riemann surface of genus The Teichmiiller Teichmiiller space closed spaceT genusgg consists ?o of genus consists g of of all all marked closed closed Riemann surfaces surfaces of genus g. g. It is of genus of is verified that T ?,g has has aa canonical complex complex manifold structure, structure, and and it is canonical is aa branched branched covering covering manifold of of spaceMg. Mn.Its covering transformation group group is the moduli space Its covering is called called the the Teichmiiller Teichmiiller group M Modo which corresponds correspondsto the modular group odg which the change changeof of markings. markings. It turns out out that M Mng is is identified with the quotient space space Tg/Mod that TofModr,g, which has has aa normal complex analytic space spacestructure. complex spaceT The Teichmiiller space h* appeared appearedimplicitly implicitly in the continuity arguments arguments 4 g has Felix Klein and and Henri Poincare, Poincar6, who who studied Fuchsian of Felix groups and Fuchsian groups and automor1880s.Robert Fricke, phic functions from the 1880s. Fricke, Werner Werner Fenchel Fencheland and Jakob Nielsen Jakob Nielsen constructed T as aa real (69 -- 6)-dimensional Tc constructed real (6g 6)-dimensional manifold. Fricke Fricke also also 2 2) as g (g k ~ asserted that T asserted ?,g is aa cell. cell. Their Their method was was based based on the uniformization theotheosurfaces due due to Klein, Poincare, rem of Riemann surfaces Poincar6, and and Paul Koebe: Koebe: every every closed closed genus gS (~ surface of genus (> 2) is Riemann surface is identified with the quotient space space H /f r I of the upper half-plane H .I/ by aa Fuchsian Fuchsian group r f which is is isomorphic isomorphic to aa fun.R. Then each damental group of R. each point [R, I] in T corresponds to a canonical ?,g corresponds canonical [R, E] system of generators generators of r. we see system E] is represented by a point in l- . Hence Hence we see that [R, X] is represented [.R, 6 g- 6 which is called R6g-0 called the Fricke R E]. Moreover, the Poincare Fricke coordinates coordinates of [R, Moreover, Poincar6 lR,t). f1 induces induces the hyperbolic metric on R, metric on H .R, and the conformal structure defined by this hyperbolic metric corresponds defined corresponds to the complex structure of R. .R. One of Oswald Teichmiiller's great contributions to the moduli moduli problem was was to recognize it becomes recognize that that it becomes more accessible we consider accessibleif if we consider not only conformal quasiconformal mappings. mappings but also quasiconformal mapping means mappings. A A quasiconformal means also quasiconformal = satisfies the Beltrami . a homeomorphism which satisfies Beltrami equation = J.lW A Beltrami equatiotr W pu". ut7 A Beltrami z z coefficient p J.l measures measures the magnitude of deformation of a complex coefficient complex structure or a conformal structure. Around Around 1940 discovered an intimate intimate relation 1940 Teichmiiller discovered between between extremal quasiconformal quasiconformal mappings and holomorphic quadratic differentials, entials, and asserted asserted that T g is homeomorphic homeomorphic to R6g-0. R 6 g- 6 • He also also introduced the thatTn Teichmiiller distance Ts. distance o\ on T g• of the 1950s, Lipman Bers Bers developed developed the In the end of 1950s, Lars V. Ahlfors and Lipman fundamentals of of the theory of of Teichmiiller spaces, spaces, and they gave gave rigorous proofs for Teichmiiller's results. results. They They also also showed showed that that To Tg @ (g 2~ Z) 2) has has a natural natural complex structure structure of of dimension 39 3g -- 3, and can be embedded embedded in A2(R) A 2 (R) as as a bounded domain, where ,42(R) A 2 ( R) is the space space of of holomorphic quadratic differentials of of a closed R of of genus genus g. From From the Riemann-Roch theorem, itit is closed Riemann surface surface E

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The content of each chapter of this book is now briefly described. The purpose of Chapter 1 is to give primitive motivations and backgrounds for the following chapters. First of all, we recall the moduli space and the Teichmiiller space of closed Riemann surfaces of genus 1. After presenting the model of these spaces, we construct the Teichmiiller space T g of genus 9 as the set of marked closed Riemann surfaces of genus g. We also study Tg from the viewpoints of quasiconformal mappings, and of conformal structures induced by Riemannian metrics. In Chapter 2, we construct the Fricke space Fg of genus 9 (~ 2) which represents Tg as a subset of R 6g-6. We show that every Riemann surface except for a few types is represented by a quotient space H / r of the upper half-plane H by a Fuchsian group r. In particular, each marked closed Riemann surface [R,17] of genus 9 (~ 2) is identified with the corresponding canonical systems of generators of r. This gives the identification of T g with Fg • In order to show this fact, we also explain briefly the uniformization theorem of Riemann surfaces, Mobius transformations, and Fuchsian groups. Chapter 3 deals with the construction of T g (g ~ 2) from the viewpoint of hyperbolic geometry induced by the Poincare metric. The fundamental method is to decompose Riemann surfaces into a set of 2g - 2 pairs of pants by simple closed geodesics. Then the Fenchel-Nielsen coordinates on Tg are defined by geodesic length functions of 3g - 3 simple closed geodesics and twist parameters along these geodesics. We also study in the problem of what geodesic length functions of simple closed geodesics determine the points of T g • Chapter 4 is devoted to fundamentals of quasiconformal mappings. First of all, we define quasiconformal mappings, using two analytic procedures and a geometric method. Then we prove two fundamental theorems due to Ahlfors and Bers, i.e., the existence of a quasiconformal mapping satisfying a given Beltrami equation, and the holomorphic dependence of solutions on Beltrami coefficients. In Chapter 5, the Teichmiiller space T(R) of a closed Riemann surface R of genus 9 (~ 2) is constructed by using quasiconformal mappings. A Teichmiiller mapping of R means a quasiconformal mapping which is "locally affine" in certain sense, and obtained from a holomorphic quadratic differential on R. The essence of Teichmiiller's idea is that the extremal quasiconformal mapping in

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These deformations are closely related to each other. The essence of these studies is to investigate Teichmiiller spaces from these various points of view, and to clarify their relationships.

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(i) deformations of Fuchsian groups, (ii) deformations of conformal structures induced by hyperbolic metrics, and (iii) deformations of complex structures induced by quasiconformal mappings.

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known that A 2 (R) is a complex (3g - 3)-dimensional vector space. Furthermore, the action of the Teichmiiller modular group M odg on T g turns out to be a discrete group of biholomorphic automorphisms of Tg • In these studies, at least the following three descriptions of deformations of Riemann surfaces were used: IX

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Preface Preface

x

those which which determine determine aa given given point point of of "(E) T( R) is is aa Teichmiiller Teichmiiller mapping. mapping. Then Then itit turns out out that that Q Tg k(g >~ 2) is homeomorphic to to the space space Ar(F) A 2 (R) _of of holomorphic turns quadratic differentials on .R. R. Hence, Hence, ?s Tg is homeomorphic to to R6c-0. R6g-6. We also show quadratic that "(.R) T(R) is complete with with respect respect to to the Teichmiiller Teichmiiller distance. distance. that In Chapter Chapter 6, using the Schwarzian Schwarzian derivative, we construct the Bers In ofT(R) into a bounded domain in in ,42(.R.), A 2 (R*), the space space of of holomorphic hoiomorphic embedding of "(R) into quadratic differentials differentials on ft*. R*. Here, E* R* denotes the the mirror mirror image of of .R. R. By By the the quadratic A 2 (R*) is also also identified with with the (3g -- 3)-dimensional 3)-dimensional Riemann-Roch theorem, Az(R-) space C3r-3. C 3g - 3 . Using this embedding, embedding, we see see that that "(ft) T(R) has has a complex Euclidean space natural complex manifold manifold structure structure of of dimension 3c 3g -- 3. It It is also also proved that that natural the Teichmiiller Teichmiiller modular modular group M M odo odg is a discrete group of of biholomorphic biholomorphic autoautothe "0. of T properly discontinuously discontinuously on T This shows shows that that the ?r, morphisms of g • This g , and acts properly moduli space M g =Ts/Modc T g/ M odg has has a normal complex analytic space space structure structure of space Mo moduli dimension 3C 3g - 3. dimension Chapter 7 treats the Weil-Petersson Weil-Petersson metric on T 4.g • The holomorphic tangent with the dual space point space of T point [R, E] with space of of ,42(R). A 2 (R). Then is identified at a of To space g [.R,X] ?n' induces the Weil-Petersson Petersson scalar product on A 2 (R) induces Weil-Petersson metric on T on.42(R) scalar product the Petersson g• We give two two proofs for the fundamental that the Weil-Petersson metric metric is fundamental fact that Kahlerian. Both Both of due to Ahlfors. of them are a.redue Kihlerian. In Chapter 8, we establish beautiful formula due Wolpert, which due to S. S. Wolpert, establish a beautiful representation states that Weil-Petersson Kahler form on T has a simple representation h* Kihler the Weil-Petersson states that 4g with respect to Fenchel-Nielsen coordinates. coordinates. Fenchel-Nielsen with respect with Schiffer's We also deals with Schiffer's interior variA deals give two appendixes. Appendix A appendixes. Appendix also give Ahlfors' conation from quasiconformal mappings. We explain Ahlfors' viewpoint of quasiconformal from the viewpoint was , struction of the complex structure for which was the first construction of its for T Ts, g with respect degeneranatural complex structure. We also discuss variations with respect to degeneraalso discuss natural compactifications of Riemann surfaces. we explain briefly the compactificasurfaces.In Appendix B, we tion of moduli spaces. spaces. books and notes of books At the end of each are bibliographical notes there are each chapter, chapter, there is complete. articles to which we referred in the text. The bibliography is not complete. There we referred articles hope spaces. is a vast literature relating to the theory of Teichmiiller spaces. We hope that is a vast papers. omissions this list helps the reader to begin to explore these research papers. Any omissions these research explore reader helps ignorance. of references, reflects only our ignorance. theorems, reflects attribute theorems, references,or failure to attribute recThe authors are extremely grateful to Professor Osamu Takenouchi who recgrateful Professor extremely are genacknowledge ommended that we write this book. They also gratefully acknowledge the genalso book. we ommended erous and colleagues colleagues Makoto Masumoto, Hiromi friends and erous contributions of our friends manuread the original manuOhtake, Hiroshige Shiga, and Toshiyuki Sugawa, a^nd Sugawa, who read Ohtake, Hiroshige Shiga, and improvements. script, and made many helpful mathematical suggestions and improvements. suggestions script, and made many

=

Yoichi Imayoshi Yoichi Imayoshi M asahiko Taniguchi Taniguchi Masohiko

October, 1989 October, 1989

111 117

LII III q0I Z6 LL

92 105

77 77 , t

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Quasiconformal Mappings 4.1 Definitions and Elementary Properties 4.2 Existence Theorems on Quasiconformal Mappings 4.3 Dependence on Beltrami Coefficients 4.4 Proof of Calder6n-Zygmund Theorem Notes sturddel4l

luturo;uocrsen$ 7 raldeq3

Chapter 4

67 71 75

9L

u

seloN

Poincare Metric and Hyperbolic Geometry Fenchel-Nielsen Coordinates Fricke-Klein Embedding Thurston's Compactification Notes

L9 69 I9 I9

uotlecgrlceduoC s(uolsrnr{I t'8 Sutppaqurg ulaly-e{?q{ 8'8 sel€ulProoc uaEalN{erlcued 7,'t frlauroag cqoq.raddg Pue clrlel{ eJe?ulod I'8 cqoqJad'{g pue f-rlauroag

3.1 3.2 3.3 3.4

Hyperbolic Geometry and Fenchel-Nielsen Coordinates

59 51 51

sa?BurprooC

uaslarN-Iaqcual

g .ra1deq9

Chapter 3

seloN acedgerlct.rg 9'Z Eapow uslsqrnJ v'z suolleIuroJslle{l snlqg'Itr t'Z sSutrer\o3Iesra^Iuo z'z ueroar{J uoll€zrurroJlun I'c acedg eryltr

47 50

Frike Space 2.1 Uniformization Theorem 2.2 Universal Coverings 2.3 Mobius Transformations 2.4 Fuchsian Models 2.5 Fricke Space Notes

09 LV 8t t8 LZ 9Z 96

33 38 27

25 25

6 .ra1deq3

Chapter 2

20 24

vz OG 9I vl 8 I

16 1 8 14 1

I

seloN sernlcnrls l"urroJuoC Pue sernlcn.llS xalduro3 9'I acedg .relpurqcleJ Pue s3urddel4l pruro;uocrsen$ V'l f snueS;o acedS re1nwqcleJ, 8'I 4'I I snueg;o acedg rellnuqttal seceJJnsuu€tualu I'I aaedg Jallnurqcral

Teichmiiller Space of Genus g 1.1 Riemann Surfaces 1.2 Teichmiiller Space of Genus 1 1.3 Teichmiiller Space of Genus g 1.4 Quasiconformal Mappings and Teichmiiller Space 1.5 Complex Structures and Conformal Structures Notes f snuag;o

1 raldeqS

Chapter 1

slua+uoc

Contents

Contents Contents

XII

C h a p t e r 55 Chapter Teichmffller Spaces Spaces Teichmiiller Analytic 5.1 Analytic Construction Construction of of Teichmiiller Teichmiiller Spaces 5.1 Spaces 5.2 Teichmiiller Teichmiiller Mappings Mappings and and Teichmiiller's Teichmiiller's Theorerms 5.2 Theorerms Proof of 5.3 Proof of Teichmiiller's Teichmiiller's Uniqueness 5.3 UniquenessTheorem Theorem Notes Notes

119 119 119 119 127 r27 135 135 144 144

Chapter 66 Chapter Complex Analytic Analytic Theory Theory of of Teichmiiller Teichmiiller Spaces Complex Spaces 6 . 1 B e r s ' E m b e d d i n g 6.1 Bers' Embedding Invariance of 6.2 Invariance of Complex Complex Structure 6.2 Structure of of Teichmiiller Space Space 6.3 Teichmiiller Modular 6.3 Teichmiiller Modular Groups Groups 6.4 Royden's Theorems Royden's Theorems 6.4 6.5 Classification of Teichmiiller Teichmiiller Modular Transformations 6.5 Classification of Transformations Notes Notes

146 146 147 r47 152 r52 162 r62 167 r67 'l'71 171 179 179

Chapter 7 Chapter Weil-Petersson Metric Metric Weil-Petersson PeterssonScalar 7.I Petersson Scalar Product and 7.1 and Bergman Bergman Projection 7.2 Infinitesimal Theory of Teichmiiller Spaces 7.2 Spaces 7.3 Weil-Petersson Weil-Petersson Metric I\{etric 7.3 Notes Notes

182 t82 183 i83 189 189 loo 199 217 2t7

Chapter 8 Chapter Fenchel-Nielsen Deformations Deformations and Fenchel-Nielsen and Weil-Petersson Weil-Petersson Metric Metric 8.1 Fenchel-Nielsen Fenchel-NielsenDeformations 8.1 8.2 A Variational Formula for Geodesic 8.2 Geodesic Length Functions Functions 8.3 Wolpert's Wolpert's Formula Formula 8.3 Notes Notes

219 219 219 219 224 224 226 226 232 232

Appendices Appendices A A B

Classical Variations on Riemann Surfaces Classical Surfaces Notes Compactification Compactification of of the Moduli Moduli Space Space Notes

233 233 243 243 244 244 253 253

References References

254 254

List List of of Symbols Symbols

271 271

Index fu~x

2U 274

uopcnporluluv

An Introduction secBdsrellntuqrlel ol

to Teichmiiller Spaces

'eJnlcnJls xalduroc leuolsuetulp-auo qlrm ecedsJJopsneH pelceuuoc € srreeruplottuout xe\ilu.tocr)uorsueurp-euo to 'U uo emlxnJts xalduoc e saugeP }! }€tl} acn{.tnsuuoul?rg f)uoNsueurtP'?uo v s 91r3f{ (!z'!2) } te{t {es all pu€'U uospooqroqUfnu aToutp.tooc {o u.ta1sfi,e 'ursrqdrouroauroqcrqdrouroloq e ''a't 'Surddeur crqdrouroloqlq e sI

We say that { (Uj, Zj) hEJ is a system of coordinate neighborhoods on R, and that it defines a one-dimensional complex structure on R. A Riemann surface or one-dimensional complex manifold means a connected Hausdorff space with one-dimensional complex structure. is a biholomorphic mapping, i.e., a holomorphic homeomorphism.

n Uk)

Zk(Uj

n Uk)

(r2U !2)tz *

('VlU h)lz : rJzotz - lrz Burddeur uo-rtrsuertaql 'g * qnu rn U (IlI) 'aue1d

Zkj = zkozjl: Zj(Uj

-+

(i) Every Uj is an open subset of R, and R UjEJUj. (ii) Every Zj is a homeomorphism of Uj onto an open subset Dj in the complex plane. (iii) If Uj n Uk =P ¢, the transition mapping

xalduroe aq1 u1 f6, ?asqnsuado ue oluo f4;o ursrqdrouroeuoq e s lz ,treag (rr) 'g;o 'fpt>ln lesqns uedo ue sr f4 fra.tg (r) U pue

=

aq1 3ut,$st :(t'f 'El.l aas) suorlrpuoc eerql 3uu'ro11oy -les rrf{ (fz'fg) eq A P1 'slrolloJ i ,(tf.ttl e ql,ta ecedsgropsn€H Pelceuuoc e 'sSurddeu crqdrotuoloqlq fq s'p sr ef,"Jrns uueruelg Jo uoIlIuSeP PrsPuels eq; (fla^lqn?ul peqcled are qf,rqa{ aueld xelduro? aql uI suleruoP Jo uoll?allo3 3 sl }I 'acnttns uuounty e pell€c sI ploJlueur xaldruoc Palceuuo? I€uorsuaulP-auo Y

A one-dimensional connected complex manifold is called a Riemann surface. Intuitively, it is a collection of domains in the complex plane which are patched by biholomorphic mappings. The standard definition of Riemann surface is as follows. Let R be a connected Hausdorff space with a family { (Uj, Zj) hEJ satisfying the following three conditions (see Fig. 1.1): sacBJrns uuBrrrarlr Jo uol+Ilrsac

'I'T'I

1.1.1. Definition of Riemann Surfaces

s a J e J J n su u E r r r a r u ' I ' T

1.1. Riemann Surfaces 'aceJJnspeluelJo u€ uo crJlaru usruusrueru e fq pecnpur ernlrnrls leuroJuo) eql Pue alnlcnrls 's3urddeur xelduroc eq? uaealeq drqsuorlela.raql sl€ert g uollces leuro;uoctsenb r; ,(pn1s am 'p uorlcas uI 't snue3 ;o seceJrns uuetuelg ;o lulodarall eql urorJ pesolc pe{r?tu IIe Jo tes eql se I uotlcas uI PaugaP st t5 eceds renntuq)Ial eqJ 'I 'd t; snuaS;o eql aceds reilnurqttal frerltqre snuaS Jo lePoul e sa,il3 qllq/tr ;o 'salduexa aceds rellnruqcleJ ar{} pue eeeds Inpour aq} q1^{ sl€eP A uol}?es lecr '1sr1g -df1 auros pue sac€Jrnsuueruarll uoltrugap aq1 errrSar* '1 uorlcag ut Jo 'fqc1e4s 'pattluo eq II€I{s etuos Pu€ Jeql€r are s;oord aurog 'sraldeqc 3urno11o;eq? roJ spunorSlceq puts suotls^Ilou 'raldeqc s-rql uI arlr3 pue '6 snua3 ;o t; aceds rellnurqclal aql lcnrlsuoc aal

In this chapter, we construct the Teichmiiller space Tg of genus g, and give motivations and backgrounds for the following chapters. Some proofs are rather sketchy, and some shall be omitted. First, in Section 1, we give the definition of Riemann surfaces and some typical examples. Section 2 deals with the moduli space and the Teichmiiller space of genus 1, which gives a model of the Teichmiiller space Tg of arbitrary genus g. The Teichmiiller space T g is defined in Section 3 as the set of all marked closed Riemann surfaces of genus g. In Section 4, we study T g from the viewpoint of quasiconformal mappings. Section 5 treats the relationship between the complex structure and the conformal structure induced by a Riemannian metric on an oriented surface.

Teichmiiller Space of Genus 9 f snuag Jo aJBdS JaIInurIr.raI

I ra+dBtlc

Chapter 1

1. Teichmiiller TeichmfillerSpace Spaceof Genus Genus9g 1.

2

z a- Plane

ar-plane

Fig. F i g . 1.1. 1.1. (U, z) of aa Riemann surface A coordinate coordinate neighborhood neighborhood(U, surface R .R is aa pair of an open open set U ,R and aa homeomorphism homeomorphism zz of U into into the complex plane such such that that for [/ in R any element Uj iI ¢, (Ui, Zj) ri) of aa system system of coordinate coordinate neighborhoods neighborhoods with with un element (Uj, U nU1 $, the mapping z o z i L : Zj(U zi(U n z(Un zozt: Uj) Uj) nU ) - --+ - z(U nU i)

is biholomorphic. This U also called called aa coordinate coonlinale neighborhood neighborhoodof R. r?. Such [/ is also Such aa homeomorphism local parameter on U of R. z is said said to be be aa local local coordinate coordinale or ot aa local homeomorphism Z ([/, z) with A coordinate neighborhood called aa coordinate coorilinate neighborhood neighborhood with p E neighborhood (U, e U is called around p, and or local parameter around p. local coordinate local parameter arounil p. around p, and zz is is called called aa local coorilinateor Local analysis on aa Riemann surface surface R ,R is reduced to analysis on domains in parameters. For example, the complex function example, aa holomorphic holomorphic funclion complex plane via local parameters. on R ,R is aa function I/ on R l? such such that loz-1 f oz-L is holomorphic on z(U) for any (U,z) coordinate neighborhood z) of R. S neighborhood (U, surface,9 ft. A mapping If of R into aa Riemann surface is said mapping if wof oz-r1 is is holomorphic for all coordinate coordinate said to be aa holomorphic mapping if wo/oz(U, z) of R and (V, (V, w) neighborhoods u) of SS with with I(U) biholomorphic neighborhoods (U, /(U) C V. A biholomorphic --- SS means mapping onto S which has mapping I: means aa holomorphic mapping If of R has the Ronto,S f : R -+ -11 - R. holomorphic :: S surfaces Rand l? a"ndS Two Riemann surfaces holomorphic inverse mapping mapping 1S are are S -+ ft. Two f biholomorphically equiualenlif between R biholomorphically equivalent if there exists exists aa biholomorphic mapping between .R write and S. In this case, case, we regard Rand ,? and S ^Sas as the same same Riemann surface surface and write = S. R = also that Rand have the same same complex compler structure. slruclure. Complex S. We say say also R and SS have structures, biholomorphic mappings, mappings, and biholomorphic equivalence equivalencemay be be and are said to be conformal confortnal structures, straclures, conformal mappings, mappings, and and are actually often said (see§1.5). conlormal respectively conformal equivalence, equiaalence, respectively(see $1.5). Remark. A Riemann surface surface is aa two-dimensional two.dimensional real-analytic real-analytic manifold, manifold, and the Cauchy-Riemann determine its orientacoordinates determine Cauchy-Riemann equation implies that local coordinates

The holomorphic function z = w 2 maps biholomorphically both the upper half-plane H = { wEe I 1m w > O} and the lower half-plane H* = { w E C I 1m w < O} onto the domain D C - L, where L {x E R I x ~ O} is a cut on the z-plane. Take two copies D', D" of D and paste crosswise along their cuts L', L" as in Fig. 1.2. Then we get a two-sheeted covering surface Rover the z-sphere. Since the function f( w) = w 2 induces a homeomorphism F of the w-sphere onto R, we can define the complex structure of R from the complex structure of the w-sphere in such a way that F: C --+ R is a biholomorphic

crqdroruoloqlq e sl U r- C : Jr leql ferrr e qcns uI araqds r } = ? e r e q a ' I - C - O u l " u r o pe q } o t u o { 0 > r r r , - I = H aueld-}1eq I C > ^l = *H aurld-;1eqra/(ol eql pus {0 < rnurl I C > t} raddn aq1 qloq fllecrqdrotuoloqrq sderu .rn - z uollcunJ ctqdrouroloq aq;,

=

=

'z'T'ttd Fig. 1.2.

eueld-or

w-plane

z-plane

aueld-z

z=w'l

-4

y

~ R D"

'euo JeuJoJ eqt al"ts er* 'ala11'uollsnulluoc ct1f,1euefq ro Pue ln)r' ,,e1sed - (n)l - z 'dleotsselC 'panle^-e13utssr "n Jo Poqlau aql fq Pelrnrlsuoc sr 1r uorlcunJ crqdrouroloq eqlJo uorl)unJ asrelur aql qllqa uo e?"Jrnsuueruelg aql sI (lxaN slql '4 - rn uorlrunJ cre.rqe3peql Jo eeeJrnsuuetuelll at{} aes sn lal 'Qlt'{o} p"n (z'3) spooqroqq3raueleurProo?o'n1fq Peusep ?) sr C uo eJnlcnJls xaldurbc V'eceJJns uueurelg e osl€ sI'3 aueld xelduroc aql '{-} go iorlecgrlcedtuoc lutod auo eql q qc$Iir{ n C = ? ataqds uuvu'ery aq; '(r'O) eleulproo, auo fluo fq uarrrSsr O uo ernlcnrls xalduroc e pooqroqq3rau 'flaurep 'e?€Jrnsuueruolg e st aueld-z xalduroa aqt ul O ul€ruoP ,traaa'1p;o 1sltg

First of all, every domain D in the complex z-plane is a Riemann surface. Namely, a complex structure on D is given by only one coordinate neighborhood (D, z). The Riemann sphere C = C U {oo}, which is the one point compactification of the complex plane C, is also a Riemann surface. A complex structure on C is defined by two coordinate neighborhoods (C,z) and (C - {O}, liz). Next, let us see the Riemann surface of the algebraic function w = vIZ. This is the Riemann surface on which the inverse function of the holomorphic function z f( w) w2 is single-valued. Classically, it is constructed by the method of "cut and paste" or by analytic continuation. Here, we state the former one.

=

sa"BJrnS uueurarlr 3o salduruxg

=

'Z'T'I

1.1.2. Examples of Riemann Surfaces '[86-Y] '[OO-V] re3ut.rdg 1e3ar5'[37-y] ueur.re3uts Pus '[ga-V] €rN pue se{re.{ '[ZZ-V] ,tqoC pue seuof '[Ol-V] Suruung '[ag-V] ratsrog '[gt-V] s.reg '[g-y] oIr€S pue sroJIqV 'ecuelsut roJ (llnsuol 'sace;tns uuerualg 'pa1e1n3u€Ir}aq u"c pue slas uado;o ;o ,{roeq1 1e.raue8eql pue s}teJ esaq} .rog 'uoll€luelro sr$q elqelunot € seq e?eJrns uueuralg ,t.ra,ta1eq1 u^{onl-llaa s-ItI '.rageara11'uor1 sr eceJrns uusruarg € letll etunsse airr slql qtla paddrnba s,tea,r1e

tion. Hereafter, we assume that a Riemann surface is always equipped with this orientation. It is well-known that every Riemann surface has a countable basis of open sets and can be triangulated. For these facts and the general theory of Riemann surfaces, consult, for instance, Ahlfors and Sario [A-6], Bers [A-13], Cohn [A-22], Farkas and Kra [A-28], Forster [A-32], Gunning [A-40], Jones and Singerman [A-48], Siegel [A-98], and Springer [A-99].

saf,"Jrnsuu?uraru'I'I

1.1. Riemann Surfaces

3

1. 1. Teichmiiller Teichmiiller Space Space of of Genus Genus g9

4

=

mapping. This This RR isis the the Riemann Riemann surface surface of of ww = t/7. -JZ. (See (See Ahlfors Ahlfors [A-4]' [A-4], Chap. Chap. mapping. 8; Jones Jones and and Singerman Singerman [A-48], [A-48], Chap. Chap. 4; 4; and and Springer Springer [A-99], [A-99], Chap. Chap. 1.) 1.) 8; Note that that the the Riemann Riemann surface surface RR of of the the algebraic algebraic function function ww -= 1/7 -JZ isis also also Note 2 = z. u,2 equation by the regarded as the algebraic curve defined by the equation w z. defined regarded as the algebraic curve Finally, we we see see elliptic elliptic curves, curves, i.e., i.e., tori tori from from the the viewpoint viewpoint of of algebraic algebraic curves. curves. Finally, For any any complex complex number number )A(# (:;l: 0, 0, 1), 1), Iet let .R R be be the the algebraic algebraic curve curve defined defined by by the the For

=

equation equation 2

((1.1) 1.1)

w ( z -- 1 1)(z ) ( z --. \A). ). w 2 = zz(z

In other other words, .R R consists consists of of all points points (z,w) (z, w) eE C C xx C C satisfying algebraic algebraic In oo). We can define equation (1.1) and the point point pPoo == (oo, (00,00). define the complex structure structure equation of ,? R by by the complex structure structure of of the z-sphere z-sphere so that that the projection projection r: 7r: E R ---+ of covering C, r(z,w) 7r(z, w) = z, z, is holomorphic. This This r? R is a two-sheeted two-sheeted branched branched covering e, z-sphere with with branch points 0, 1, 1, I'A, and oo. 00. The mapping surface over the z-sphere surface written as is function = This f: R --+ e, C, f(z,w) holomorphic. f as u, w = is holomorphic. w, / : R fQ,u) f z( z - 1)( z - A) and R R is a Riemann Riemann surface on which which the algebraic function function \rc=W]

=

=

V

=

w -= {z(z vz(z _tG - 1)(z -, - A) is is single-valued. single-valued. u The Riemann Riemann surface surface ,R R defined defined by algebraic algebraic equation (1.1) (1.1) is rega.rded regarded topoThe of logically as as a surface surface illustrated illustrated in Fig. 1.5. copies of the Riemann 1.5. Take two copies logically (fig' 1.3). 1'3)' and 00 m (Fig. between,\ 1, and and between and 1, A and spheres between 0 and cuts between Sz with cuts ,ph".", Sl, St, S2 The join 1.5). (Fig. 1.5). place them face cuts (Fig. along their cuts Place along and 1.4), and face (Fig. 1.4), face to face Hence, R looks surface -R. resulting R. Hence, homeomorphic to the Riemann surface surface is homeomorphic resulting surface torus. A torus surface a torus. like such a Riemann surface We call such surface of a doughnut. We like the surface (see comes from the elliptic integral (see is name comes Lhis name curue; this elliptic curve; called an elliptic is also also called §1.4). $1.4).

°

00

00

Fig. F i g .1.3. 1.3.

1.1. Riema.nn Surfa.ces

5

sef,"Jrns uu"uall|I'I

Fig. 1.4.

'r'r'ttJ

Fig. 1.5.

'e'r'tt.r sa?BJrns rruBtuaru pasolc '8'I'I

1.1.3. Closed Riemann Surfaces uoll€ler l€lueu"PunJ eql sogsll€s Ptre 6g t6, t"''rg Iy uro.r;pecnpul luql'luVJ ' "' ' llgr] ' lly] sess€lcfdolouroq eq1 fq pale.rauaEsr od lurod eseq q?lar U 1o (od'g1ttt dno.r3 leluaurePunJ eqtr '(f't ;St.f) seprs dy ql1,u uoE,tlod xaruoc e o1 crqdroruoeuoq ul€tuoP e 1aBar* ' " ' 'r€l(I}t selrnf, uerll 'g'I 'EU ul se od lutod as€q I{lI^r ug'6V Pasolcaldurts od e4e; e lurod 3uo1e g, lnt pue d snuat Jo U ac"Jrns uueuell{ Pasolc e uo 'acottns uuourery uado ve Pe11eo y 'snua! allug Jo ac"Jlns uuetuerll pesolc € sI st a?"Jrns uueruarg lceduroc-uou u^rou{-lle^{ sl U'I snuaS;o $ snro} e Pue /tJa^a a?€Jrnsuu€r.uaru l€qt lceduroc '6 snuaE;o sr araqds uuerualg aql 'f snua6 to ecottns uuDut?NAPesop e PelIe? sl g'I '3lJ uI sB solPu"q d qtl,u alaqds e o1 crqdrouroeuoq ec€Jrnsuueuary Y

A Riemann surface homeomorphic to a sphere with 9 handles as in Fig. 1.6 is called a closed Riemann surface of genus g. The Riemann sphere is of genus 0, and a torus is of genus 1. It is well-known that every compact Riemann surface is a closed Riemann surface of finite genus. A non-compact Riemann surface is called an open Riemann surface. Take a point Po on a closed Riemann surface R of genus 9 and cut R along simple closed curves AI, Bl, ... , A g , B g with base point Po as in Fig. 1.6. Then we get a domain homeomorphic to a convex polygon with 4g sides (Fig~ 1.7). The fundamental group 71'1(R,po) of R with base point Po is generated by the homotopy classes [Ad, [Btl, ... , [A g ]' [B g ] induced from AI, B I , ... , A g , B g and satisfies the fundamental relation 6

9

'(rlunaqr) r = r-[rB]r-tfvltrsltrvlL[ Il[Aj][Bj][Ajtl[Bjtl = 1

(the unit).

r=f

j=l

t=f{ !g'!V} lo srolor?u?6lo an1sfrsf)?ruouoc n

1ec e16

We call {[Aj],[Bj]U=1 or {Aj,BdJ=1 a canonical system of generators of 71'1(R,po). ro t=r1VAl'llV)}

'(od'a)rv

l.1. Teichmiiller Teichmiiller Space Space of of Genus Genus g9

6

. 6 . ((gg = Fig. 1.6. = 33)) F ig. 1

: 3) (g = 3) 1.7. (g Fig. Fig. 1.7.

of Tori Tori 1.1.4. Representations of Group Representations Lattice Group L.L.4. Lattice

r

plane C C by by complex plane of the the complex C/lf of quotierrt space spaceC We the quotient asthe torus as represent aa torus shall represent We shall meromorphic a single-valued = is aalattice group r. Since w(z) = J z( z 1)( z A) is a single-valued meromorphic lattice group l-. since ur(z) fiG4Q=U complex the complex considerthe we can can consider (1.1), we function by equation equation (1.1), definedby torus R R defined the torus on the function on = th9 on 'R,the p (z(p),u''(p)) point integral any of 1/w(z) along paths on R. For any point p = (z(p), w(p)) on R, For paths on ft' along integral of Ilw(z) u(z) function algebraic of branch elliptic integral 0, they are linearly independent over the real number field R. We call such r a lattice group for R. A lattice group r for R is regarded as a subgroup of the analytic automorphism group Aut( C) of C. In fact, every 'Y = m7l"1 + n7l"2 E r is identified with a translation 'Y(z) = z + m7l"1 + n7l"2 of C. We say that two points z, z' E Care equivalent under r if there exists an element 'Y E r with z' = 'Y( z). Denote by [z] the equivalence class represented by z.

Fig.l.S.

'e'r'ttJ

The quotient space C/ r of C by r consists of all equivalence classes [z]. This quotient space C / r is realized as a surface obtained by identifying sides

saprs Eurr(;rluapr fq peul"lqo aceJJnse ss Pazllear sl J/C eceds luarlonb sq; '[z] sasselc ecuele,rrnba lle Jo st$suoc.7 ,{q C P JIC eceds luarlonb aq;

Space of of Genus Genus I9 Teichmiiller SPace l.1. Teichmriller

88

A with with A' A' and BB with with 8' B' in in the the lattice lattice of of Fig. Fig. 1.8 1.8 by by the the translations translations 7r1,tr2, 11"1,11"2, ,4 respectively. respectively. Now, we we define define aa complex complex structure structure of of C CI/ f . Let Let r11":: C C +-+ C CI be the the projecprojecNow, / f be tion, i.e., i.e., "(r) 1I"(z) - lzl [z] fot for zZ €E C. C. Introduce Introduce the the quotient quotient topology topology on on C/f , which which tion, is defined defined as as follows: follows: asubset a subset U U of C open ifif the the inverse inverse image image r-r(Lr) 11"-1 (U) is is is open otC/f is open in in C. C. ItIt is verified verified that that C/f CI is a connected connected topological topological space. space. open [a], [b] €E Cf l,wewe can take take neighborhoods neighborhoods 7o,V6 Va, Vb of of a,b For any two two points points [o],[6] For with r(I/") 1I"(Va) nn r(%) 1I"(ltb) -= {.4>. Since Since z11" is an open mapping, mapping, this this shows shows that that C/iwith Hausdorff space. Moreover, Moreover, for any point point [c] [a] eE C/f , taking taking a sufficiently is a Hausdorffspace. Va of of a, we see see that that n11" gives a homeomorphism homeomorphism of of v" Va into into small neighborhood neighborhood vo small Ua = r(Vo) 1I"(Va) and Za: (Jo Ua --+ Vo Va be a homeomorphism with with zo(lzl) za([z]) = z. and zo: C/f . Let Uo Then (t/",2o) (Ua,za) gives gives a coordinate neighborhood around lalin [a] in C/f - Thus Thus C/f Then becomes a torus, i.e., i.e., a closed closed Riemann surface surface of of genus genus I1 such that that the projecbecomes 11": C ---+ Clf CI is holomorphic. The The triple triple (C,r,C/f) (C, 11", CI F) gives gives an example of of tion zr: tion universal coverings, coverings, considered considered in $2.1 §2.1 of of Chapter 2. universal '--' C lf As is known in the theory of of elliptic elliptic functions, the mapping [@]: [4>]: r? R -+ I we see Hence biholomorphic. sending a point pER point [4>(p)] E C Hence we see point C/lis p to a point sending e R [O(p)] e c/lr r that a torus defined defined by equation (1.1) (1.1) is represented represented by a Riemann surface surface CI that for a lattice group l-. In In Chapter 2, we we shall show show that that every torus is represented represented it is 2.13). Conversely, conversely, it by a lattice group l- in C (see the Corollary c (see corollary to Theorem 2.13). elliptic biholomorphic known that such a Riemann surface C I is always biholomorphic to an elliptic surface Riemann known that /f refer to Ahlfors [A-4], curve details, we refer algebraic equation (1.1). For details, defined by algebraic curve defined [A-4], Siegel 3; Siegel Chap' 3; 2; Jones and Singerman [A-48], Chap.7; Clemens [A-21], Chap. 2; Singerman Jones Chap.?; Clemens [A-21], [A-48]' Chap. [A-98], Chap.l. 1; or Springer Springer [A-99], Chap. 1; [A-99], Chap.1. [A-98], Chap.

r.

=

r

r

Iris

clr,

clr,

clr

clr,

clr.

=

clr.

=

clr

r

Iris

r. r

r

r

1.2. Genus 11 Space of Genus Teichmiiller Space 1.2. Teichmiiller genus 1. 1. Let us space of genus us construct the Teichmiiller space

1.2.1. of Tori Tori The Moduli Moduli Space of L.2.1. The c/f, surface C Riemann surface We I r, is represented represented by aa Riemann every torus is the fact fact that every use the we use 2.13). Theorem where r is a lattice group on C as in §1.4 (see the Corollary to Theorem 2.13). (see corollary group in on c as where ]- is a lattice $1.4 assumefrom from we may assume r* ZI1I"1 On necessary'we z ........ zf 4,if, if necessary, On performing the transformation Z I and ones ca"nonical I a,re the the beginning that the generators 11"1 and 11"2 for r are the canonical ones 1 and 12 Lor generatols ?r1 and the the beginning that Tr with 1m T > 0, respectively. respectively. with Imr ) 0, Now, group lattice group consider aa lattice Now, consider

rr m,€ nZ E} ,Z}, f " == { {, j== mm * n+ r lnT m ,I n group rr I} the lattice lattice group = { wa.sseen seenin in §1.4, where As was C II 1m ImrT > 0}. As > O}. H = where Tr E €C $1.4,the €H {rT E = Cf l, surface R, corresponds to a subgroup of Aut(C), and the Riemann surface R = CI rr isis and the Riemann ,Aul(C), corresponds to a subgroup of r the f, has thal cf Notice to c/f,. aa torus. Denote by 11" r the projection of C to C Irr. Notice that C Irr has the projection of c torus. Denote by r, the structure group. additive group. of an an additive structure of

'(z'dtsalH =tw

M1

~

H/PSL(2,Z).

'(Z'Z)lSd fq g;o eceds 'sl teqt luarlonb eql qtl^\ PeUIluePtsl I,f41}€tll sarTdurrI'I uraroeqJ'IrolJo sassBl?af,ual -earnba rrqdrotuoloqlq IIe Jo tas eqt ''a'r'r.uoqto aeods,Ppout eqt aq rW P"l 'g reddn aql aue1d11eq 'ilno.r,6Jolnpou eq1 ;l ,L fra.rrg

the modular group. Every, E PSL(2, Z) is a biholomorphic automorphism of the upper half-plane H. Let M 1 be the moduli space of tori, i.e., the set of all biholomorphic equivalence classes of tori. Theorem 1.1 implies that M 1 is identified with the quotient space of H by PSL(2, Z), that is,

3o ursrqdrourolne crqdrouroloqlq e s\ (Z'7,)'IS1

( l P + t ' c- , ' ' l = ( z ' 7 , ) r s d Q ) L| I\ t = " 9 - p o p u e z ) p ' ? ' q ' "ll '?7+=! o

PSL(2, Z) = { ,(T) =

:;:~ I a,b,c, dE Z

and

ad - bc = 1 }

) dnorS aq1 1ec a,u 'aaog

Now, we call the group

0

tr sl ,U -

'l'(P - ([z])/ fq ua'rt3 + n)) ,'A I t Surddeu crqdroruoloqrq e uaql 'splotl (8'I) ;t 'f1asra,ruo3 'I = cq - pD s^eq a1ll

we have ad - bc = 1. Conversely, if (1.3) holds, then a biholomorphic mapping f: RT, given by f([z]) = [(CT + d)z].

RT

IS

-lD + tcl '0< (, ,nD;q ,rwr _d

1m T'

= ICT + dl 2 (1m T) > 0, ad- bc

/-1

0

/(1)

=1

f = (t[or-;f

',! - (,t){or-/ pu" esuls 'I+ = ?q - pe l3q1 aas e^a ' (eJoureqlrr\{ 'srafielur eJ€ suotlela.reqt urorJ /p Pue ,?' ,9' ,D alaqlvr

where a', b', c', and d' are integers. Furthermore, from the relations and /-l o/(T') = T', we see that ad - bc = ±1. Since

a'T' +b' c'T' + d"

,tP*'rP -" rQ* ,'t,o 'r-! o, lueurnS.reetues eql 3ut{1ddy 1aBen T=-...,..-----,-,-

Applying the same argument to

/-1, we get aT+b cT+d

'Ptt? - " ' 9*te

T=--.

,

ur€?qo ein 'a.ro;ereqa 'sre8alur ar€ P Pue

'? 'q'D aleqlr

where a, b, c, and d are integers. Therefore, we obtain 'P+tc=n=(1)l

/(1) = a = CT + d, 'q+tP=1a-(,t')l

/(T') = aT' = aT + b, aleq a^{'ecua11'? rapun 0 = (0U o1 luap,rrnba are (1)/ pue (,-r.)rfqloq snq1, '0= d ecueq pue'0 = (0)1 leql erunsse,teur eaa'.re,roaro141 '(9'6 eufrua1 ,lc ra = (,)! lc) 0 + lc pue sreqlunu xelduoc 5rc ! pue ereIIA\'d + '(7'6 rue.roaql 15) s€ ualllr^a sr / uaqa 'j t, o. 'crqdrouroloqlq q .f esn€ceg 'l ,!)Lo! qanf 3 - C,! Eurddeurcrqdrouroloqe'sr 1eq1 P ! Wle lo"o-wrll 'pelcauuoc t(1durs sr acurg 1i slsrxe eraql teql salldtq ureroeqf fuorPouoru aq1 '1srtg 'ig oluo ,"A /oo"l4' Io / Surddeu crqdrouroloqlq e q arerll l€t{} alunss?

Proof. First, assume that there is a biholomorphic mapping f of RT, onto R T. Since C is simply connected, the monodromy theorem implies that there exists a lift / of f, that is, a holomorphic mapping /: C - C such that 7r T o/ = f07r T I ~f. Theorem 2.4). Because f is biholomorphic, so is f. Then / is written as f(z) = az+f3, where a and f3 are complex numbers and a:f. 0 (cf. Lemma 2.8). Moreover, we may assume that /(0) = 0, and hence f3 = O. Thus both /( T') and /(1) are equivalent to /(0) = 0 under FT' Hence, we have

'c 'q 'o e.taym 'I = cq - p,Dqwn sta,aTut arD p PUD

Theorem 1.1. For any two points T and T' in the upper half-plane H, two tori R T and RT, are biholomorphica//y equivalent if and only if T and T' satisfy the relation , aT+b T=--, (1.3) cT+d where a, b, c, and d are integers with a'd - bc = 1.

,Plrc _ t , g*tp uoxlnpt, tf, fi,1uopuD fi Tuapamba frllocttlilloutopqrq erD 'tg puo at17filsr7os p puv t lt .r,edilneql w / pao t squr,oilony fiuo rol '11- tuaroaql uol onl 'g auold-{1ot1

(e'r)

'Z'I I snue5;o aoedg rall+urqrral

1.2. Teichmiiller Space of Genus 1

9

1. Teichmiiller Teichmiiller SPace Space of of Genus Genus g9 1.

10 l0

is known known that that the the quotient quotient space HIPSL(2, is aa Riemann Riemann surface surface (cf. (cf. PSL(2,2)Z) is spaceHf ItIt is (cf. of §2.4 of of Chapter Chapter 2) 2) and and that that aa fundamental fundamental domain domain (cf. $a.2 §4.2 of Chapter Chapter 2) 2) for for $2.4 PSL(2, Z) is is the the shaded shaded area area in in Fig. Fig. 1.9. 1.9. Intuitively, Intuitively, we we get get the the Riemann Riemann surface surface PSL(2,2) HIP S L(2, Z) bV by identifying identifying the the sides sides of of this this fundamental fundamental domain domain under under the the H/PSL(2,2) -lfz transformations zz >1--+ zz +1 + 1 and and zz e1--+ -liz as as is is illustrated illustrated in in Fig. Fig. 1.9. 1.9. Hence Hence transformations we see see that that the the moduli moduli space space of of tori tori is is biholomorphic biholomorphic to to the the complex complex plane. plane. For For we more details, see, see, fot for example, Ahlfors Ahlfors [A-4], [A-4], Chap.7; and Jones Jones and Singerman more [A-48], Chap. 6. [A-48],

-1

o Fig. 1.9. Fig.1 .9.

parameter A(# ,\(f Remark. (1.1) depends depends on a complex parameter Remark. A torus given by equation (1.1) tori S>. S1, such tori 51 and S>., 0,1), It is well known that two such ,Sr. It denoted by S>... 0, 1), which is denoted are exists aa linear fractional if there there exists and only if equivalent if and biholomorphically equivalent are biholomorphically set oo }} of S>. transformation which takes 1, A, }, 00 Sr to the set set of branch points {O, takes the set { 0, 1, Chap. example, of branch of S>" (see, for example, Clemens [A-21], Chap. (see, for Clemens points {0,1,A',00} ,91, 1,^',m} branch points [A-21], {0, and only equivalent if and 2.7). are biholomorphically equivalent 51, are we see see that S>.. 51 and S>..' 2.7). Thus we if A' numbers: one of the following numbers: )/ is is equal equal to one 't l A-I l-1 11 1 A, 1A, ) , . \ . ~' ' + , 1 - 1-A-' ,\-1 ) ,r r - ) A' ' = l/A 1/) and and generated by gl(A) Now, group of of order order 6, 6, generated finite group be aa finite let G G be Now, let cr()) = -.\ = This fact C 1 of D = C - {O, g2(A) = 1I - A which are automorphisms of are analytic analytic automorphisms {0, I}. }. This fact Sz(\) = (cf.§2.4 quotient space of D Dby G (cf. spaceof shows where DIG by G meansthe the quotient Df G means D/G,where that M ML= showsthat $2.a 1 ~ DIG, ---+ which C, F : D of Chapter 2). Moreover, we find a biholomorphic mapping F: DIG --+ C, which mapping of Chapter 2). Moreover, we find a biholomorphic lG = f(A) is with uy F([A]) r([.1]) = is defined defined by /(.\) with

( ,2\ -2 -A^+ + 1)3 l)3 (A = A2(A _ 1)2 /()) = f(A) 1)2 t2()

'Z'I I snuaD1o aoedgrallnurq)ra;

1.2. Teichmiiller Space of Genus 1

11

I snuaC go acedg rallntuqcral

1.2.2. Teichmiiller Space of Genus 1 'Z'Z'I

'(Ot't'3t.f pu€ .{1a.,rr1cedsa.r',r. aas) I o? (,r)/ pue (1)1 '{q spuas (1)//z r T between lattice groups. Thus we can consider that the difference between 'I

r

l.1. Teichmriller Teichmiiller SPace Space of of Genus Genus g9

12 12

and r'corresponds r' corresponds also also to to the the different different choices choices of of generators generators of of ur1(R',po), 1r1(RT,po), i.e., i.e., and

[B 1(r)]}] and and {{/.([A I. ([B 1(r'))) ]} (see (see Fig. Fig. 1.10)' 1.10). 1(r)], [B'(")] 1(r'))), /-([s'("')]) /.(Fr("')l)' {{[A F'(")1, generators E, Now, for any torus R, take canonical system generators E p = {{[A [B 1 ]}] of system take a canonical any torus ft, 1 ), [Br] Now,for Fr]' pair (r?,Xo). Such the of the fundamental group 1r1(RT,p) of R, and consider the pair (R,E ). Such and consider group of R, r1(.R,,p) p of the fundamental = and .Do' Ep ]} a E is called a marking on R. Two markings E = {[A ), [B and Epl = markings r?. Two on p a p called a Xo is 1 {[1t]'1 [Bt]] _= curve continuous Co exists a { [A~], [Bi]] [Bn} are said to be equivalent when there exists a continuous curve Co when there to be equiaalenl said are { [,4i], on R R f.oln from pP to to p' p' which which induces induces the the isomorphism isomorphism Ts": Tc o : T1(R,p) 1r1(R,p) --+ t1(R,pr) 1r1(R,p') ot = an element sends Here, ?c'" = with [A~] = Tc ([A )) [BU = Tco([Bd). Here, Tc sends [C) of of Tc"(lBr)). and Tc"(lAi) 1 with [Ai] [C] o o [Bi] 'C,] . product of the definition 1r1(R,p) to [C;l . C· Co] of 1r1(R,p'). For the definition of product For ol r{R,p'). C an element r{R,p) to [Co-1 of curves, curves, see see $2.2 §2.2 of of Chapter Chapter 2. Next, Next, two pairs (R, E) E p ) and (S, -Do) E q ) as above above of '-+ R h: S mapping equivalent if only if biholomorphic mapping h: are a biholomorphic exists if there and only if equiaalent arc = o i s e q u i v a l e n ttto = ) such that h.(E = h.({[A~],[B~]}) = {h.([Am,h.([Bm} is equivalent t . ( { t h a t h . ( D ) such q lA',,1,[B't]]) {h-([Ai]),h.([Bi])] call of (R,D). We class E {[A ], [Bd}. Denote [R, E ] equivalence class of (R, E ). the equivalence = by Denote p p p 1 Ep [fi,X0] 1[Ar],[Bt]]. such a [R, torus. The The Teichmiiller space space T1 T1 of of genus genus 1 consists consists of of p] a marked lorus. Do] lR, E all marked tori. tori.

=

Theorem 1.2. every point E H, H, let let E(r) E(r) == {{[A [B 1(r)]} be be the the point rr € For euery L.2. For 1(r)), [Br(t)]] Theorem [,41(r)], and, r in to I conv,sponil for which (r)] (r)] correspond on R = CIF [A [B 1 and r marking and R,T c/1, T for 1 1 marking [Br(r)] lA1(r)l = r r' . only if if and in T1 , respectively. Then [RT,E(r)] F = [RT"E(r')] T if ifr = r'. T rvspecliuely. Then [-R",X(r)] 1 1,, lR,, , E(r')l is a biholomorphic there is Proof. Assume biholomorphic Assume that [RT,E(r)] Prool. fR,,,E(r')]. Then there [R",](r;1 = [RT"E(r')]. = is - R mapping RT, RT t h a t h.(E(r')) h . ( X ( z ' ) ) = {h.([A s u c h that h: R , such , , -+ m a p p i n g h: { h . ( [ . 4 11(r'))),h.([B ( r ' ) ] ) , h . ( [ , B11(r')))} ( r ' ) ] ) ] is = h([0]) that assume equivalent E(r) = {[A h([O)) [0] W" may assume 1(r)], [B 1(r)]}. We equivalent to I(r) [0] [Bt(t)]]. { [41(r)], - h([z)) by replacing if necessary. necessary.Then, the definition h(tO]) if n(lrl) -- h([O)) hr(ltl) with h replacing h with 1([z)) == that h.([A h.([.41(ti)]) implies that of equivalence h.(X(r')) implie~ X(r) and h.(Elr')) equivalence of E(r) 1(r)] and 1(rJ)) = [A [,41(r)] = = az for some some h() with h(0) O. 0. Then h(:) = h.([B Take a lift h of h with_h(O) h.([,B1(r')]) 1(r)]. Take 1(r'))) = [B [,B1(r)]. = r. = = ar' and h(r') h(rt) = ar' = d 1, and complex 1, h(1) a we conclude concludethat h(l) Hence we number a. Hence complex number Therefore, we 0tr converse is obvious. T' , The converse we have have r = r'.

=

=

=

=

some r in Since E p ] is is represented representedby [R marked torus [R, every marked T , E( r)] for some Since every [.R',X(r)] [ft,Xo] H, I/' 7r1 is is identified with H. shows that T f/, this theorem shows diffeo' Another method to mark tori is orientation-preserving diffeorealized via orientation-preserving is realized groups. generatorsof fundamental groups. morphisms systemsof generators instead of systems between tori instead morphisms between pair (S, (S,/) = {[Ad, Then any any pair on,R. For [B 1 ]} on R. Then f) purpose,fix E = fix aa marking marking E that purpose, For that {Ft],[.B1]] defines aa R--+ Ss defines of S and diffeomorphism I: orientation-preserving diffeomorphism an orientation-preserving torus.g and an of aa torus f :R = {/.([A ([.B1]) on S. S. marking 1 )) }] on ma.rkingI.(E) 1 )), I. / ([B f.(21 = { /.([At]), -- S, - S' be S' be R -+ let I: R -+ S,g: Theorem S, and g: R tori, and and let be tori, anil S' St be Let R, R,S, L.3. Let Theorem 1.3. f :R = [S',g.(E)] in T if T1 orientation-preserving Then [S,/.(E)] diffeomorphisms. Then 1 if orientalion-preserving diffeomorphisms. [S',9-(t)] in [S,/-(t)] = - S'. --* h: St. mapping S and : S -+ S' is homotopic to a biholomorphic mapping h: S -+ a biholomorphic to homotopic is .9 S' if go/-l andonly only if Sof-L:

points r,r' I/ for for r,r' E = [S',g.(E)]. two points € H Proof. Take two that [S,/.(E)] Proof. Suppose Supposethat [S',g.(t)].Take [S,f.(t)] = = E(r), where .R", R,', which [~,E(r)] and [S,/.(E)] [RT"E(r')), where R , RT" E(r), and E]: T which [R,E] fR7,D(r')], [S,/.(tI lR,,x(r)] [,R, to of ~ E" to diffeomorphisms of and r') are as diffeomorphisms and gg as a^sabove. above. Regard Rega-rd1 defined as are defined and E( ':(z') / and r', 1, and and r', and rr to to 0, 0, 1, l, and RT,. send 0, 0, 1, and g their lifts lifts jI and that their assumethat may assume .R,,. We We may f send by setting setting and'g respectively. between jf and homotopy between we obtain obtain aa homotopy Thus we respectively. Thus i by

=

=

8I

I snueC 1o acedg rapurqslal

1.2. Teichmiiller Spa.ce of Genus 1

13

'r; I;0 'c) "

'Z'I

Ft(z) = (1 - t)j(z) + tg(z),

z E C, 0 ~ t ~ 1.

'(r)0r+(z)!(?-t)=Q)'4

Putting Ft([z)) = [Ft(z)], we have a homotopy Ft between I and g. Hence, go/-l: R.,.I --+ R.,., is homotopic to the identity. Conversely, suppose that go/-l: S --+ B' is homotopic to a biholomorphic mapping h. Two mappings hoI and 9 from R to B are homotopic. Let Ft : R --+ B' (0 ~ t ~ 1) be a homotopy between hoI and g, and let Po be a base point for the marking E. Let Co be a continuous curve on B' from hol(po) to g(po) which is given by Ft(po), 0 ~ t ~ 1. By the isomorphism Teo: '1r1(B', ho/(po)) --+ '1r1(B',g(po)) induced by Co, we see that markings (hof)*(E) and g*(E) on B' are equivalent, which implies that [B.!*(E)] = [B',g*(E)]. 0

'1ue1e'unbaele '5i] '[(f)'t'rS] =l(S)'l teql saqdurr qcrq'u tr s3ull.reur t"qt ees et 'og.{q pacnpur ((a)0',g1rt, ,g uo (g3)*f pu€ (3').(/"q) "c; rusrqdroruo$ aqt fg 't j ? ; 0'('d)tg fq uarr3 q qtlq/'a * (("it)1or1'r,S)r:r : ruor; ,S' uo elrnc snonurluo? e eg. oC ?e'I'3 Euqreu aqt ('d)6 oq ('i\!rrt rog lurod aseq e aq od 1e1pue 'f pue /or1 uear$leq ,tdolouroq 3 3q (I i t i O) rf, p"I'ctdolouroq e.reS: ol U uror; f pue ;l'or1sEutddeur o/'al 'V Eurddeur ,S * U : 'd1asra,ruo3 * g : crqd.rouroloq.rq€ of crdolouroq sI 1eq1 esoddns rS ,-to6 'f1r1uapreql ol ctdolouroq sr ,? - 'tg : r-to6 '[(z)rd'] = (lr))',t 3q?tnd (ecuag 'f ptre ;f uaenlaq rg fdolouroq e a^eq ear

Surddeu reeutl eql teql '"2f = 'S] pue aas eal 'l(")S'oA) = k'U] rsqr qsns l? ) !'o! slurod [(".r.)3' log = [d3'g] ret{t q?ns S' +- A i;f urstqdrouroa; om1 3ur1e1 ',(1en1cy't(f)T'S] -;rp Suur.rasard-uor1e1ue1ro lre PuU "^ '[d3'5] snrol pe{reur f.rerltq.re rr€ roJ

For an arbitrary marked torus [B, Ep ], we find an orientation-preserving diffeomorphism I: R --+ B such that [S, Ep ] = [B,I*(E)]. Actually, taking two [R.,., E(T)]' we see points To, T E H such that [R, E] [R.,.o' E(To)] and [B, Ep ] that the linear mapping

=

=

z E C,

'c)z'6ffia=Q)'!

'(Z'dlSa l,;igaeeds /(ff); qrp pagltuaPrsI uol;o s\ (Z'Z)IS1'acuepuodsalrocslql qnpoutrr eql pu€'(U)J.to lc€ ol PelaPrsuoc (U)Z;o ursrqdrourolneerqdrouroloqlqaql ^A'(Z'Z)lSa I t ol Sutpuodsarroc sI *[fr] uaql '[r1oo"t''A) = (l't ''U])-["] fq (U),, uo uoll?e EI eugePPu" ""A = 'r-(("')L1o"'rt) =.'! tnd '(U)J A loursqdrouroaglPe sr qctq^r !1as1roluo ui 1t"llr;"""rrti'!,Ul = l(')tS,(r)!'] ulelqo "^ r("t)Llo"tqo'! = (')r;'o'r1ecurg 'l(i)'ri = (Ir])'rl fq ua,rrE'A teql /$oqs ol lu?Icgns sr 1r lB ',S] eleq e.rlrsnqa 'ctdolouoq er€ rd '16' Pue / feqt flrsea a,rord ,Sl = [/ uec a \ 'aue1d aql ul {s!p lrun aq} o1 crqdrouroeuoq $ U urorJ fg pue fy 1e Eur -1e1apfq peurc?qo uretuop aql aculs 'uorlrusap aqt fq [td'S] = [6',9] teqt atoN

Note that [5', g] = [5, gl] by the definition. Since the domain obtained by deleting all A j and B j from R is homeomorphic to the unit disk in the plane, we can prove easily that f and gl are homotopic. Thus we have [5,/] = [5', g]. To prove the surjectivity, it is sufficient to show that for any [5, E'] E T g there exists an orientation-preserving homeomorphism f of R onto 5 for which [5, E'] [5, f*(E)]. This is a well-known fact, but we shall give a construction of such an f. First, we may assume that all Aj, Bj,Aj, and Bj in E = {[Aj ], [Bj] and E' = { [Aj], [Bj] are simple closed smooth curves. Let P~ be the base point for E'. Further, set

=

U=I

U=I

6

6

9

= U(Aj U Bj),

Ro

= R- C,

r=! =oA,CAn!,V))=,C'(gnlil)=

"

r=f

j=1

C' = U(Aj U Bj),

5 0 =5-C'.

.,C-S=oS,C-A

9

C

j=1

Surf;sr1es{odl - ,C '1xap oluo {0d} - C Jo 3;| usrqdrouoagrp Sur,traserd-uotleluelro ue a1e1 '(,2'S) uor; fear aruesaq? ur peurclqo {srp lrun pesolc st{} "y fq elouaq 'uy {slP }Iun Pasol, eql o1 crqdroruoaJlp sl d leqt asoddns feur arvr'4,;o xalra,r qcee punor€ ernlcnrls alqerlueragrp Surceldag '(f 't '4.f ;c) aueld eql uo saprsf6 qll/'^ d uo3flod Pesolc e pue oU ueearleq ursrqdrotuoagtp Sur,trase.rd-uorleluetro ue s1srxe eraql ueqtr,

Then there exists an orientation-preserving diffeomorphism between R o and a closed polygon P with 2g sides on the plane (cf. Fig. 1.7). Replacing differentiable structure around each vertex of P, we may suppose that P is diffeomorphic to the closed unit disk .dR. Denote by .d s the closed unit disk obtained in the same way from (5, E'). Next, take an orientation-preserving diffeomorphism II; of C - {Po} onto C' - {p~} satisfying {'d}

- !g)21 lS = (oit}

fI:(Bj - {Po}) = Bj - {p~} -

fI:(Aj - {Po}) = Aj - {p~}, ' { o d } - t , V= ( o d }

- !y)s1

for all j = 1, ... , g. Here we assume that II: preserves the orientation of all loops A j , Bj, Aj, and Bj. Let i be the restriction of fI: to the boundary aLJ.R of .dR. Take a sufficiently small neighborhood U of Po in R which is diffeomorphic to a closed disk, and consider the set E on aLJ.R corresponding to C n U. Then we can construct a diffeomorphism aLJ.R -+ aLJ.s for which coincides with on aLJ.R - E. It is shown that this extends to an orientation-preserving diffeomorphism F: .dR -+ .d s (Hirsh [A-42], Chap. 8, Theorem 3.3). Now, projecting F to R- U, we obtain an orientation-preserving homeomorphism g: R- U -+ 5 - g(U). This g is not necessarily smooth in a neighborhood of each A j and B j . Thus, using a smoothing theorem (Hirsh [A-42], Chap. 8, Theorem 1.9), deform g homotopically to an orientation-preserving diffeomorphism gl : R - U -+ 5 - g(U). Furthermore, take a neighborhood UI of Po which contains U and is diffeomorphic to a closed disk. In the same way as for construct an orientation-preserving diffeomorphism go on UI from gl on aUI . Define an orientation-preserving diffeomorphism g2: R -+ 5 so that g2 = g on R - UI and g2 = go on UI . This g2 is not necessarily smooth in a neighborhood of aUI , and hence by using again the smoothing theorem, deform g2 homotopically to an orientation-preserving diffeomorphism f: R -+ 5. The construction of f shows that f*(E) and E' are equivalent, and hence [5, f*(E)] = [5, E']. 0

'[f 'S] =l(5).{ '5'l acuaq pue'1ua1e,rtnbe are ,3 Pu€ (g)Y tnql E s,r,roqs/ Jo uollrnJlsuol aql 'S - U : / usrqdroruoagrp Sur,trasard-uot1e1uat.ro ue o1 .{llerrdolotuoq zf ur.ro;aptlualoaql Eurqloours aq1 ureSeSursn .'tqecuaq pu€ 'rng z, slq;,'r11 uo 06 = 7'6pue lo pooqroqq3reu e ur qloorus dl.ressaeaulou sI rn - A uo f - z6 WqI os S r- g : zf ursrqdrouoegp Surrrrasard-uoll€lualJo u€ augeq 'r1lg uo Id uorJ tp uo o6 usrqd.rouroagrpEut,rreserd-uolleluelro ue lf,nrls -uoc '3[ ro; se fe,r,r atu€s eq? uI '{qp pesol? e o1 crqdrouroagp q Pue 72 sul€}uoc qcrqaiodgo t4 pooqroqq3tau e ar1e1'aroruraqlrng '(tD| - S * If -y :16 tustqd -rouoesrp Surarasard-uolleluelro ue o1 flectdoloruoq f ruro;ap '(6'1 ureroeql 'g 'd*qC '[ZfV] qsqg) ueroeql Surqloorus e Sursn 'tnqtr 'lg p* fy qcea;o r

the ratio of the major axis to the minor axis of this ellipse is

u asdrlla srr{}Jo srxe rounu aqt o} srxe roleur aq] Jo ol]er el{l

'l"l(l(o)'rl - r)l(o)"/l + r)l(o)"/l i l(o)zl 5 l,l(l(o)'tl sarlrlenbeutaqt ,tg '(tt't '8t"f) aueld-rn aqt ul asdrlla ue o1 aueld-z aql ur 0 raluec qt-ra elcrlc e spues 7 deur 't > r€aurteqt 'raaoero141 l$)"t /(O)ttl = l(g)r/l pue O * @)"1 1eq1saqdunqcrq,ra

which implies that fz(O) =P 0 and 1J-l(0) I = Ifz(O)/fz(O)1 < 1. Moreover, the linear map L sends a circle with center 0 in the z-plane to an ellipse in the w-plane (Fig. 1.11). By the inequalities

'o< - .l(o)"/l = (o)/r .l(o)"/l sagsrles0 - z le (6)f uerqocel s1t 'urstqdrouroagrpSutrlraserd -uorleluerro ue sr / acurs '0 - z Ie / ;o uorsuedxa ro1,te; eql Jo tural rapro lsrg eqt aq z(iltt + z(g)'t = G)l 1a1 'aue1d-rnxalduroc eqt ul /O uIPruoP e oluo aueld-z xaldruoc aql ul 6 urSr.roeq1 Surureluor O ureluop € Jo tuuqd.rouroagrp Surarasard-uorl€luarroue sr / l€ql etunssearra'spooqroqq3reualeurpJoo?Surraprs -uoc 'srql easoI 'sluer)lgeoc tusrtleg;o Surueeurcr.rlatuoa3eq1 ureldxa a.tr'1srrg

First, we explain the geometric meaning of Beltrami coefficients. To see this, considering coordinate neighborhoods, we assume that f is an orientation-preserving diffeomorphism of a domain D containing the origin 0 in the complex z-plane onto a domain D' in the complex w-plane. Let L(z) = fAO)z + fz(O)z be the first order term of the Taylor expansion of f at z = O. Since f is an orientationpreserving diffeomorphism, its Jacobian Jj(O) at z = 0 satisfies 1.4.2. Quasiconformal Mappings s8urddetr4l lBruroJrrocrsen$'6'7'1

',t1t1eur.royuoc uorJ 3[ 3o uotletaap aql ernseauro1 pesn sl / Jo luerrlgaof, lruprllag aql pue 'g uo arnlrnrls xaldtuoc eql '(U)"f ul Jo uorleruroJepe sluasardar (y)"6 ul U'S] lurod e leql su€errrlr ef,uag {y :l 7eq1 Wl'lAl = [/'^g] ?eql s^roqssrq;'Surddeur crqdrotuoloqlq€ q,S * pue 'tusrqd.rouoeJrp Surl.rasard-uolleluelro ue sr /U * A :p? deu flrluepr eq1 'tl)D{ 's1as se (Io"*'("1)r-t) 1eq1 U = /U leql eloN } spooqroqq3raueleurp 'deilr slqt uI -rooc ;o uals,ts qlurr paddrnba IU ateJJns uu€uIeIU A\eu e aleq ea,t 'U uo arnlf,nrls xaldruoc " seugep v>a{(toDm'("1)vt) } spooqroqq3raueleu ',9 - g : ursrqdrouoeslp Sut,uasard-uolleluelro ue roJ -rprooc rt ;o ualsIs B puts S uo vl"{ ("*'"A) } spooqroqq3rau eleutp.roocJo ualsds e rog '.Lrop

which is called the Beltrami coefficient of f on R. Now, for a system of coordinate neighborhoods {(Va' wa ) }aEA on 5 and for an orientation-preserving diffeomorphism f: R ----> 5, a system of coordinate neighborhoods {U- 1 (Va ), waoJ) }aEA defines a complex structure on R. In this way, we have a new Riemann surface Rj equipped with system of coordinate neighborhoods {U- 1 (Va ), waoJ) }aEA. Note that Rj = R as sets, that the identity map id: R ----> R j is an orientation-preserving diffeomorphism, and that f: Rj ----> 5 is a biholomorphic mapping. This shows that [5, f] = [R j , id] in T(R). Hence it means that a point [5, f] in T(R) represents a deformation of the complex structure on R, and the Beltrami coefficient of f is used to measure the deviation of f from conformality.

'U uo / 1o Tuata$aocnaDr?Iegeql pallec$ q)nl^t

(e'r )

dz J-lj = J-l dz'

(1.5)

''P ,t - trl

zp

where Zkj = Zkoz;l. This shows that the set of Beltrami coefficients of f on coordinate neighborhoods of R induces a differential form of type (-1, 1) on R. Thus we denote this differential form of type (-1, 1) simply by

,tq ,{ldurrs (t't-) ed{1 ;o ruroJ l"rluereJlp slq} a}ouap e^r snq;, 'U el€ulprooc uo ad{1 (I'1-) Jo urroJ leltuareJlp € se)npul U Jo spool{ro,Q{31au - l{z areq^l uo /go sluerrlgeoc rur€rllagJo les eqt leql s.lroqsslql'rizors

(~~j) / (~~:)

'(trut2)tz uo (#) on

Zj(Uj n Uk),

(r r)

trl

J-lj = (J-lkOZkj) .

(1.4)

l@).(rzotrl)=

e^tsqa,lr'Q * qnU ln '(qz'qn) pue rurerllag uaq,11',,(learlaadsar ol qll/'^ Jo sluarf,lgeoc (lz'fn) leadsar ./ eqt eq 'trl pup ld lr,1't1 1 (r2)l pue ln > (dI ?€qt qcns g p (tn'tn1 '(!m'11) spooq.roqq3reu eleurproocpue gr 1o (tz'qn) '(tr '.!2) spooqroqq3rau

neighborhoods (Uj,Zj), (Uk,Zk) of R and coordinate neighborhoods (Vj,Wj), (Vk,Wk) of 5 such that f(Uj) C Vj and f(U k ) C Vk. Let J-lj and J-lk be the Beltrami coefficients of f with respect to (Uj, Zj) and (Uk, Zk), respectively. When Uj n Uk =P ifJ, we have

'7'1

1.4. Quasiconformal Mappings and Teichmiiller Space

17

a*dg

L1

ralnurqf,ral pu? s8urddul4l purro;uorrsen$

1. Teichmiiller Space Space of of Genus Genus g 1. Teichmriller

188 1

This shows shows that that any infinitesimally infinitesimally small circle with with center center 0 is mapped by /f to This ellipse whose whose ratio ratio of of the major major axis axis to the minor minor axis is K(0). K(O). an ellipse

L(z) L(z)

I'

-I'

--t ~

cp rp -= a := a bb =:

1

8=Zar o : l a r egutL(O) $)

8+arg/,(O) 0+aref,Q)

( 1 +l p ( o ) l ) r l l , ( o ) l (l+ltL(O)I)rl/,(O)1 (r-lp(0)l)rlf,(o)l (l-ltL(O)I)rl/,(O)1

Fig.l.ll. Fig. 1.11.

This statement holds we also also call the Beltrami Beltrami holds at every every point in D. Thus we coefficient , \

ffz(z) t(r)

p t Q=) =fz(z)' f f i , zED, z eD , IlI(z) if /f the complex of /f at z. As we we saw before, III saw before, dilatationof complen dilatation Ft = 0 on D if and only if quasiconfonnal D. We call f a quasiconformal mapping mapping of D to is a biholomorphic mapping on a We is a f D' if f satisfies satisfies Dt if f = sup .". sup11l*+ IIIlI(z)1 Kr . 00. lrrl'J! KI ' = < t ?()I zED z) l , e b r -- l pIII coefficient Ill. Further, ff is called quasiconformal mapping We mapping with Beltrami coefficient called aa quasiconformal Lrt.W" call K I d,ilatationof ff.. call K1 the maximal dilatation quasiconformal mappings. mappings. We We shall shall In this chapter, we only consider consider smooth quasiconformal chapter, we quasiconformal mappings 4. study more general general quasiconformal mappings in Chapter 4. value IIlI(z)1 absolute value Transformation (1.4) implies implies that the absolute tansformation formula (1.4) lprl(z)l of the = diffeomorpJQ)dzldz an orientation-preserving Beltrami = III (z) di/ dz of an orientation-preserving diffeomorcoefficient III Beltrami coefficient W is aa - S l?. Thus III phism f: I II is local coordinates coordinates on R. does not depend depend on local .9 does lpy I : R --+ get we get r? is is compact, compact, we I I| < 11 on continuous and III on R. ,t. Since Since R continuous function and lpty

= sup su pIIlI(z)1 111l11i00 l py( z) l m l l :v

fldurrs e

a simply connected Riemann surface

L1={lw! e x p(-2'11"2jlog ( - 2 r 2 / l o g , \A) ) a ((iv) iv) F (> 1 et r = or a g .l < w € w h e r e l o g z d e n otes l}. Define'll": exp(2'11"ilogzjlogA), (z) = e x p ( 2 t r i l o g z / l o g . \ ) , where logz denotes A bby y r'II"(z) e f i n e r ; H - --- + A 1 ). D its principal principal branch. Then Then 11 H becomes becomes a universal covering surface of of the its annulus ,4. A. e /, a n d llet e t r'II"b be (v) bee aal alattice byy 1 aand and n d aa ppoint o i n t r €E IH, t t i c e ggroup r o u p ggenerated e n e r a t e db ( v ) tLet et 4T b C is a universal projection of of C onto onto the quotient quotient space Then space C fj rfr.T • Then the projection covering surface of the torus C/ C j lr.T • covering surface of

=

r

r

=

'1, R - R wittr '1ro^l is-called Any biholomorphic mapping "I: 1"0"1 = z'II" is called a coaering covering fr, E with Any (R,r, denote by (R,r,R). given covering covering(ft, transformation '11", R). For a given '11", R), denote covering(R, transformalion of a covering mappings, rl' the set covering transformations. By the composition of mappings, set of all its covering group (,R.'r,,R). forms a group, which is called the covering transformation group of (R, 11", R). transfonnation coaering called lgroup of (R, (-R,'r,.R) In particular, particular, we 11", R) lransformation group uniaersal covering coaering transformation we call I the universal if R is a universal covering surface of R. r?. if ,R

r

r

Example 2. We give covering transformation groups (i)', ... , (v), of coverings (i), ... , (v) in Example 1. The notation ("11,' .. ,"In) expresses the group generated by "11, ... ,"In' - *2tri. with "I1(Z) (i)' 2'11"i. (i)' r f = ("11) (rr) with 71(z) zz + (ii)' (ii)' r f ==( r("11) r ) w with i t h 7"I1(Z) 1 ( z )== Z z+ * 11.. - zz exp groupof order n. ordern. finitegroup is aa finite (iii)' r (2'11"ijn) , which is exp(2riln),which with "I1(Z) ('rt) with 1= ("11) 71(z) = )2. with "I1(Z) (iv)' AZ. (iu)' r r = ("11) (zr) with 71(z) "fz(z)= Z z+ a,nd"I2(Z) where"I1(Z) (T,72) -- r f,,T , where + r. (")' r (v)' 7 = ("11,"12) * 11 and 7{z) = zz +

=

=

= = =

= =

=

=

=

2.2.2. Construction Surfaces Covering Surfaces of Universal tJniversal Covering Construction of means aa path on surface R R means First of on aa Riemann Riemann surface definitions. A path we need need several several definitions. of all, all, we points and C(0) 1]. The continuous The points C(O) and the interval interval [0,1]. where.II is is the I -- R, R, where curve C: C: 1 continuous curve [0, also points of respectively. We We also of C, C(l) lern'inal points C, respectively. initial and and terminal to be be the the initial are said said to C(1) are confusion book, if if no no confusion the book, say path from Throughout the to C(1). from C(O) c(0) to c(1). Throughout that C is aa path say that c is same letter letter C. C. is by the the same possible,its also denoted denoted by is also its image image C(I) C(/) is is possible,

s8urraaogl"sra^ru1'Z'Z

2.2. Universal Coverings

29

'd = (ilu tlA u\ il,"r;3:"9:r|"'; A uo CWed e sI Ar uo,, qled e o't11llv 61 pre6 sl q!. q d lurod y 'U ecsJrnsuueruelg e;o Sutrerroce eq (g')L'A) p"l '0),C lurod leururrel eql pue (0)C lutod prltut aq1 qf!^A U uo ,C . C rlled e 1aBaar 'rg;o lurod FIlluI eqt qq^a C;o lutod Ieunurel eq1 Eurlcauuoc ,(q '(g),9 = (t)C lsq? qcns A ao tC Pue C sqled oarrlrog

For two paths C and C' on R such that C(1) = C'(O), by connecting the terminal point of C with the initial point of C', we get a path C . C' on R with the initial point C(O) and the terminal point C'(1).

Let (R, 'Tr, R) be a covering of a Riemann surface R. A point p in R is said to be lie over a point P in R if 'Tr(p) = p. A lift of a path C on R is a path G on R with 'TroG C. Now, by using paths we shall construct concretely a universal covering surface of a Riemann surface. Fix a base point Po on a given Riemann surface R. Let (C, p) be a pair of any point p on R and any path C on R from Po to p. These two pairs (C, p) and (C', p') are equivalent if p = p' and C is homotopic to C' on R. Denote by [C, p] the equivalence class of (C, p). Let R be the set of all these equivalence classes [C, p].

=

= 4z turl eql }eql ees err,r')Lodz xelduroc parrnba.raq1 splarf {(!z'!2)},tgurel -les 'gg ur ureurop pelcauuoc fldurrs e s\ dn l€qt qrns (or'on) pooqroqq3rau al€urprooc e e{sl 'U p [d'C] - gf lurod fue ro;'1ce; u1 'Surddeur ctqdrouroloq '1xe11 e setuof,aqA * U:1, l€qt os Ar uo alnltnrls xalduroc 3 euueP a,n 'deur e Eurra,roc Jo uorlrpuoc aql sagql"s pue U oluo Ur yo Eutddeur (uolllnrlsuof, eql ,cg 'a : (la'gl)r. rq eas o1 fsea q snonurluoc € sr l 1l 1eql ua,rr3 uorlceto.rd eql ee U * A:)L p1 'sceds 1ecr8o1odo1 JroPsneg € seuoteq g ueql 'A u\ 4 Jo spoor{roqq3rau pluaurepunJ Jo uralsfs e ouuep s^\ 'd2 aseql p;" dn uea/$leq aauapuodsauoc euo-ol-euo Isf,Iuouef, e a^€LI e,t 'uteutop ig,'h palcauuoc ,tldurrs e sr.d2 acurg 'D o1 d uroq d4 ur pauteluoc qled frerlrqre ue st d4 ';2 ul .{eirrr e qcns ul ur e sr b pw 1eq1 lutod 2I lb'bC.9] slurod II3 Jo les aqt dn pooqroqq3rau e 4p fq alouaq 'U ul ul€ruop pelceuuoc fldurrs-e $ qtul^r d lo '9, ol Peau an '1srrg e ecnPor?ul rog uo ,(3o1odol {ue e{el 'g lurod { Jo ld'Cl 's,lrolloJ se qqt leqt ees eA\ e satuoreq eceJrns Surra,roc U l"srelrun Jo B '[d'g] sasselcacuale,rrnba aseq?ileJo les eql eqU lef '(d'C) Jo ssplt acuele,unbaeqlld'CJ dq aloueq'g uo tC ol crdolouroq sr j pue d - d !\Tuapanba erc (d' ,g) pue (d'9) srted oarrl aseq; 'd o1 od uror; A uo C r{led fue pue U uo d lutod fue 3o rted e eg.(d'C) 'ecsJJns ulretuelg e 'U eosJrns uuetuell{ ualrE e uo od Jo lurod es€q 3 xld te1 'alotr1 ecsJrns Surre,rocl"srellun e i(lalarauoc lcnrlsuoc lleqs arrrsqled Sutsn fq

We see that this R becomes a universal covering surface of R as follows. First, we need to introduce a topology on k For any point p = [C,p] of R, take a neighborhood Up of p which is a simply connected domain in R. Denote by Up the set of all points [C· C q , q] in R in such a way that q is a point in Up and C q is an arbitrary path contained in Up from p to q. Since Up is a simply connected domain, we have a canonical one-to-one correspondence between Up and Up. By these Up, we define a system of fundamental neighborhoods of pin k Then R becomes a Hausdorff topological space. Let 'Tr: R -+ R be the projection given by 'Tr([C,p]) = p. By the construction, it is easy to see that 'Tr is a continuous mapping of R onto R and satisfies the condition of a covering map. Next, we define a complex structure on R so that 'Tr: R -+ R becomes a holomorphic mapping. In fact, for any point p = [C, p] of R, take a coordinate neighborhood (Up, zp) such that Up is a simply connected domain in R. Setting Zp = Zpo?!, we see that the family {(Up, zp)} yields the required complex structure on R. 'ur uo ernlcnrls

R,

'p?peuu@ fiylul'ts puD pep?uuoc st 'aaoqo pautoldxa sD peptulsuoc 'g acottns ?qJ 'T'Z Btutuarl

Lemma 2.1. The surface

constructed as explained above, is connected and

simply connected.

fdolouroq € eleq am'f11eutg 'I ) n roy (n1)"g = (n)(c't)p fq g'uo (t't)p qled e ausep a,r'7 x / f (s'?) fue ro3'uaqJ,'1 3 s due .lo;'d = (I)"f, = (0)'J pue 'C = rI 'oI - og sagsrles tl'{ ("'.)uf = 'd } leql q?ns uaql uee^.rleq rldolouroq € eq U * 1 x I ii p1 'o7' o1 crdolouoq sI C leql su€etu q?Iqtl 'l"d'Cl,tq peluasarde.r 'q1ed pasoll " $ 'fod'Cl -C ecurs lod'ollleqt epnpuoa am sl C Jo lurod leurturel eql puie 'od lutod es€q IIII^a g uo qled Pasolee sI 5t uaql 'ei" '[od'olTol crdolouroq q C tnd [od''I] lutod eseq q]p\ g uo g qled Pesop ',r,r,o11 fierta 1eq1 aes ol luelrgns fl 1I 'Paleeuuos ,(ldturs q f€qt b,rord arr,r !f '[d'g] ol 'pelceuuoc q U leql satldtut qclqar lod'oll uro+ U uo I qled € a^eq al,r '7 ) s fle,ra iog [(s)g'"C] = (s)g 3ur11ag'I ) 1ro; (ls)j = (rt"C rq A uo'C Wed e eugep'1 I s q?ea ro,{'U uo qled e Aqlod'o1f ql.I^{ pal?auuoc sl lI f [d'g] lurod i(ra^a teql A{oqsol sjcgns }l 'U Jo ssauPa}ceuuoc aq1 a,rord ol1[t'0] - I ) ?,(ue ro; oa = (t)"1,(q paugap g-uo qled eq] aq oI p"I'too.t'4

Proof. Let 10 be the path on R defined by Io(t) = Po for any t E I = [0,1]. To prove the connectedness of R, it suffices to show that every point [C, p] E R is connected with [Io,Po] by a path on k For each s E I, define a path C$ on R by C.(t) 9(st) for t E I. Setting G(s) [C.,C(s)] fo: every s E I, we have a path Con R from [Io,po] to [C,p], which implies that R is connected.

=

=

Now, we prove that R is simply connected. It is sufficient to see that every closed path G on R with base point [10 , Po] is homotopic to [Io,Po]. Put C = 'TroG. Then C is a closed path on R with base point Po, and the terminal point of G is represented by [C, Po]. Since G is a closed path, we conclude that [Io,Po] = [C, Po], which means that C is homotopic to 10 , Let F: I x I -+ R be a homotopy between them such that {F. = FC. s) }.El satisfies Fo = 10 , F 1 = C, and F.(O) F.(1) Po for any s E I. Then, for any (t, s) E I x I, we define a path C(t,.) on R by qt,.)(u) = F$(tu) for u E I. Finally, we have a homotopy

=

=

30 30

2. 2. Fricke Fricke Space Space

F1r,"; E between b"t*""n C and[Io,Po] F(t, s):; I Xx I -- t R i and by setting settingF(t, F1t,"; s) = [Get,.), F.(t)] for for llo,polby [C1r,,;,r"(t)] (r, s) s) E 1 1. Therefore, we (t, I x I. Therefore, we conclude that R is simply connected. 0B conclude E is connected. € putting these these observations observations together On putting together with the uniformization theorem, theorem, we we obtain obtain the following following theorem. theorem. ,

Theorem Theorem 2.2. For Fo'revery eoerg Riemann surface surfaceR, there there exists exislsa universal uniaersalcovering coaering surfaceR which is is biholomorphic surface R of R, which biholomorphic to to one one of the the three three Riemann surfacesC, Riemann surfaces A, C, or H. C,orH. Throughout Throughout this section, section, as 6 ao universal universal covering covering surface surface R E of aa Riemann surface R r? we we always surface always take take the one one constructed constructed above. above. From the construction of such such aa universal universal covering, covering, it is get the is easy easy to get following lemma by an an argument argument similar to that used used in the case case of analytic continuation (Ahlfors [A-4], 8). [A-4], Chapter 8). (Existence and path) For any Lemma Lemma 2.3. (Existence and uniqueness uniqueness of of aa lift path lift of of aa path) any path point p, p, on with point C on R initial and for any P R initial and for ang there exists erists a unique R oaer there unique F of Rover lifl C wilh initial initial point p. lift C of C with fi. (Litt of Theorem Ttreorem 2.4. (Lift of aa mapping) mapping) For Riemann surfaces surfaces Rand let R and S, let (R,Tn,R) (S,trs, (R, 1fR, R) and 1fs, S) be and (5, be their universal uniaersal coverings coaeringsconstructed construcledas as explained etplained preaiously,respectively. giuen an previously, respectiaely.Then Then given an arbitrary arbitrary continuous continuousmapping mapping ff:: R R-- t S, S, there exists etists a continuous there -t 5 continuousmapping mapping it R fr,--with f01fR osoi. This Thit mapping mapping S with forp== 1fSoj. - (1, where is uniquely uniquety determined tleterminedunder iI is untler the the condition conditionthat that i(pd wherePI fr. and € Rand i@) = ql, fu E = ff(1fR(PI». are such such that that 1fs(qt} rs(Q1) = S are iiI e 5 4r E brn(Fr)) Moreover, Morvooer, if if ff is differentiable differentiable or holomorphic, holomorphic, then then if is also also differentiable differentiable or holomorphic. holomorphic.

i:

i:

This mapping -t 5 f: R mappine i t RFl.ir called called a lift lift of f: ^9is .R -* t S.

= =

= =

Proof of Theorem Theorem 2·4· 2./. Setting and ql get a Setting PI we get fu = [CI,pd 4t = [DI,f(pd]' lCr,pr] and [Dt,/(pr)], we 1 .f m a p p i n g defined d e f i n e dby b v i([C,p]) p o i n t s . mapping = [D f(Cdf(C),f(p)] for all points [C,p] ( C t ) t ( C ) , / ( p ) l . a l l f(lc,pl) [C,p] _ l D tI. f Jor - ql and f01fR - 1fSoj. R. Then it it is is obvious obvious that that i(pd in k Since 1fR and 1fs nsol. zrp a.nd n5 Since /(f1) {1 fSrp are locally biholomorphic and f/ is continuous, are continuous, / must be continuous. continuous. It It is is also also that ifif f/ is differentiable j. the The uniqueness uniqueness trivial differentiable or holomorphic, then so trivial that so is /. 2.3. 0 assertion assertion follows follows from Lemma 2.3. D

i

(Uniq:reness of Remark. (Uniqueness universal covering) covering) For any two universal universal coverings of universal Rernark. (R,r,R) (R, 1f, R) and (r?1 R I ,,11,R) 1fl' R) of a Riemann Riemann surface R, there there exists exists a biholomorphic biholomorphic surface r?, mapping g cp of R to r?q RI with with 1fIOcp example, Ahlfors and Sario [A-6], Trog - 1f. n. See, See,for example, [A,-6], Theorem 18A of of Chapter l. L

l

=

Proof. To prove (i), suppose that 7I"(p) = 7I"(if) = p. Then we ~ave p = [C1 , p] and if [C2 , p] for some paths C 1 and C 2 on R. Putting Co C 2 . C 1 1 , we see that I = [Co]. satisfies if = I(P)· To see (ii), take a point pER, and set P = 7I"(p),p = [C,p]. Choose a neighborhood U of p in R which satisfies the condition in the definition of a

e Jo uorl-rugep ar{} ur uorlrpuo? aql sessrl€s qclq,$ u ur d 3o 2 pooq.roqqStau e esooqo 'ld'Cl = 4'@)" - d 1as pue 'g f gl tutod € a{€t '(rr) aes o5 " '(g),0 - i sagsp, 'l"C) -- L ', - o5l 3ur11n4 'A uo zg pue IC sqled euos rc1 t€rl? aas am IC . 7'C fd'z7l = p pue [d'rCj = 4 a,re{ er'ruaql 'd = (!)y = (4)a 1eq1esoddns '(r) a.,'o.rdoa '{oo.t4

=

=

'Q* X u ( y ) r p q l q c n s J ) L s l u a u a l a f i u o u t Q a T g u { I s o l l t?' o a . t ' oa " r ' a q y ' g sSaoi (rrr) lo y TasqnsTcoilutocfiuo.to!'st IDW:ry uo fipnonu4u@srp fr.1.r,ailo.ti1 'sTurodpat{ ou soy fiyquapt.ayy.tol Tilacxa - J ) L f r ^ r a a"ar o t Q = n U ( d L e ) D e ' " t o l n c q t o dq ' { p ! } 1 lo yueu,a1q ? D q l 1 l ? n sU u ! 4 ! o 2 p o o t l . r o q q \ n u a l q l p n s o f l e r e q l ' U ) g f . r a a a i o g ( n )

(i) For any P, if E R with 7I"(p) = 71" (if) , there exists an element I E r with if = I(P)· (ii) For every pER, there is a suitable neighborhood U of p in R such that I( U) n U = ¢ for every I E r - {id}. In particular, each element of r except for the identity has no fixed points. (iii) r acts properly discontinuously on R; that is,for any compact subset K of R, there are at most finitely many elements I E r such that I(K) n K f. ¢.

'@)L= D uo s?srseatayT'(p)tt.= (4)v q?!nU ) !'q fruo.tog o ltlln J a L \uau.ta1a

Lemma 2.6. The universal covering transformation group Riemann surface R satisfies the following properties:

:sa4.tado.td6utno11otaq7 sa{stqos g acottns uuvur?tg eqJ 'g'Z BtutrroT 6ut.r,eaoc o Io (U':r-'A) to .1 dnotf uorTout.tolsuo"tT losJeaNutu

r

of (ll, 71", R) of a

'a,rr1calrnssr acuepuodsa.rroc O srql ecueq pu€'.[C] - l, teql seqdrur7'Z ruaroeqJ'(d'A)ro Jo luetuele u€ = ["d'Cj = (l'd ''I])t reqr s^\oqs g'U €urureT snql sr [9] ecurg '(t'd'"tl).[C] 'flarrrlcadsar'g;o s1u1od pue', pue pue aql ar€ Ierlrur Ieurural ['d'oI] I'd'Cj C l€I{} Jo lJll € sl , acuaH'od lurod eseq qlu{ g. uo qled pesol? s sI Col, sarldurr y JLov uorleler aql ueql '(l"d'"tl)L ollod'oll uroq Ur rio qled e al Q 'a,rr1celrns slq] 1eq1 aaord o; sr ecuapuodsarJoc 3 ,L luaurela ,tue a1e1 'err,r1celur st eeuepuodseJJo?sltl]

where 10 homotopic to 10 , and hence [Co] is the unit element of 7I"l(R,po). It follows that this correspondence is injective. To prove that this correspondence is surjective, take any element I E r. Let C be a path on R from [Io,Po] to 1([10 , Po]). Then the relation 71"01 = 71" implies that C 7I"oC is a closed path on R with base point Po. Hence C is a lift of C, and [Io,Po] and [C,Po] are the initial and terminal points of C, respectively. Thus Lemma 2.3 shows that 1([Io,PoD [C,Po] [C].([Io,PoD. Since [C] is an element of 7I"l(R,po), Theorem 2.4 implies that I = [C]., and hence this correspondence is surjective. 0

=

=

=

Ie.I'J

l€qt sA^olloJII'ed'a)rL Jo luatuale lrun eqt q [u ef,uaq Pu€'07 o1 ctdolouroq sr op 'snq;'[t'O] = 1 ) t fue to1 od - (l)'l reqr qrns U uo qled eq] sl o1 ereq!\

= [Co, Po] = [Io,Po], is the path on R such that Io(t) = Po for any tEl = [0,1]. Thus, Co is

'lod' o " o If = fod' Cf = (l"d' tl).|' Cl a^eq a^r ueql 'J Jo lueurele lrun eql q -[?] 1eq1asoddns 'arrrlcefutq ]l ]€q] e,rord oa '.7 o1 (od(g)tv;o rusrqdroruouor{e sr ecuapuodserrocsrq} }tsqt pI^Ir} sl rI'loord

[Co].([Io,PoD

we have

r. To prove that it is injective, suppose that [Co]. is the unit element of r. Then Proof. It is trivial that this correspondence is a homomorphism of

71"1 (R,

Po) to

'(A'o'U) 0ut.taaoc losrearun n to 1 dnot'6 uotTotu.totsuo.r,T |uuaaoc losraarun?ql oluo A Io ("d'A)rv dnot6 pTuau,opunt aql uo spyaffi-['d - log)acuapuodsauocaaoqDeyJ'g'Z rraroaqJ to tustrlilrotuost.

Theorem 2.5. The above correspondence [Co] 1-+ [Co]. yields an isomorphism of the fundamental group 7I"l(R,po) of R onto the universal covering transformation group r of a universal covering (R, 71", R).

'(A'v'A) Jo uorleruJoJ -sue.r1Sur.rar'oce sr 1r 'sl leql '..,1o1 s3uolaq -[op] srql 'uo1]-ruuepeql fq 'fpea13

Clearly, by the definition, this [Co]. belongs to formation of (R, 71", R).

r, that is, it is a covering trans-

'ld'c.'c) = ([d'c1).1'c] ry> la'c) [Co].([C,pD

= [Co' C,p],

[C,p] E

R.

fq g uo -[op] uorlce eql eugep e^{'(od'U)rv>['C] ]ueuele fue rog 'f, 1o (od'g)rY eqt o1 crqd.rouroslsl J dnor3 Eurraaoclesrelrun slt l€q? aas dnorE leluaurepunJ 'p ace;rns uueuerg e II€qs alll Jo (g'.u'9,) ece;rns EutrarrocIesJeAIunuarrrEe rog

For a given universal covering surface (R, 71", R) of a Riemann surface R, we shall see that its universal covering group r is isomorphic to the fundamental group 7I"l(R,po) of R. For any element [Co] E 7I"l(R,po), we define the action [Co]. on R by 2.2.3. Universal Covering Transformation Groups sdno.rg uor+BrrrroJsuer;, EurreloC

31

lesrallun'e'Z'Z

2.2. Universal Coverings s8urra,ro3l"sra^rufl'Z'Z

IT

g2 32

2. 2. Fricke FrickeSpace Space

I covering map map in in §2.1, and denote (U) covering denote by by U the connected connected component U the component of of 7rr-r(J) $2.1, and containing p. Actually, it is containing Actually, it is sufficient to take a simply connected domain sufficient to take a simply connected domain U f. U c o n t a i n i n gp. f o r s o m e containing If I(U) n U f:. if; for some I E r, then f t h e n there p o i n t sPI, t h e r e are a r e points e.lt1(0)n0 e 0U + { 1€ , f u , fihu E with ih Since7r0 ro7 we get get 7r(pd n, we T(fr1) = 7r(lll), r(4), and with for and hence henceiiI f1 = I(PI). l(it). Since l = 7r, Q1 PI, fi1,for r7r is is biholomorphic biholomorphic on = on U. Thus we U. Thus we have have I(pd id(Ft), where where id fd is is the the identity. identity. l(Ft) = id(pd, - id. By Theorem Theorem 2.4, we conclude 2.4, we conclude that lhat I1 = By id. Finally, to to verify verify (iii), (iii), assume assumethat that there there exists Finally, exists aa sequence sequence{{ In consisting Z" }~=l }f,r consisting of mutually mutually distinct distinct elements elements of of r l- such such that of that In(I O. and ad- Dc ) 0. Therefore, this 7I is written written in in the form form (2.5). 0 tr

=

=

For For more more on the the fundamental properties properties of of M6bius Mobius transformations, such as transformation transformation of of circles into into circles, circles, and the the invariance of of the crms cross ratio ratio under them, them, we we refer, refer, for for instance, instance, to to Ahlfors Ahlfors [A-4], [A-4], $3 §3 of of Chapter Chapter 3; 3; and and Jones Jones and and Singerman Singerman [A-48], [A-48], Chapter Chapter 2. 2. Now, Now, for for every every 7I eE Aut(e) Aut(C) given given by by

'slueruel€?s3urno11o;eql e^€rl a,r,r'uor1e1nc1ec aldurs e ,(g 'oz - (oz)L Surr(;sr1es oz > Jo tas aqt C IIs 'r(1r1ujpreq} oz slurod pexgJo las aql eq (t)xrg 1a1 }ou sr qcrqn

which is not the identity. Let Fix(,) be the set of fixed points Zo of" that is, the set of all Zo E C satisfying ,(zo) = ZO° By a simple calculation, we have the following statements. 'sl

ler{}1'L1o

, I = c q- p D , ) ) p , c , q , o

_ az + b ,Z( ) ---, cz +d

a, b, c, dEC, ad - bc

= 1,

, ' . 1 1 , i= Q*zo

@)L

Let, be a Mobius transformation given by

.{q ue,tr3 uor?euroJsuer} snrqory e aq I 1a1 'z'e'z

2.3.2. Canonical Forms of Mobius Transformations suorlBurroJsuBrl

snlqgl trJo srrrroJ lBcruouBc

'{1aar1aadsa.r '(1'1) atnTvufts to dno.r,6fil,opun lonads eq1 '1)29 pue (U'6)79 araqirr plle 7 aa.r6ap lo dno"r6nautl yonads lDa, eqt are (1

where SL(2, R) and SU(l, 1) are the real special linear group of degree 2 and the special unitary group of signature (1,1), respectively.

' { t+} lfi 'r)ns= G't)nsa = (v)wv ~

PSU(l, 1) = SU(l, 1)/{ ±I},

pue

}/(u'?,hs= (ll'z)tsa = (n)pv

PSL(2,R) = SL(2,R)/{±I}, '{(e)t"v

~

'{ r+

Aut(Ll)

and

Aut(H)

a^eq a^{ 'flrelrurrg ) Ll 6oLor-.{ } = (ra)lny reqr pue

and that Aut(pl) = {F-Io,oF Similarly, we have

I, E Aut(C)}.

' = -, :; lz:ii';il li:1, | l Zl] [ Zo

t--+

A [Zl] Zo

CZI

+ dz o

= [azl + bZo ] ,

Remark. When the Riemann sphere C is identified with the one-dimensional complex projective space pI, an element of Aut(C) corresponds to a projective transformation of pl. Indeed, define a biholomorphic mapping F: pI --+ C by F([zo : zd) = zo/Zl, where [zo : zd is a homogeneous coordinate of pI Then we see that an element ,(z) = (az + b)/(cz + d) of Aut(C) corresponds to a projective transformation

uorleuroJsuerl a.,rr1ce ford e o1 spuodser.roc(3)tnv Jo (p + zc)l&+ zD) - (z)1, lueurale ue 1eq1 ees e/tr uaq;,'rd Jo eleulProoc snoeueSouoqe sr [tz : 0z] araqrrr'rz/02 = (ftz : ozl)g * rd :J Surddeur crqd.louroloqrqe euuep 'paepul 'rd Jo uorleruroJsu€rl f,q ? a,rrlcefo.rde o1 spuodsauoc (3)lny Jo luetuela ue '1d aceds err,rlcalo.rd xalduroc Ieuorsuaurp-euo aql qlra pagrluapl q C a.raqdsuuetuerg eql ueq1\ llrDureq 0

'(c'dts

We set PSL(2, C) SL(2, C)/{ ±I} and call it the projective special linear group of degree 2. An element A of M- I (,) for an element, E Aut(C) is called a matrix representation of ,. Note that, is represented by two elements ±A in SL(2, C).

ul yT sluauala o,rl!,1 ,tq peluasarder sI ,L 1eq1elop "L Jo uorlD?uesa.tdat rt.tTou e pelle)c$st(g)l"y 3 ,L luaruele ue rog (1,)r- W lo V lueuala wtr '4 aa.rfaplo dno.t6 = (C'Z)lSd toauq lotcads aar,Ttato.td eql tr lpc pue {1+}/6'dlS tas aIA

' { r + } / 6 ' z ) t s= ( q ) w v

of the special linear group SL(2, C). Conversely, we have a homomorphism M of SL(2, C) onto Aut(C) defined by M(A)(z) = (az+b)/(cz+d) for A E SL(2, C), where a, b, c, and d are the entries of A as above. Then the kernel of Mis {±I}, where I is the unit matrix. Hence, by the homomorphism theorem, M induces an isomorphism Aut(C) ~ SL(2, C)/{ ±I}.

ursrqdrourosrue secnpur (ruaroeq? tusrqdrouoruoqeql dq 'ecua11'xrr?Burlrun aql sI 1 ereq^r ltl '{ 'c'g 'o ereqrrl 'eloqe se y f + 1 sl W Jo loura{ aql ueqtr, Jo saulueeq?er€p pue '(C'Z)1S )>y .ro;(p+zc)/(q+zo) = (z)(y)W tq paugep(q)ny oluo (3'6)79 yo ;;4rusrqdroruoruoqe e^eqem'r(lasrarruo)'(C'dlS dnor3 reauqletcadsaq1;o

A=[~

:]

lp c1 l'^ -l =V L q D)

qll^{ j

luauele u3 aAeq e^r

= cz+ az + b, d

,I =cq-pp

we have an element D+z) (1__-i_ (z\L 9*zo

a, b, c, dEC with ad - bc

> p'?'q'D

,(z)

= 1,

suorl"urrolsu"rJ snrqol{'t'z

2.3. Mobius Transformations

35

9t

3366

2. 2. Fricke Fricke Space Space

o.

(i) The The case case where where oo 00 € E Fix(7),i.e., Fix("Y), i.e., cc == 0. (i) If a/d aid = 1, that that b, is, ca - 6f d === *1, ±1, then then 7"Y has a sole sole fixed point point oo 00 and it it is lf written in in the form form written

=

=

1'(z)=z+b, 1 e)=z*b, On the other other hand, ifif a/d aid f:j:. 1, then 7"Y where b is a non-zero complex number. On another fixed fixed point point zo, and is represented as as has another uw- -z oZo= = \ ( A(Z z - z -o )zo), ,

w =lQ), = 1'(z), and )A is a complex number equal neither to to 0 nor to to 1. where to

(ii) The The case case where oo 00 f~ Fix(7), Fix(/), i.e., i.e., cc:j:. o. (ii) f 0. point = If (a + d)2 4, then "Y has a sole fixed point Zo and it is written written as as zo and it If * d)2 7 has sole

=

1 1 W -- % Zo= ; : Z1-+ Zo w o'

---=--+0', (c*+ d)' If (a where to w = 1'(z), and a a is is a a non-zero non-zero complex complex number. number. If d)2 :j:. 4, then then l' where 7 # 4, lQ), and the form has two fixed points Zl and Z2, and it is represented in points 22, and it is represented z1 and fixed has w Z - Zl W - Z Zl t ^, Z7 ---Z' -l , --=A W Z - Z2 " , - tZ2r -

L. where A is equal neither to 0 nor to 1. is a complex number equal where W w = "Y(z), 1(z), and ,\ contobe Aut(X)'conjugate Now, E Aut(C) are said to be Aut(X)-conjugate or conare said Aut(A) elements"Y1,"Y2 Now, two elements e 7r,jz that "Y2 6o"ho6-t such that jagale in Aut(X) jugate Aut(X) if Aut(X) such ob- 1 ,, element b 6E if there exists exists an element e Aut(X) 1z = bO"Y1 where H, or ..:::1. A. C, C, H, where X X is is one one of C, This following lemma. leads us us to the following This leads two f,xed id) has has one otueor two Lemma fixed points tmnsformation "Y(:j:. Mdbius tronsformation 2,9. Every Eoery Mobius Lemma 2.9. 7(f id) tmnsformation "Yo: on following Mobius M|bius tronsformation the foltowing antl is Aut(C)-conjugate Aut(e)-conjugate to the on C, e , and 7o: , a a* 0:j:. . s o m eaa €ECC, (i) +da for z+ ( i ) If"Y t, h e n"Yo(z) T.Q)= Z h a sa sole s o l efixed I f t has f o r some f i x e i lpoint, p o i n tthen \ C , ) (ii) points, then "Yo(z) = AZ for some A E C, A :j:. 0,1. p o i n t s , ) , 2 s o m e t h e n T o ( z ) € ( i i ) If"Y h a stwo t w o fixed I f 7 has for +0,1. fited

o.

canonical representations of canonical We Matrix representations canonical form We call this "Yo form of "Y. 7. Matrix 7o aa canonical forms and (ii) correspond correspond to forms in (i) and

0] [;?], [tf..;>.o ,f.^), 11";>"

respectively. respectively. = RU{ points are fl = RU{ 00 m }} a.rein in R A whosefixed fixed points fd) whose transformation "Y(:j:. A real real Mobius Mobius transformation 7(l id) ) of a matrix a or that the entry is such that the entry a or A of a matrix such canonicalform form "Yo to aa canonical is Aut(H)-conjugate Aut(H)-conjugate to 7o representation '''Yo is real number. number. is aa real representation.To

'peuueplla,lr '1, sr sll a.renbs (1,)rr1 Tnq fq flanbrun peurrurelap olw peretlesr ec€r1slr ueql'V- f,q pacelda.r sr y 1ousr (l).r1 snql'(p+D)'.{. uorleurroJsusrlsnlqgl4le 'p ;1 Jo nD4 " pelpc q qcg/'r * o (l)r1 1nd a14

We put tr(r) = a + d, which is called a trace of a Mobius transformation r. If A is replaced by -A, then its trace is altered into -(a + d). Thus tr(r) is not determined uniquely by r, but its square tr 2 (r) is well-defined.

' T= ? q, l \ ' ^ l= , ,C PD ) p,c,q,o L 9 DJ

A =

[~ ~],

a, b, c, dEC, ad - be = 1.

Now, a Mobius transformation r which is not parabolic is Aut(C)-conjugate to a canonical form ro(z) = AZ (A E C, At 0,1). This A is called a multiplier of r. We have two choices for the multiplier of r, i.e., A and l/A. It depends on the choice of a fixed point of r corresponding to the attractive fixed point of roo Its multiplier A satisfies the equation (a + d)2 = A+ l/A + 2. Note that a + dis the trace of a matrix representation of r:

:,L;o uorleluasa,rdarxrrleru " Jo ecerl eql np+p tsql atoN 'Z+y/I+y= e , ( pa r ) u o r l e n b ea q l s e g s l l e sy r a q d r l p t u s 1 1 'ol,;o lurod pexg e^r1?€r11ea{} o1 Eurpuodsarroc,L;o lurod pexg sJo alloq? aql uo spuadep yy/I pue aqt roJ sarroqr o,lrl e eq a64'l;o 1''a'r'l,yo.rar1dr11ruu .taqd41nut, e pallsc sl y slql'0'O* y'C ) y) zy - Q)'L uroJ Iscruouer € ot apSnluoc-(g)tnv o. cqoqered lou sr rl?rqar ,L uorleurroysrrerl snrqotrtre 'alo1q 'g uo sTurodpae{ omy soq L fipo puo ctloqtedfitl s! ,t (H) ta tt, .zz = rz puV , *H 'H 'rz slutod par{ on\ sorl L ) zz ) rz pUl qtns zz tt fi1uopuo tt, ctTdqla st' L 'g uo sr L Tutod pat{ alos o soy L lr fiyuo puo lt cqoqn.r,od

is is H", (iii) r is

(r) (r)

(i) (ii)

r r

parabolic if and only if r has a sole fixed point on ft. elliptic if and only,jf r has two fixed points Zl, Z2 such that and Zl = Z2' hyperbolic if and only if r has two fixed points on ft.

Zl

E H,

Z2

E

Lemma 2.10. Let r be a real Mobius transformation which is not the identity. Then the following hold: :p7or16utmo71olaq7 uayl 'fr,7t7uapt ?ql pu s, q)rqn uotgou.t.r,olsuorl sntqory loar o eq L pT 'Ot'Z eurtuarl

's11nsaresaql Surulqurop 'uor?ross€ aqt urelqo ea,r 3ura,ro11o; /tls"a 'cqoqradr(q ro 'cr1dt11a ro 'cqoqered reqlla sI ','[1t1uapt ro (11)7ny Jo luetuale fra^a 1€ql ^\oqs ol fsea st 11 'fla,rrlredsar '1,

Note that there are Mobius transformations which are neither parabolic, elliptic, nor hyperbolic. An example is given by r(z) = Az (A E C, IAI # 1 and A fi. [0,00)). A Mobius transformation which is not the identity is said to be loxodromic if it is neither parabolic nor elliptic. In this book, we assume that the loxodromic transformations include hyperbolic ones. Let Zl and Z2 be the fixed points of a loxodromic Mobius transformation r. Suppose that Zl and Z2, respectively, correspond to the fixed points and 00 of a canonical form ro(z) = AZ for some A with IAI > 1. Then Zl and Z2 are called the repelling fixed point and the attractive fixed point of r, respectively. Denote by r-y and a-y the repelling fixed point and the attractive fixed point of r, respectively. It is easy to show that every element of Aut(H) or Aut(~), which is not the identity, is either parabolic, or elliptic, or hyperbolic. Combining these results, we obtain easily the following assertion. eql lou sr q?rq^r'(V)lnV

go paxg Suqledar eq} !o pue r; ,(q alouaq pexg a^rlrer11e a{l pue lurod lurod 'flarrleadser 'L elql pve Turotl pax{ 0ur11ada,eq} pell€, 1o yut,odpar{ aatyco.tqqo elre ez pue rz uaqJ, 'I < lVl qll^ y euos to! zY - Q)'L ruroJ l"tluouec e Jo qurod pexg eql o1 puodsarroa 'fla,rrlcadser'zz pu€ Iz 1eq1 asoddng oo pu" 6 'l, uorleur.ro;suer? snrqoq cnuorpoxol e;o slurod pexg aql aQ zz pue rz lo.l 'seuo ?rloqredr(qapnlcur suorleurroJsusrl c[uoJpoxol eql airr '4ooq slql uI 'crldqe rou ttloqersd .raqltau sl ?l JI cruoJporol l€ql eurrrnss€ eq ol pres sr flrluepr eql lou s! qrlqa uorleurroJsuerl snrqory y'((oo'O] / V zV : ue,rrEsr aldurexauy 'cqoq.redfq rou 'cr1dq1a pu€ > I f) I lVl'C Q)L.{q 'cqoqered raq?reu are qf,rqra suorl€ruroJsu"rl snlqol are eraq+ teql aloN I

°

' I + y ' 0 < y e r u o s r o Jz y = ( z ) o L u o r t e l l p e o l a l e 3 n f u o c s r l I ; r c t l o y , a d f i qq f ( U ) '(7 3 u) ttuT - (r)'L uorlelor e o1 ele3nluoc sr ycr,Ttlqla q t (I) ]I * 0'1g )f g auos tol zsp

(i) r is parabolic if it is conjugate to a translation ro(z) = z + Q' for some Q'EC,Q'#O. (ii) r is elliptic if it is conjugate to a rotation ro(z) = eie z for some () E R, () # 2mr (n E Z). (iii) r is hyperbolic if it is conjugate to a dilation ro(z) = AZ for some A> 0, At 1.

'o+n'c>p

eurrros roJ a + z -

(z)ot uorlelsuerl e o1 ele3ntuoa s! 1l y cqoqotod q f (t)

'f1r1uapr eql tou $ qcrrlAr uorl€ruJoJ -suerl snrqontr e eq ,L 1a1 'sadf1 earql otq suorl"ruroJsusrl snlqotr tr fSsselc all

We classify Mobius transformations into three types. Let r be a Mobius transformation which is not the identity.

snlqg,trtrJo uorlBcgrssBlc '8,'8'z suorlBrrrroJsrrB.LT.

2.3.3. Classification of Mobius Transformations suorl"urolsuerl snlqgl I 't'z

2.3. Mobius Transformations

37

LT

38 38

2. 2. Fricke Fricke Space Space

By a simple calculation, we we see seethat Mobius transformations transformations are are classified classifiedby trace. ' trace. Lemma be a Mobius transfonnation which which is not the the identity. Lernma 2.11. 2.LL. Let r7 be M6bius transformation idenlity. Then Then the following hold: hold: the following if and and only ift-?(r) if tf (7) = 4. a. (i) r7 is parabolic parabolic if if and only if if 0 ~ tf (1) < a. (ii) r7 is elliptic if < 4. f t-?(r) (iii) rr is hyperbolic hyperbolic if if and and only ift-?(r) if tf Q) > > 4.. (iv) (iu) rr is loxodromic lorodromic if if and and only only ift-?(r) if tf (1) E e C -- [0,4]. [0,4].

=

Finally, we we define define the axis axis of a hyperbolic real real Mobius transformation r. 7. - )z with A ,\ > 1, by a Suppose that r7 is conjugate conjugate to a canonical canonical form ro(z) > 1, Suppose that U@) = AZ = 6oroo6-1. real 6. Namely, suppose that r7 = 6oloo6-1. The half-line Mcibius transformation 6. Namely, suppose real Mobius joining 0 and = { geodesic, L = iy I 0 < y < 00 } in the upper half-plane H is the geodesic, joining oo} half-plane .Il is < < {iy | 2 I1 (see 00, I(Im Z)2 on H (see §3 of Chapter 3). with respect respect to the Poincare Poincar6 metric IdzI z)2 3 3) . oo, with $ ldzl2I [m denoted by A-y. Ar. Then The image L) of Lunder .L under 6 is called called the axis oris of r7 and is denoted image 6( 6(L) joining the fixed A-y geodesicjoining r,, and a-y a., of r, is characterized characterized fixed points r-y ,4', is the geodesic 1, which is as real axis. axis. Similarly, r., and a-y o., and is orthogonal to the real as a semi-circle semi-circle which joins T-y we Aut(A). we define axis A-y A, of a hyperbolic transformation r7 in Aut(Ll). define the axis

2.4. Fuchsian Models Fuchsian Models First, we whose universal universal covering covering surface surface is is not we show that a Riemann surface surface whose show that biholomorphic to the upper half-plane one of C, e , C, C -halfplane H f/ is is biholomorphic to one properties of discrete discrete subgroups subgroups {O}, we study some some fundamental properties { 0 }, or tori. Next, we groups. of Aut(H), i.e., Fuchsian Fuchsiangroups. Aut(H), i.e., 2.4.1. Type Surfaces of of Exceptional Exceptional Type 2.4.L. Riemann Riernann Surfaces

Let us whose universal universal covering covering surfaces surfacesare are biholosurfaceswhose us determine determine Riemann surfaces morphic to either C or C. morphic either 0 biTheorem R has a universal uniuersal covering coaering surface surface R fr. bisurface^ 2.L2. A Riemann Riemann surface R has Theorem 2.12. holomorphic to the Riemann sphere C if and only if R itself is biholomorphic is biholomorphicto holomorphic lhe Riemann sphereC if and only if R

C. C. transformation Proof. that R .E = C. e . Since Sitt"" every element r7 of its covering transformation Prool. Assume that Flom Lemma is a Mobius transformation, it should have fixed points. From group r should have fixed is M o g ) o q + z - ( z ) o L r y q 1 e r u n s s ef e u r e a ru a q l , c q o q e r e ds o , L ; 1 'cqoqrad{q

Proof. We may assume that r =1= { id}. Take an element 'Yo E r with 'Yo =1= id. Since 'Yo has no fixed points on H, Lemma 2.10 implies that 'Yo is parabolic or hyperbolic. If 'Yo is parabolic, then we may assume that 'Yo(z) = z + bo (b o E R, bo =1= 0) by conjugation in Aut(H). It is easy to see that an element 'Y E Aut(H) is commutative with 'Yo if and only if 'Y is written in the form 'Y( z) = z + b for some b E R. Since r acts properly discontinuously on H, there exists a positive number b1 such that is generated by 'Y1 (z) = z + b1 . Now, suppose that 'Yo is hyperbolic. Then we may assume that 'Yo(z) = AoZ (A o > 0) by conjugation in Aut(H). It is easy to show that an element 'Y E Aut(H) is commutative with 'Yo if and only if 'Y(z) = AZ for some positive number A. Since r acts properly discontinuously on H, there exists a positive number A1 such that r is generated by 'Y1 (z) = A1Z. 0

r

@*'q'U

'H uo slurod pexg ou seq ol, acurg o,L .ro crloqered sr saqdurr etuure1 1eq1 0I'Z 'p! oL qtl,lr '{pgl1 o,L .too.t4 luaurale ue a{€tr * J 3 * J leqt arunsss,(eu e11

'ct1cfics, u?Vl 'uoqaqos? 'H uo snonut?uocsrp lN J II fr4.rado.rd s! J to uorl?o eW llUI q?ns puv g uo sTutoilpac{ ou sly {p?} - J .VTZ BururaT lo Tuaua1e fi.raaa Toqgqcns (g)nv {o dnolfiqns D eq J pI

r be a subgroup of Aut(H) such that every element of has no fixed points on H and such that the action of r is properly discontinuous on H. If r is abe/ian, then it is cye/ic.

r - {id}

Lemma 2.14. Let

'adfi7 TouorTdnr? lo eq ol pres sr rrol ro '{ O} - C 'C 'C Jo euo o1 erqd.rouroloqlqq qcrq^\ ac€Jrnsuueuer}I V

A Riemann surface which is biholomorphic to one of C, C, C - {O}, or tori is said to be of exceptional type.

' .t / C ot cryd.toutoloqrq s! A pW q?ns J dno.tf ac47ol D slstr,? a.r,aq1'g sn"to7fr.tana"lo3r .z(.re11o.ro3

r.

Corollary. For every torus R, there exists a lattice group biholomorphic to C /

r

such that R zs

'3 o1 crqd.rouroloqlq sr snrol € Jo aceJJns D 'dno.r3erlc{c e aq pFoqs 3ur.ra.,loc I€srelrun e e)ueH J e qcns leq} slras$ qcrrl^r (p1'6 eurtual) eurural 3ur,rlo11og eqt s1?rp€rluoc srqJ .g. ;o dno.r3 FluauepunJ eq? o1 crqdrourosr sl J roJ (6 4uer;o dno.r3 uerlaqe aarJ € eq lsnur J uar{} ,.Fl o1 ctqdrouroloqlqq U JI'snrol e q g 1eq1asoddns'fleurg'C = U leqt ^rou{ a , $ ' I ' Z $ y o 1 e l d u r e f g u r u a e ss e A rs V . { O } - C = A 1 a 1 , } x a N. C = A 1 a Ba m 'palcauuoc 'C fldurrs =A fl ?eql aurnsselsrg,asra,ruoc aq1 lroils o1 C ecws 'flalrlcadsa.r 'snro1 e pue '{ '9 o1 crqdrouroloq 0} - C '(II) pue '(ll) ,(r) sasecur ,e.ro;araqa U ac€Jrnsuueuerg eql

Therefore, in cases (i), (ii), and (iii), the Riemann surface R = C/ r is biholomorphic to C, C - {O}, and a torus, respectively. To show the converse, first assume that R = C. Since C is simply connected, we get R = C. Next, let R = C - {O}. As was seen in Example 1 of §2.1, we know that R = C. Finally, suppose that R is a torus. If R is biholomorphic to H, then must be a free abelian group of rank 2, for is isomorphic to the fundamental group of R. This contradicts the following lemma (Lemma 2.14) which asserts that such a r should be a cyclic group. Hence a universal covering surface of a torus is biholomorphic to C. 0

r

-lq sl J/C

r

-

'U re^o luapuedepur fpeauq er€ qcrq^r C f r g ' o g a u t o sr o y t g * z = ( z ) r L p r r s o g* z = ( z ) o l a r a q a r ' ( r t ' . t ) = J ( l l l )

(i) r={id}. r = ('Yo), that is, r is generated by a translation 'Yo(z) = z + bo for some bo E C - {OJ. (iii) r = ('Yo,'Y1)' where 'Yo(z) = z+b o and 'Y1(Z) = z+b 1 for some bo,b 1 E C which are linearly independent over R.

'{o}-c)oq

oruos roJ oq*z

(sr - (z)ol uorlelsuerl e {q pele.reueErl ,('tl leq} J

--,t

(tr) (ii)

{ p t l = . t (r)

:(1'deq3 Jo Z$ '[t-V] sroJlqy ';c) sesecaerql 3ur,rao11og aql rncco ereql l"ql elo.rd uec e^{ ueql '(9'6 eruuel ;c) g uo flsnonurluocsrp 'g uo slurod pexg ou seq -,f > f fpadord slc€ J teql [eeer em '.ra,roaro1,1i {pl } f u e a s n e r e q ' l = D ' r a q l r n g ' ( O f " ' C f g ' o ) q + z o - ( z ) L t u r o Je q l u r uellrr^,l\sl J ;l,L fre,re g'A"uure.I fq,(g)lnV;o dnorEqnse sl J acurg.dno.rE uol+€ruroJsue.rl Sur.ra,rocl€sralrun st! eq J lel 'C = U leqt aunssy '{oot4

Proof. Assume that R = C. Let r be its universal covering transformation group. Since r is a subgroup of Aut(C), by Lemma 2.8 every 'Y E r is written in the form 'Y(z) = az + b (a, b E C, a =1= 0). Further, a = 1, because any 'Y E r - { id} has no fixed points on C. Moreover, we recall that r acts properly discontinuously on C (cf. Lemma 2.6). Then we can prove that there occur the following three cases (cf. Ahlfors [A-4], §2 of Chap.7): sIePoI{ u"rsrlf,nJ 't'z

2.4. Fuchsian Models

39

6t

2. 2. Fricke Fricke Space Space

40

2.4.2. Fuchsian Fuchsian Models Models and and f\rndamental FundaIllental Dornains Domains 2.4.2.

The following is an immediate immediate consequence consequence of of Theorems 2.I2 2.12 and 2.13. 2.13. The surface fr, Theorem 2.L5. 2.15. A A Riemann covering surface R biunioersal couering surface R has has a universal Riemann surface Theorem and only lype; that is, if holomorphic to H if and only if R of exceptional type; is, if is not exceptional if R H if holornorphic ^of if R is not biholomorphic to anyone of C, C, { 0 }, or tori. C o r t o r i . o n e o f C , C , is not biholomorphicloany if {0},

If a universal universal covering surface E R of of a Riemann Riemann surface ,R R is the the upper upper halfhalfIf plane fI, H, we call its its universal covering covering transformation transformation group Ir a Fuchsian Fuchsian rnodel model of .R. R. In In this this case, case, fr is asubgroup a subgroup of of Aut(H). Aut(H). However, However, identifying identifying I/ H with with 4, ,,1, of we sometimes sometimes consider consider a F\chsian Fuchsian model fr as as a subgroup of Aut(A). Aut(Ll). Remark 1. By By an argument similar similar to that that in the proofs of of Theorem 2.13 2.13 and Remark -1. Lemma 2.I4, 2.14, we see see that that the fundamental group of of a Riemann surface surface R is commutative ifif and only only ifif .R R is biholomorphic to to one of of C, C, C, C -- {O}, tori, { 0 }, tori, r } . unit disk orr aannuli n n u l i {{ zz E < Iz < }. i s k ,,1, . 4 , ,,1 4 - - {{ 00} }, tthe he u nit d e C 11 l z Il < | 1< , o

geometric image image of correspondence correspondence between between a Riemann In order to obtain a geometric surface R and its Fuchsian model f, r, we use use a fundamental fundamental domain domain for f. r. An An satisfies for f if F open set F of the upper half-plane H is a fundamental domain r if satisfies domain 11 of open fundamenlal the following three conditions: with 1 =f (i) ,(F) f with, every 7 E oFF = ¢ e r / for every, { id. z({) n (ii) If 11, then closure of Fin .F in H, .F is the closure If F

a H

,r(F). =U [J ,(F). 'YET 7el

with respect respect to the (iii) The relative boundary of measure zero zero with has measure 0F of F in H has two-dimensional measure. Lebesguemeasure. twodimensional Lebesgue as is considered consideredas These surface R = H //fr is us that that the Riemann surface These conditions tell us l.. points on of covering group r. under the covering dF identified under

F F with

y'. For we define definesimilarly its Example coveringgroup r l- in Example 2 in §2.1, each covering For each Emmple 4. $2.1,we give examples of fundamental (i)", ... (t)" give examplesoffundamental fundamental domain. The following (i)", ..., , (v)" 2, respectively. respectively' domains groups of (i)', ... (t)' in Example 2, . . . ,, (v), covering groups domains for covering

(i)" ( i ) "F I mzz< 2211"}. r =- {z r}. e C | 0 < 1m { , Eel - {z Ee H I| 00 < Re (ii)" ( i i ) "Fr = R ez z< 11}.} . {, (iii)" a r s z< 2211"/n}. ( i i i ) "F r/n}. F =- {z | 0 < argz { 0 } 10< { , Ee cC -- {O} (iv)" ( i v ) F" .=F {z - { zE eHH11l I 1, then Cn - C as n - -00. Since ab ::f 0, A > 0, and A ::f 1, we see that {Cn }~=-oo consists of distinct elements. Therefore, by Lemma 2.16, is not discrete, which leads to a contradiction. 0

I r o l =' I'qt'; j "t ueqt'I > y > g y'acua11 o1 sa3ra.iluoc

Hence, if 0 < A < 1, then C n converges to

o'l = ' I 'I | . ( , r ,- I ) e D; i

C_[10 n -

ab(l- A 1

2n

)]

.

- ug 3ur11as'arrolq'flalrlcadsar

respectively. Now, setting C n = BAn B- 1 A-n for any positive integer n, we get arrrlrsod{ue.ro; u-VrBuVB 1eBea,r'u.ra3a1ur

, o* q e , u ) 9 , o

A=[~ A~l]' B = [~ a~l]'

a,b E R,

ab::f 0,

,

g] -

[,;,

, r + y, o < y , l ' ; . A > 0,

A::f 1,

i]

g

_V

Proof. Assume that neither (i) nor (ii) holds. By Aut(H)-conjugation, we may assume that Fix(I) {O, 00 } and Fix(I) n Fix(6) 00 }. Then matrix representations A, B of I' 6 are given by

dq uerrr3an g'L lo g'V suorle?uas '{ -e.rda.r xr.rleur ueql oo } = (g)*1.{ U (l)xrg pue { m'O } = (l)*tg t€rll eunsse feur aar 'uorle3nfuoc-(Hhnv fg 'sp1oq (rr) rou (r) raqlrau l€ql eunssv 'loo.r,4

=

={

.0 = (g)xr.{ u (r)4.{ (r) (i) Fix(I) (ii) Fix(/)

n Fix(6) = - oo. 00. Hence, Hence, .Ar An converges converges to to Ao, A o , which which ant tr the discreteness discreteness of of l-. r. 0 contradicts the contradicts

Theorem 2.22. Eaery Every element element of of a Fuchsian Fuchsian model model of of a closed closed Riemann surface surface Theorem of genus genus S g (|=2) (~ 2) consists of the the identity and hyperbolic hyperbolic elements. elements. consislsonly of of

r-{

Proof Since Since every element 7I eE f - { id id}} has no fixed fixed points points on I/, H, itit is parabolic parabolic Proof. that f contains contains a pa"rabolic parabolic element element 10. By Aut(H)Aut(H)Assume that or hyperbolic. Assume lo.By suppose that 10(Z) -= z*1. z+1. From Lemma2.20, Lemma 2.20, any element element lhatT"Q) conjugation, we may suppose .y I (+ (:f id) of of .i- with with f(m) 1(00) = oo 00 is parabolic, which is written writ ten in in the form form = oo} = is a j f = 6. Hence, ]-I(Z) Z + b for some b. Hence, {I E 11(00) oo} number e some real z r(*) *b I { ilz) assume we may with element, if necessary, we assume if necessary' element, with another cyclic group. Replacing 10 7, - ((az L, a a z *+bb)/(cz generator Since every I(Z) add - b cbc= 1, v e r y7 e n e r a t o rffor or ft. S i n c ee ) l @ z * d+) d), , tthat h a t 10 (z) 7 o iiss a g we Thus 1. that 2 shows belonging to l0, 2.21 shows that lei ~ 1. we 2.21 Lemma 0, ,i-- satisfies satisfies c :f belonging lcl +

r

=

r

r - roo

obtain

=

roo =

=

roo.

Iml(z) rmTQ)~S

r

=

=

1

1 ~511 (Imz)1e1 1r-;pp2 -

distinct any two distinct 2}. Then any 1m z > 2}. for all zz with 1m 1. Set Uo e H Illmz Imzz > Set U ) 1. o = {z {z E - roo. l--. Thus the quotient points on U 0 are element of r f are not equivalent equivalent under any element [/o corresponds r?. Since space Ro since 10 = U space Do = Uo/f* o/ roo is biholomorphic to a domain R o in R. 7o corresponds closure R R,o .r?,the closure to aa non-trivial E o of R element of the fundamental group of R, non-trivial element disk in R is Do is is biholomorphic to the punctured disk connected. Since Since Do is not simply connected. ( I}, t o {z € C |0 < b e homeomorphic h o m e o m o r p h i cto {z must w e infer that R 1 } , we i n f e r that o u < Izi 4m s t be e C |0 < { z Eel l"l < { z Eel Izi ~ I}. This contradicts that R is compact. 0tr is compact. contradicts | ltl S ). geometry disdisRemark. using the hyperbolic geometry obtained by using is also also obtained theorem is Remark. This theorem 2 = 2/(lmz)2 be be present dsz cussed of Chapter 3. We present its outline. Let ds = IdzI its outline. 3. We Chapter cussedin §1 ldzl2l(Imz)2 $1 - H/r. H/f . on R = the hyperbolic metric on which induces induces the hyperbolic on H, 11, which the Poincare Poincar6 metric on positive number a, o, 1. For For any any positive Assume f has has aa translation 10(Z) * 1. that r Assume that 7o(z) = zz + fo segment La tro joining ia denote of the segment which is is the the image image of ,Rwhich on R closedpath on by C denote by C"a au closed of -.R. length of the hyperbolic hyperbolic length and (ia) by R. Let C a ) be the projection 11": Let f( l(C")be r: H H ->by the the projection and10 1o(ia) we see seethat that Then we C metric. Then to the the Poincare Poincar6metric. with respect respectto of La .Lowith length of the length a, i.e., the Co,i.e., we have sequence have aa sequence f(C being compact, compact, we + 00. .Rbeing - 00 as other hand, hand, R oo. On the other On the as nn ->t(C") a ) ->+ 00, + Po + po oor as r(l'o") oo and ) {an };;"=l of positive numbers such that an ->00 and 1I"(ia ->as nr, ->that dn positive such numbers of o" n }L[r { which connecteddomain domain U u which where simply connected we take take aa simply point on Hence,if if we po is on R. rt. Hence, where Po is aa point n. large n. for sufficiently sufficiently large path Can in U contains is included included in [/ for C,. is po, then the closed closedpath then the contains Po, - id, This = a contradiction. id, a contradiction. implies 10 This implies lo

L?

acedg arpug '9'7

47

2.5. Fricke Space

oreds a{rl{,{ '9'U 2.5. Fricke Space

:suorlrpuoc uorlezrleurou aq1 esodurl e,!\ ,U uo 3 3ur4.reurue,rr3e o1 J Ieporu uersqrnd e ,,{lanbrunu8rsse o} rapro uI .ile,lr se Ur aures eqt Jo lapour uersqrnd e sl dnor3 eq1 '(g)wv fue ro; ,s1 ryql :(11)7ny go sursrqd ) ,.7 9 r_9J9 -rouoln€ .reuur{q pasnec ,{lrn8rqure egl seq ,lop A Io J Iapour uersqcnd e '6'"''Z'I = ! q c e e r o J ' f 1 a , r r 1 c e d s,e( .ord , A ) t o q [fg] pue [fy] o1 SurpuodsarrocJ Jo stueuela oq1 ld pue fo fq alouap ,9.6 utrreroeqlur pelels A lo J Iepou u€tsqcnd e pue (od,U)Iz uae.nleqursrqd.rourosr aql repun 'f snuaS;o U e)eJJnsuuetuerg pesol) e go (oa,g') t.u dnor3 plueur€punJ t=f{ aql srolerauaS;o uralsds q ''e'l 'U uo 3urr1.reu n rl Jo Ierruouec [lS],[!V]] 'f snuaE sec"Jrns uu€urarg pesol) pa{retu s}srsuo) 3' eraq^r Jo [3''U] II€ }o tg aceds rellnuqrrel eqt ,I raldeq3 ur peugap se^.rsv kZ) 0 snuaS;o Jo t$ 'QZ) 0 snue3go se)eJrnsuueuarg pesolc r; ecedsrefintuq)rel eql uo sel€urprooc e{]r.U pall€c Jo 'slapour u€rsqcr\{ sroleraueS;o ue1s.{s -os eugap fteqs ellr Jo l€)ruorr€c e Bursn fg

By using a canonical system of generators of Fuchsian models, we shall define socalled Fricke coordinates on the Teichmiiller space Tg of closed Riemann surfaces of genus 9 (~ 2). As was defined in §3 of Chapter 1, the Teichmiiller space T g of genus 9 (~ 2) consists of all marked closed Riemann surfaces [R, E] of genus g, where E = {[A j ], [Bj] }]=1 is a marking on R, i.e., a canonical system of generators of the fundamental group 11"1 (R, Po) of a closed Riemann surface R of genus g. Under the isomorphism between 11"1 (R, Po) and a Fuchsian model r of R stated in Theorem 2.5, denote by OJ and j3j the elements of r corresponding to [A j ] and [B j ] in 1I"1(R,po), respectively, for each j = 1,2, ... ,g. Now, a Fuchsian model of R has the ambiguity caused by inner automorphisms of Aut(H); that is, for any 8 E Aut(H), the group r' = 8r8- 1 is a Fuchsian model of the same R as well. In order to assign uniquely a Fuchsian to a given marking E on R, we impose the normalization conditions: model

r

r

'1 1e lurod paxg a^rlcerl?e sll ser{ t^a (I) ',{1anr1cadsa.r 'oo pue 6 1e slurod pexg a^r}ceJ}le pue 3ur11eda:s1 seq td (l)

(i) j3g has its repelling and attractive fixed points at 0 and (ii) a g has its attractive fixed point at 1.

00,

respectively.

'(tt) p u n s u o r l r p u oc ( r ) uorlezrl€rurou aq1 f;sr1es ud pun to lr.I? etunsse.r[eura,ll ,r(ressaceu y,(g)Wy ul Surle3nfuor'.raq1.rng'Q = (6d)xllU(to)xrg seqdrur0U.Z€unueT ,arrrlelmuuroc ..,ir ud pun to acurg 'cr1oq.red,tq arc 6j pue to q1oq,Z7-(,tuaroeqtr dq ,1cegu1 lou ere 'suotllpuoc uor?ezrlsruJouaql segsrlss rIJlqA\u Jo lepo{u u"rsq)nJ 3 s}srxesferrrle araql 'f snuaS;o a)eJrns uu€urerg pesop e uo 3' 3ur>1reuuarrr3e Jod .tlJDurey

Remark. For a given marking E on a closed Riemann surface of genus g, there always exists a Fuchsian model of R which satisfies the normalization conditions. In fact, by Theorem 2.22, both a g and j3g are hyperbolic. Since a g and j3g are not commutative, Lemma 2.20 implies Fix(a g )nFix(j3g) = d'. -- b'· c'· = 1| 0' a'· o'i,our,C, ) 0, eF., cj fiCi 1 1 1 1

1

=

, i ,2,. .... . , 0, 9- -, 1. for f o r each e a c jh= 1 1, 6 -* R Now, we define g- 6 by R6e-6 coorilinates:F Fo:Tn Fricke coordinates g : Tg --+ we define the Fricke d ' s- r ) . a ' o- 1 ,C~_l' c ' n -r , d~_l)' ce - t , dgdg - 1r ,, a~_l' . . . ,,ag-l, as- | t Cg_l, a l , c~, c \ , d~, d 1 ,... ( 4 1 ,Cl, c 1 ,d 11, ,a~, :Fg([R, l ] ) = (al, f o ( [ R , E])

=

surfaces closed Riemann surfaces The image F space of closed the Fricke Fricke space is called the g(Tg ) is fo(To) g = :F Fn Fo topology of genus g. The topology of F is introduced by the relative topology of F Fo genus 9. g in g is introduced connected R 6g 6. In §2 of Chapter 5, we shall verify that F is a simply connected domain is asimply that.F, 5, we shall R6c-0. g $2 6 g- 6 • By the following theorem (Theorem 2.25), :F is a in R 2.25), Fsg is a bijective mapping theorem R6g-0. with F -Q ?,g with of identifying T on T g. Hence we define define aa topology on g to FFo. g under Hence we ?o of T Qg by identifying Riemann a closed of :F Therefore, a topology of the Teichmiiller space T(R) of a closed Riemann • space ?(.R) Teichmiiller of d.g Therefore, a we book, we rest of of this book, In the rest surface of T ?n. from that of genus 9g is is induced induced from of genus -R of surface R g • In assume that T and T(R) are equipped with these topologies. topologies. with these are equipped "(.R) assume that Tn g and 6 ---+R is injective. injectiae. Theorem g- 6 is R6'-6 coorilinales:F fo: g --+ Fricke coordinates g : TTo 2.25. The The Fricke Theorern 2.25.

=

= (al,cl,dl, ( o r , " t , d v , .... ..,a 's-r, p o i n t :Fg([R,E]) Proof. ,a~_l' e v e r y point fo(lR,t]) t o show s h o w that t h a t every W e need n e e d to P r o o f . We genof system canonical dg-l' d~_l) in F determines uniquely the canonical system {aj, {3j } of genthe uniquely determines Co-r,ils-1) in F,g {oi,0i } point E)eTo. erators ofthe normalized Fuchsian model r for the point [R, E] E T • the Jfor model F\rchsian normalized eiatoriof the g [8, : 1t a i dji --bjcj bici = r e l a t i o najd = 1,2, t h erelation For , g-1), o b t a i n e dfrom f r o m the i sobtained (j = I , 2 , .... ..,9 - l ) , 6 i bj is e a c hjj (j F o reach the same BV by with 0, and hence aj is determined uniquely by :Fg([R, E]). By the same fo(1R,4)' uniquely oi is determined hence and with Cj cr' > 0, ) argument, = 1,2, is also alsodetermined. determined. 1) is 1,2,..... .,,gg -- 1) argument,{3j 0i (j U=

6'

2.5. Fricke Space

aredg e:1crrg'g'g

49

What remains to show is that both O:g and {3g are determined by .rg([R, ED. By the normalization condition (i) for r, we have {3g(z) = ,\z with ,\ > 1. By the normalization condition (ii) for r, O:g has its attractive fixed point at 1, and hence

eeuaq pue'I 1e lutod pexu e^rlcertle sll mq to'J roJ (ll) uorlrpuoc uorl€zllsruJou aql {g'I y q}l^{ zy = (z)6! el"q a.&t'J.loJ (r) uorlrpuoc uorl€zrl€ruroueq} /tg < '(g'Ul)ot,{q paururralap ers 6d p* to qloq }sq} q ^roqs ol sureuer leq1\

Q'z)

(2.7)

=

'lld'fall=[U IeS'rldo6po6g= 6noLaltsqe/$ '.raq1rng lfd'!plr=!6lJ uolleler pluatuepunJ aql urorJ

id, putting ,

=

Set

, P + z c_ Q ) L 9tzo

a, b, c, d E R,

ad - bc = 1.

' I - ? q- p D

,(z)=az+b, CZ+ d

.

,L 3ur11nd 'pl

nJ;:;[O:j,{3j], we have ,00:g = {3goO:go{3"9

1

=

'opa6c=6q+6o

Further, from the fundamental relation rU=l[O:j,{3j]

'g)

p'e'q'o

:ploq suorlenba 3uu*o11o;aql 1eql aurnsse .[eur 'p- po* '"- 'q- 'r- ,(q p pue 'a 'g 'o Surceldag arrr 'fressacau ;t '.,f1arr1cadser

Replacing a, b, c, and d by -a, -b, -c, and -d, respectively, if necessary, we may assume that the following equations hold: (a - l)ag + bCg = 0, ca g + (d - ,\-l)cg = 0,

(orz) (o'e) (s'a)

(2.8)

'0=6?q+6e(t-o)

(2.9)

'0=ta(r_y-p)*6ec

cbg + (d - l)dg = O.

(2.10)

.0=tp(I-p)+6qc

'(Ot'Z) pue (9'6) uro.rg 1aBaan '16l pauru.ra?ap e^eqa^,recuag '(p - i/0 - o) - y pue 'I 'ZZ'ZvrarceqJq?rpsrluoc sSlJ 'cqoqeredu ,l * p'l * e ryql s^rolloJll snql qclq/'{'}- (f)zrl ecuaqpue'I = p ueq}'I = DJI'(p - I)y = I - p 1eq1sarldurr ol€q a^\ (O'Z) p* (9'6) uro.g 'qsrue,rlou saop 6c ro 6o Jo euo lseal le aculs

Since at least one of a g or cg does not vanish, from (2.8) and (2.9) we have a-I ,\(1 - d). If a 1, then d 1, and hence tr 2 (,) 4, which implies that , is parabolic. This contradicts Theorem 2.22. Thus it follows that ai-I, d i- 1, and ,\ = (a - 1)/(1 - d). Hence we have determined {3g. From (2.8) and (2.10), we get

=

ag

=

bCg = --, I-a , D _ I - ocq -

=

(rrz)

=

(2.11)

cb g

(zrd_

-(2.12)

= 1 _ d'

.-P\ -al - -ro ^ dg

san€ (2'6) otq (UI'Z) pu€ (II'Z) Jo uorlnlrlsqns

Substitution of (2.11) and (2.12) into (2.7) gives I-q+D

a+b-l _c+d-l 1 - a cg 1 - d bg. .onP-I 'I-p+c

(erz)

(2.13)

-0"

o-L

Here, if c + d = 1, then we have a + b = 1, because cg i- 0 by Lemma 2.24. Thus, from the relation ad - bc 1, we find that a + d 2, and hence , is parabolic. Again this contradicts Theorem 2.22. Therefore, we have determined O:g by .rg([R, ED. 0

paururalap a^eq ellt.'aro;araq; 'ZZ'Z ureroeqtr slcrperluoc srql ureSy 'cqoqered sr ,L acuaq pu€ '6 = p D pug e,lr 'I = cq - pD uorlelel aq1 uror; 'snq; + leql ' V 6 ' Z e : U u . U u i e l f q 6 c a s n e c a q ' I = q + p a ^ e q e ^ , ru e q ? ' I = p a a OI ;r'are11

=

n

=

'(ls'al)U rq,u"

5500

2. 2. Fricke Fricke Space Space

Notes Notes For historical historical and and expository expository accounts accounts of of the the uniformization uniformization theorem, theorem, we we refer refer to to For universal covering Abikoff [2], [2], and and Bers Bers [29] [29] and and [36]. [36]. The The original original idea idea of of using using universal covering Abikoff surfaces is is due due to to H. H. A. A. Schwarz Schwarz (cf. (cf. Bers Bers [29], [29], pp.264-265). pp.264-265). Complete Complete details details of of surfaces covering surfaces surfaces are are contained contained in in the the books books on on Riemann Riemann surfaces surfaces listed listed in in the the covering notes of of Chapter Chapter 1. notes The notion notion of of a F\rchsian Fuchsian group was was first first introduced by by Fuchs Fuchs in in the the study study of of The ofthe equations analytic continuation continuation ofsolutions of solutions ofcertain of certain ordinary ordinary differential of the analytic second order (cf. Ford [A-31], [A-31], Chapter XI). XI). See See also also Yoshida [A-113]. [A-113]. For more second Fuchsian groups, groups, we refer to to Jones Jones and Singerman Singerman [A-48], [A-48], and Lehner details on F\rchsian called Kletnian [A-66] [A-67]. subgroups of PSL(2, C) called Kleinian groups, gro 1J,ps, are of PSL(2,C) subgroups Discrete and [A-67]. [A-66] intimately related to to the theory theory of of Teichmiiller spaces. spaces. It It is most which are intimbtely that this interesting subject cannot be covered. covered. Concerning Kleinian Kleinian regrettable that groups, see see Beardon Beardon [A-11], [A-ll], Berset Bers et al. al. [A-15], [A-15], Ford [A-31], [A-31], Krushkal" Krushkal', Apanasov Apanasov groups, GusevskiI [A-61], [A-61], Lehner [A-66], [A-66], Magnus Magnus [A-70], [A-70], and Maskit Maskit [A-71]. [A-71]. For and Gusevskil between Kleinia"n Kleinian gloups groups and 3-manifolds, 3-manifolds, we we also also refer refer to Epstein [A[Arelation between Bass and Morgan 25] and [A-26], Laudenbach Poenaru [A-29], Bass [A-76], Po6naru and Laudenbach Fathi, 25] [A-76], [A-29], [A-26], McMullen [154], Thurston [231]. Poincare [A-90] collected works works on McMullen [A-90] is his collected [231]. Poinca"r6 [154], and Thurston Fuchsian groups functions. groups and automorphic functions. Fuchsian groups, see see Nicholls For the interaction between between ergodic Nicholls discrete groups, ergodic theory and discrete Velling [A-86], Bowen and Series [47], Morosawa [158], Series [195], and VeIling and Series Morosawa Bowen Series [195], [158], [47], [A-86],

Matsuzaki Matsuzaki [241]. [241]. Fricke and Klein [A-33]. Fricke and appeared in Fricke spaces first appeared Fricke spaces [A-33]. For modern treatBers and Gardiner [42], ments, Goldman and Magid [A-36], Bers and Abikoff [A-1], see Abikoff ments, see [42], [A-36], [A-1], Keen Saito [186], and Weil [243]. and Keen [110], Saito [243]. [186], [110], group .9cfioltky group we can can use use aa Schottky For aa representation surface' we Riemann surface, representation of aa Riemann space instead instead of aa instead we obtain aa Schottky schottky space and we F\rchsian group, and instead of aa Fuchsian and Sato Sato Teichmiiller space. Bers [35], is discussed discussedin Bers space. This topic is [98], and [35], Hejhal [98], [188] and [189]. and [18e]. [188]

sa{s4os V V '1'g uorlrsodor;

Proposition 3.1. (Schwarz-Pick's lemma) Every holomorphic mapping f : ..1 ----+ ..1 satisfies : t |utddoru ctryl"toutoloyfr"taag (wlo ru:al s6{crd-z.re,*qcg)

'zz pup- Iz uee/r leq ??uDlslp?rDourod eq1 (zz'rz)d llet e^\.aclr€lslp Jo $uorxe aql seuslles d ryqI u^roqs sr.Il'zz pue rz ?ceuuoct{clq,$ 7 ut se^rnf, elqeul}leJ II€ sa^ou 3 'arag

Here, C moves all rectifiable curves in ..1 which connect Zl and Z2. It is shown that p satisfies the axioms of distance. We call p(Zl, Z2) the Poincare distance between Zl and Z2. , l 'zi l' - t 'o i'

rf c / - l u' = l (zz'tz)d

lzplz J

'V tlz slu-tod orr,r.1fue log ) zr las alrlr eJo lapotu e lrnrlsuoc ol clrletu stt{l pesn ?J€culod 'H

H. Poincare used this metric to construct a model of a non-Euclidean geometry. For any two points Zl, Z2 E ..1, we set 'drlauroaS ueaprlcng-uou

. z Q l '-l i = "tP "1'P1tr

ds = (1 _ Iz12)2 . 2

41dzl 2

The unit disk ..1 = {z E C Ilzl < I} has several "natural" metrics. One of them is the Euclidean metric ds 2 Idzl 2 dx 2 + d y 2, and another important one is the Poincare metric

xuleu 9rv?utod eq} sr euo luelrodrur leqloue pue 'rftp * "*p = ,ltpl -- es'pclrletu ueepqcng aql sl ureqlJo auo'srrrla{u ..letrnleu,,Iere^es seq {I > lrl I C ) zI = 7 }tslP }Iun eqtr

=

=

crrlatr l ?rBcurod 'T'I'8

3.1.1. Poincare Metric 'I'g

3.1. Poincare Metric and Hyperbolic Geometry l(llaruoag

pue rlrlatr tr ?rerutod

rrloqrod/tH

'uolsrnql

In this chapter, we shall discuss some aspects of the hyperbolic geometry on Riemann surfaces which is induced by the Poincare metric on the unit disk. First, in Section 1, we define the Poincare metric and study basic properties, especially those concerning geodesics. Using hyperbolic geometry, in Section 2 we define a system of coordinates, called Fenchel-Nielsen coordinates, on the Teichmiiller space of a closed Riemann surface. In Section 3, we discuss an embedding of the Teichmiiller space into an Euclidean space by means of geodesic lengths, which has its origin in classical investigations of Fricke and Klein. Finally, in Section 4, we give a sketch of the construction of a notable compactification of the Teichmiiller space, which was recently proposed by W. Thurston.

'A\ fq pasodorddlluacars€^rqcrq/!\'acedsrellnurq)Ial aqt uotlecyrlceduroe Jo q)le{s e e,rr3 aaa.'p uorlcag ut 'fleutg elq€1oue uorlf,nrlsuot eql Jo Jo 'sq1Eua1 'urely pup a{?tJd suorle3tlsa,rul Jo Ieclsselcut ur3trosl! seq qctq,lr crsepoe3go sueeru,(q acedsueaprlcngue olur acedsrallntuqclel eql JoSutppequa ue ssncsrpa,n 'g uorlces uI 'e?€Jrnsuu€ureru pesoll e 3o ecedsrallnuq?Ial 'saleurprooc uralsdse auuaPe1'r aql uo 'saleurp.roolueslerNleq?uad ;o Pallec 'd.rlauroaE esoql fletcadsa crloqradfq 3urs11'scrsapoe3 Sutu.recuof, 6 uorlces ur 'serlredordcrseq{pn1spue elrlau ereculodaql eugepaiu.'1uorlcagut'1s.rtg '{srp uuetuelg }run eql uo crr}eruersculodaql dq pecnpulfl tlclq^rseceJrns uo frleuroa3 cqoqredfq aq1;o slcedseaurosssnc$pII€qsa^\ 'reldeqc sql uI

salBurProoc uoslarN-Iaqruad puB rt.rlauroa.D rrloq.radfll

Hyperbolic Geometry and Fenchel-Nielsen Coordinates

t raldBrlc

Chapter 3

52

3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3.

1f'(z)1 lf'.(:)l

1

=.~ 1+, -lzI - l " r2-' I1--If(z)12 lf(r)l' =

z €E a. Ll.

point in Moreover, ifif the equality equality holds holds at one one point in A, Ll, then then ff is a biholomorphic biholomorphic Moroaer, automorphism of A, Ll, and the equalitg equality holds at any point point of of A. Ll. aulornorphism of

Fix a point point z in in A Ll arbitrarily, arbitrarily, and set Proof. Fix w *z w+z .tt\w) =TlZw, 'n (w) = 1 + zw' / \ w-f(z) . l12(W) z t w t == ,W- f- z f(z) 1w' 1- f(z)w

11 and 72 12 belong to to Aut(A), Aut(Ll), and ,F(to) F(w) == J2 12 o 0 ff "0 lr(w) 11(W) is a holomorphic Then 71 Ll into into 4. Ll. Since Since .F(0) F(O) == g0 attd and mapping of 4 ' l - l , l 22

'( ) 1 - Izl f'() F,(o)= F 0 = 1 _ If(z)12 ffif'(r), z,

otr

we have have the assertion assertion by Schwarz' lemma. Schwarz'lemma. we

denote by /-(ds2) j*(ds 2) the pull-back of the Poincard Poincare metric ds2 ds 2 = we denote When we 2 2 implies that 4IdzI 3.1 implies Proposition 3,1 aldzl2/(1 /, Proposition /(l - Iz1 lrl')')2 by f, f* (ds2) < ds2

2 if - ds and that f*(ds 2) = Aut(Ll). ro Aut(A). if /f belongs belongs to dsz if and only if that I.(dt2) ------Ll Corollary. A satisfies satisfies mapping ff : Ll A --+ holomorphic mapping Eaery holomorphic Corollary. Every p(f (zt), f (zzD 1 p(21, z2),

21,22 € A.

Remark. /{(h) of aa Riemannian Riemannian metric general, the curaalure K(h) the Gaussian Gaussian curvature Remark. In general,

h(z)2IdzI2 g i v e n by by ( h ( r ) >> 0) i s given o ) is h ( z ) 2 l d z l z(h(z) 2 4 fl2logh g K(h) , h ( h=) _~ = - F d10 h.

h2

a azaz

Poincar6 metric A simple curvature of the Poincare showsthat the Gaussian Gaussian curvature simple computation shows -1 on is on Ll. 4. equal to -1 is identically identically equal under the is invariant under Moreover, )2Idz\2 is h(z)2ldzl2 when aa metric h(z we can can see see that, when Moreover, we constant up to to aa constant metric, up action with the the Poincare Poincar6 metric, coincideni with it is is coincident by Aut(Ll), Aut(A), it action by factor. factor.

t9

,{r1auroag cuoqradifll pu" f,rrlel{ gr"f,u-Iod 'I't

3.1. Poincare Metric and Hyperbolic Geometry

53

scrsaPoaD 'z'T't

3.1.2. Geodesics

Ie

eleq e^r JI '9z ul zz pve Iz $ullcauuoc (cr.r1eru gr€oulod aq1 o1 lcadsar q1ra,r)ctsapoe0e 'V ul zz pue rz Surlcauuot'g cre pasolc elq€Urlcar€ IIef, e \'V ) zr 'rz s?ulod orrr1fue rod'(r)/ fq 1r elouap pu€'C p y76ua1cqoqtadfr,tlaql sp "[ lV, ell.'V ur , ]re pasolc alqegrlcer fre,ra rog

For every rectifiable closed arc C in .d, we call ds the hyperbolic length of C, and denote it by f( C). For any two points Zl, Z2 E .d, we call a rectifiable closed arc C, connecting Zl and Z2 in .d, a geodesic (with respect to the Poincare metric) connecting Zl and Z2 in .d, if we have '(C)l = (zz'rr)d

sl puD zz puo rz q0norqt sassodt1cn1m7uau,6as Vg fi.topunoqeql oI 1ouo0or17.to euq eql ro el?Jr?eyyto ctoqns o s, puD anbtun st 7r |teaoanory'V ul zz puo rz |utTcauuoc crsapoa0o slsNaeere1l 'V ) zz'rz fi^to4tq.toro,I 'Z'g uorlrsodor6

Proposition 3.2. For arbitrary Zl, Z2 E .d, there exists a geodesic connecting Zl and Z2 in .d. Moreover, it is unique and is a subarc of the circle or the line segment which passes through Zl and Z2 and is orthogonal to the boundary {).d of .d.

'v lo

'0 1 zz pue = rz luauela ue fq tuaql Sururrogsuert /tq 1eq1 etunsse deur 0 en'(V)1ny fq uorlce ar{l repun lu€rrelur sr crrleru ar€curod eq1 acurg /oo.l4,

Proof Since the Poincare metric is invariant under the action by Aut(.d), we may assume that Zl = and Z2 > 0, by transforming them by an element

°

i9 Z -

Zl

zlz -t

= Q) L ()

#eP

,Z = e

1-

ZlZ

eleq era 'zz pue g Surlcauuoo trts pesolc frala ro;'r".II C 'U f d elq€1lns qYal. (V)7nY P

of Aut(.d) with suitable 0 E R. Then, for every closed arc C connecting

°

and

Z2,

we have

,W"[ ",ol

-:-2-=-ld-;-z > t' _2_dx_. e 1 -lzl 2 - 10 1 - x 2

= f(C)

7 luaur3es euq eq? qlr^r lueprcurof, sr Cyr fluo pueJI (rh

if and only if C is coincident with the line segment L =

-

tr

Hence p(O, Z2)

,:;:-1

.2, I xp7,

1 = (zz'1)d acuag

[0, Z2].

0

'lzz'o)

''V LV uo slurod oall i(ue Eurlcauuoc crsapoe3fre,ra'6'9 Jo rr"qns e q uorlrsodor4 ,tq 'teql a?ou eJaH ',0 fq uorlce eql rapun luerrslur sr f,y uxe aq; 'L p'V sDr€eql palpc sr Ve oI 1euo3oq1.ro sr pue slurod asaql q3no.rq1sassed qcrqa,llluaur3as auq eql ro alcrr? eql Jo y u1 lred aql l"ql Ip?eU 'Vg uo Lo prte L.r, slurod paxg Irurlsrp otrl seq l, 'crloqredfq 4 (VhnV 3 ,L uaqal '1eq1 lpcag

Recall that, when, E Aut(.d) is hyperbolic, , has two distinct fixed points r-y and a"( on {).d. Recall that the part in .d of the circle or the line segment which passes through these points and is orthogonal to {).d is called the axis A"( of ,. The axis A-y is invariant under the action by ,. Here note that, by Proposition 3.2, every geodesic connecting any two points on A"( is a subarc of A,,(. 3ur11as,(qpaugap sr g aueld-geq .raddnaql uo {sp crrlaur ar"curod eq&'tlrout?[

Remark. The Poincare metric dSk on the upper half-plane H is defined by setting

"l'Pl

2

IdzI 2

, z(lu'l) = ,rrp dS H = (Imz)2'

'y otuo H lo (! + z)/(? - ,) = (z)1,uorleurroJsuert snlqgl i eqt ,,(qv uo zspcrrleru ?r"curod eql Jo {req11nd aql 1nq3mq1ouq qclq,tr

which is nothing but the pull-back of the Poincare metric ds 2 on .d by the Mobius transformation ,(z) = (z - i)f(z + i) of H onto .d.

Hyperbolic Geometry Geometry and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3. Hyperbolic

54 54

3.1.3. Hyperbolic Hyperbolic Metric Metric on on a Riemann Riemann Surface Surface 3.L.3.

Let itR be a Riemann lliemann surface whose universal universal covering surface is biholomorphibiholomorphiLet L1. Consider Consider a Fhchsian Fuchsian model I of of ,R R acting on /.L1. Let Let cally equivalent to 4. 11": A L1 -----+ r? R be the projection of of 4L1 onto R = = A/f. L1/ Since Since the Poincar6 Poincare metric r: ds 2 is invariant under the action by .l-, we obtain obtain a Riemannian metric ds2p dsh on ds2 R which satisfies satisfies r- (dszp)= d,s2 1I"*(dsh) ds 2 .

r,

r.

r

We call this dsft dsh the Poincar6 Poincare metric, or the hyperbolic hyperbolic metric on R. E f corresponds corresponds to an element element [Cr] [G,] of the fundamental group Now, every, every 1 e 1I"1(R,po) R (Theorem 2.5). 2.5). In particular, ,7 determines determines the free free homotopy r{R,po) of ,R class say that of the class G" where G, representative of class [G,]. say that, covers We of where is a representative class C7, C,, 7 coaers [Cr]. the closed curve G,. curue the closed Cr. j € -t, - A-, When, E f is hyperbolic, it it is seen that the closed closed curve L, A,/I < ), seen that 1l , >, When image on .R R of the axis axis A, A, by 11", geodesic (with (with respect respect to the zr, is the unique geodesic the image hyperbolic metric metric on ,R R )) belonging belonging to the free homotopy homotopy class class of of G,. L, Ct. We call L-, hyperbolic geodesic to corresponding to or the closed "7, G,. closed geodesic corresponding C.,.

r

r

=

coueringsurface surface Proposition be a Riemann surface with with universal uniaersal covering Lel R be Riemann surface Proposition 3.3. Let H, and r be a Fuchsian model of R acting on H. Let model acting on Let H , and 11l be Fuchsian

+ = ---, ,Z tk) cz+ ( )

a z * bb _ az c z * dd'

a , b , c , dE a,b,c,d e PR, -,

a d -- bc b c = 1, 7, ad

r

geodesic be element on and L, bethe the closed closedgeodesic on R corresponding hyperbolic elemenlof 11, L, be R corresponding beaa hyperbolic I , and to lenglhl(L,) I(Lr) of L, L, satisfies satisfies to ,. Then the the hyperbolic hyperboliclength 7. Then

t.'(r) - @+d)2= 4cosh2 e) 2 an {t) are Proof. tr2(7) are invariant under the conjugation of, of 7 by an t(L-r) and and tr Prool. Since Since l(L,) - AZ )z (.\ 1). also element = (A > 1). We may also we may assume assume that ,(z) element of ol Aut(H), Aut(H), we 7(z) we have this case, case,we have and d = 1/-/>.. I/\5.In In this assume t5, b = c =0,0, and assumethat ao - -/>.,

=

= =

=

dy ((L.t) =fIr^ = 210ga. ) = l o gA 2log a. = -y = + = log A

l(L,)

1

Hence we have have the assertion. assertion. Hence we

o!

3.1.4. 3 . 1 . 4 , Pants Pants

Consider which admits the hyperbolic metric by aa surface R r? which Riemann surface Consider cutting aa Riemann family of mutually geodesicson R. Let P be be aa relatively simple closed closed geodesics mutually disjoint simple compact subsurfaces.If If P contains contains connectedcomponent component of the resulting union of subsurfaces. compact connected no more geodesicof R, be triply triply connected, connected, i.e., i.e., .r?,then P should should be more simple simple closed closed geodesic homeomorphic region, say say homeomorphic to aa planar region,

cc

55

frlauroa.g rqoqraddll

'I't Pu€ f,rrlel{ grsf,urod

3.1. Poincare Metric and Hyperbolic Geometry

({i t rr- ",}^{i >rr+,r})- { z > l z l } = 0 4 Po =

({ Iz

{izi < 2} -

+ 11 ~ ~} U {Iz - 11 ~ ~}) .

'U uo rlsapoe3 pasolc elduns € sl 2I ul d Jo frepunoq e^rleler eqt Jo luauoduroc palrauuoc frarra 3r Pue Palceuuoc i(1dt.r1q d JI g' 'g e IIef, e^\ '.re1;eara11 1o sTuod;o .rted e U Jo d eD€Jrnsqnslcedtuoc f1errr1e1e.r Surppnqa.r.ro; secerdlseilerus eql Jo auo s€ pereplsuoc aq ue? d et"Jrnsqns e qcns

Such a subsurface P can be considered as one of the smallest pieces for rebuilding R. Hereafter, we call a relatively compact subsurface P of R a pair of pants of R if P is triply connected and if every connected component of the relative boundary of P in R is a simple closed geodesic on R.

4J/V:d

Fig. 3.1.

'r'8'tIJ

Fix a pair of pants P of R arbitrarily. Let r be a Fuchsian model of R acting on ..1, and 1r : ..1 ---+ R = ..1/ r be the projection. Let P be a connected component of 1r- 1 (P). Denote by rp the subgroup of r consisting of all elements ,of such that ,(P) = P. Then p is a free group generated by two hyperbolic transformations, and P = P/ p. Set P = ..1/ rp. Then P is a surface obtained from P by attaching a suitable doubly connected region along each boundary component. Hence, P is again triply connected, and rp is a Fuchsian model of P (see Fig. 3.1). Clearly, P is considered as a subsurface of P, which is the unique pair of pants of P. In other words, P is uniquely determined by rp. Habitually, P is called the Nielsen kernel of P, and P is called the Nielsen extension of P.

'd louoNeuelseueslerNaql Pellet sr d Pu€'d Jolaweq ueslerN erll Pall€l 'd st 2r 'f11en1rq€H'-dJ fq paurunalap flanbiun ,, j '.pto^ reqto uI Jo slued ',,(pee13 '2' 3o aee;rnsqns € s'e pereplsuoc sl d ;o rred anbrun eql s! plq^{ 'pelcauuoc ,{1dr.r1 '(t'g '31.{ eas) ; Jo lapour uelsqcqil e q d.7 pue 'acue11'lueuodtu6r ,t.repunoq qcee Suop uorSar palceuuof, .{1qnop ure3e sr 2' paul€tqo areJrns€ sI uaql 'a,t/V - d las elq€lrns e Surqcelp fq dr 4'uro.r; ql/d d pue'suotleurrogsuerl crloq.rad,tqom1fq pele.reue3dno.r3aa.ge s1 d.7 uaql 'd = ({)t }€tI} q?ns J Jo L dnor3qns eqt dJ fq alouaq '(d) r -! Jo lueuoduroc sluetuele yo Surlsrsuoc J Jo 1e = U aq1 aq pelceuuoce aQ V i )L pve'7 uo 3ur1ce JIV d 1e1'uorlcelord 'fpre.r1rqr" g 3o 7 slued ;o .rted e xtg U Jo lapour uersqcqE e aq J 1e1

r

r

r

56

3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3.

3.1.5. Existence Existence and and Uniqueness Uniqueness of of P'-ts Pants 3.1.5.

shall discuss the the relationship relationship between the the complex complex structure structure of of a triply triply conWe shall domain J? Q and and the the hyperbolic hyperbolic structure structure of of P, P, the the unique unique pair pair of of pants pants of of nected domain Q, induced by by the hyperbolic hyperbolic metric metric on O. Q. O, Let L1,L2, L 1 , L 2 , and L 3 be the boundary boundary components, components, which a.re are simple closed closed and..L3 Let geodesics, of of the the pair pair of of pants pants P. P. Let Let J-e model of of the domain domain geodesics, o be a Fuchsian model Q acting on A. Ll. Then'i-s Then'ro is a free free group generated generated by two two hyperbolic transO that 1'1 and may assume that 1'1 and 1'2 cover .L1 L 1 and L2, formations, say, say, 7r may assume and. 1'2. We formations, 72 71 72. respectively.

r

Theorem 3.4. 3.4. For given triple triple (ayaz,as) (a1, a2, a3) of of positiae positive numbers, numbers, For an arbitrarily giaen Theorem planar Riemann connected planar surface Q such such that there exists Riemann surfoce triply connected lhere erists a triply

=

t£(L ( L ji )) = a aj, 1,

=

ji - 1 ,1,2,3. 2,3.

it by constructing constructing O Q explicitly. explicitly. Proof. We prove it say C C2, Let Cr C 1 be the part part of of the imaginary imaginary axis in A. Ll. Fix Fix another geodesic, geodesic, say 2, 2. On the Ll such that the Poincar6 Poincare distance distance between between C1 C 1 and C2 C 2 is equal equal to a1f at/2. such that on 4 from which which the Poincar6 Poincare distance to C1 C 1 are equal other hand, geodesics geodesics on 4Ll from other a3/2 form form a real one-parameter one-parameter family family (i.e., (i.e., the family family of circular arcs arcs Cl C~ to oBf2 geodesic, Hence there exists exists a geodesic, tangent to the broken circular arc in Fig. 3.2-i)). Hence and C Cg Cz2 and say that the Poincare Poincard distance between C such that family such say C 3 is Cs, 3 , in this family equal a2f2. equal to a2l2.

i)

ii)

Fig. 3.2. Fig.3 .2.

determined by the condition Next, let Zl 4 uniquely determined z2 be be the points in Ll 21 and arrd Z2 a1

p(rr,r) p(Zl,Z2)

== 2' ?,

zzEe C2· zr Ee CCt, Cz. 1 ,Z2 Zl

and {zs, let {Z3, 22. Similarly, Let Z4} and Z6} geodesicconnecting 21 and and Z2. Similarly, let connecting Zl L\ be be the the geodesic Let L~ {rs,re]1 {z3,za} be by the the conditions conditions pairs of points uniquely determined by uniquely determined of points be the the pairs

L9

pu" rrrlel l ?r"3ulod 'I't

3.1. Poincare Metric and Hyperbolic Geometry i(rlauroag ruoqradi(g

a2 = 2'

57

Z9

' 8 8 8 8 8 C ) v z ( z C ) ,E zz - ( v z ' e s ) 6 p(Z3' Z4)

Z3

a3 tp

= 2'

E C3 , Z6 E C ll

'rC)sz'eC)sz

, 7 , = (sz(s2)i p(zs, Z6)

E C2 , Z4 E C3 ,

Zs

'eslrlurod fg Eur,rraserd les 3 go ursrqdrour ''e'l'f^2 -o1ne crqd,rouroloq-rlu" aqt qlr^r o1 uorl?auer eql;q (g'Z't = f) laadsar 't=[{!,1'!c} lh p1 pepunoq uo3exaq cqoqrad,(q pesolt eq? eq o p"r fq ('(fa'g 'ft9 eag) 'ez Pue ez Pue'?z pue 8z Eurlceuuoc scrsapoaSeq1 'f1a,rr1cadse.r'!7 pve 1I ,{q alouaq 'flelrlcedser

respectively. Denote by L~ and L~, respectively, the geodesics connecting Z3 and Z4, and Zs and Z6. (See Fig. 3.2-ii).) Let D be the closed hyperbolic hexagon bounded by {Cj, Lj }J=1' Let 'TJj (j = 1,2,3) be the reflection with respect to Cj, i.e., the anti-holomorphic automorphism of C preserving Cj pointwise. Set 'TJ2,

tebsV)-rl0

1'2 = 'TJ3

'TJ1·

'Vtoeb-zL

1'1 = 'TJ1

0

Then 1'1 and 1'2 are hyperbolic elements of Aut(Ll). Let ro be the group generated by these 1'1 and 1'2' pele.rauaEdno.rEaq1 aq o.ir1e1 '(V)l"V

.z,L pue rl, aseqt fq t" sluauele arloqrad,tu are zl, pue rl, ueq,L

Q=L1/ro Fig. 3.3.

'8'8'ttJ

It is clear that il = Ll/ ro is triply connected, and that the unique pair of pants P of il is the interior of the set obtained by identifying the boundary of DU'TJ1(D) under the action by roo Thus il is a desired surface (see Fig. 3.3). 0 '(g'g '81.{ eas) ace;rns peilsap " sr snqtr '0J ,tq uorlce eql rapun (O)tbnO tr {Ji go ,trepunoq eq1 Surf;rluapl fq paurclqo ?as eql Jo rorrelur eql sl (J ;o 2, slued yo rpd anbrun arl? lstll pue 'pelceuuoc f1du1 sr.oJ /V = U leq+ realc s-rlt

'4 perepro aq1lo st176ua7 cqoqtedfrq aqy fiq lo sTuauoduoefi".topunoq p?aunrepp fryanbrunsa 4 syuodto .ttorl o to ernlrtuls aelilutoc ?ttJ .g.g ruoJoaql

Theorem 3.5. The complex structure of a pair of pants P is uniquely determined by the hyperbolic lengths of the ordered boundary components of P.

feur e,r. 'r(.ressacau.l uorleEntuot-(g)7ny ue 3ur:1et 'asodrnd srql lsrll eutrrnssp rog'r={{{o},(q peururralap,{lanbrun are zL pue I,L }eql ^\oqs o} seclsns U 'e? 'I zL) eL sraloc o teql pue (e = 1) 17 s.rairoc{1, ?sqt arunsss feur aa,r ,_(tf 'ara11'0.7 srolereuaS;o rualsfs e aq 'I.L} 'g eueld-y1eqraddn eq} uo leT Jo {zt 3ur1ce Jo lepour uersqr\{ e eq 0J pu€ 'd Jo uorsuelxe ueslarN eql eq d ?c,-I'd d Jo (g'Z'l = f) f7 lueuoduroc frepunoq aq1 ;o q1Eua1crloqlad,(q eql eq lp p"I 'fy.rer1rq.reuaarEsr sluedgo ned s 'too.r4 teql esoddng ('IIt'[Ott] uaay'93) 2,

Proof. (Cf. Keen [110], III.) Suppose that a pair of pants P is given arbitrarily. Let aj be the hyperbolic length of the boundary component Lj (j 1,2,3) of P. Let P be the Nielsen extension of P, and ro be a Fuchsian model of P acting on the upper half-plane H. Let h1,I'2} be a system of generators of roo Here, we may assume that I'k covers L k (k = 1, 2) and that 1'3 = (1'2 0 I'd -1 covers L 3. It suffices to show that 1'1 and 1'2 are uniquely determined by {aj }1=1' For this purpose, taking an Aut(H)-conjugation if necessary, we may assume that

=

58 58

3. Hyperbolic HyperbolicGeometry 3. Coordinates Geometryand and Fenchel-Nielsen Penchel-Nielsen Coordinates

11(Z) 1 ,1, I t ( z )=,x2 = \ 2z2, , 00 < ),x< < a z * b az + b / \ .r2(z) ad-- be bc- 1,I,c 0, ) 0, 12(Z) =;ii,, ez + d' ad e>

=

=

and that that 11 is is the the attractive attractive fixed point of fixed point and of 12, or equivalently, equivalently, 12,or

=

a l b = e+d, sai, a+b

b

O fz(m) == a/e (a -- d)/(2e) d)/(2c) of points of of two two fixed fixed points of 12 has aa value value less (a lessthan than 12(00). 72 has Zz(oo). Next, write write Next,

az -1,, -u+ u Ab, ad-be =) -_-_, (, 7 3\ -1 ) -(z) ' ( z= a d - b z= - 1.1 (,3) dz*b

ez+d

(73)-l = 12 j2o0 ,7r, we may Since(,3)-1 may assume assumethat Since 1> we

6 , = a \ , b=b/>', 6 = b 1 \ , e=e>., E - c \ , d=d/>.. d=a1>,. a=a,x, particular,e> d > O. Moreover,the In particular, of the 0. Moreover, the middle-point middle-point(a (6 - d)/(2e) a11pe1of points the fixed fixed points (73)-1 has greaterthan hasaa value valuegreater than (,3)-1(00) (73)-r(x) = a/c. of (,3)-1 d < O. d/8. Hence, Hence,a a+ O. + the other On the other hand, hand, by by Proposition On Proposition3.3, we have 3.3,we have

=4cooh2 () + 1/>.)2 l/r)' =4cosh2 (>' (~1) (+) ,, 2 =4cosh (a+ d)2 d)2=4 cosh2 (a (~2) f +) ,, \ 2 / '

(a;) . \ 2 /

=4cosh2 (+) (a J:2=4 cosh2 @+ d)2

j2 are Therefore,'y1 and 12 are uniquely uniquely determined determined by {a1, Therefore, 11 and a3 }. az,as}. {or, a2,

D tr

proved that, for any triple have proved We have triple of positive numbers, numbers, there exists exists a pair of pants admitting admitting a reflection reflection (induced, (induced, for example, example, by "7d that the hyperrlr) such such that bolic lengths of of the ordered ordered boundary components components are are the given given triple (Theorem 3.4), and that 3.4), that it it is uniquely determined by the given given triple triple (Theorem 3.5). Thus we (see also we have have the following corollary (see also Fig. 3.4). 3.4). Corollary. Corollary. Eaerg Every pair pair of of pants P has has an anti-holornorphic anti-holomorphie automorphism Jp J p of of order two. two. M o r e o a e r ,the L h esset e t F Jr rp = {{z E P Il JJpp( off a all points off J pp consists Moreover, l l ffixed ixedp oinlso consists z e ( r z) ) = zz}} o of satisfyingthe of thrce three geodesics geodesics {Di}|=, {Dj }1=1 in P satisfying the following following condition: condition: For euery L,2,3), Di every j (j (j = 1,2,3), D j has has the the endpoints endpoints on, on, and is orThogonal orthogonal to, to, both both LL ij aand n d LL 1j +; r1,, w h e r eLL +4 = LL t1.. where

=

=

=

=

We call "Ip J p described described in this corollary the rc,fl,ection reflection of of P. P.

.6J

Let R be a closed Riemann surface of genus g (~ 2). As before, consider cutting R along mutually disjoint simple closed geodesics with respect to the hyperbolic metric ds1t on R. When there are no more simple closed geodesics of R contained in the remaining open set, then every piece should be a pair of pants of R. Recall that the complex structure of each pair of pants of R is uniquely determined by the triple of the hyperbolic lengths of boundary geodesics of it by Theorem 3.5. It is clear that R is reconstructed by gluing all resulting pieces suitably. Hence, we can consider, as a system of coordinates for the Teichmiiller space T g , the pair of the set of lengths of all geodesics used in the above decomposition into pants and the set of the so-called twisting parameters used to glue the pieces. Such a system of coordinates is called Fenchel-Nie/sen coordinates on Tg .

uo selDuzpron ueslely-leq?ury pelle;. sI sel€ulprooc;o rua1s.{s€ qcns 'sacerdaql an13o1 pasn sralatuered 3ut1stall pellec-os arll Jo las aql pue slued olur uorysod -ruocep e^oqe eql ur pesn scrsapoe311e;o sqlSualJo les aq1;o.ued aq1 'rg aceds rellnurqcral aql roJ sel€urproo?;o ue1s.{s e se '.reptsuocue, a^\ 'ecue11'f1qe1tns secard3ur11nsarIIe 3urn13fq palcnrlsuocer sI U leql r€elc sI lI 'g'g ura.roaq; ,{q 1r Jo scrsepoe3,{.repunoqyo sqlEual cqoqradfq eql Jo a1dtr1aq1 fq paunurelep .{lanbrun sl g. slued ;o rred qcea Jo ernlf,nrls xelduroc eq} }eql II€reU Jo '1as uado Surureurereql ul 'p. yo qued 3o .rrede eq plnoqs ecard drarraueql paureluoc U Jo s)rsepoa3pasolc alduns eroru ou are areql uaq1yg, uo {sp f,Ir}aur flenlnur 3uo1ey crloqrad.rtqaq1 o1 laadser qllr* srtsepoa3pesolc aldurts 1u1ofs1p 3ur11ncraprsuoc'eroyaq sy'(e {) f snua3Jo af,eJrnsuu"ruerg pesol?e aq Ur lerl

3.2.1. Pants Decomposition uorlrsodtuocaq

slusd'1''Z'e,

'.re1deqc$r{} Jo rapulqurer aq1 ut fleeq suollJasseeql esn 'g .ra1deq3 lr]un rueroer.{lsqtJo;oord e Sur,rr3auodlsod a6 [eqs e,lr q3noql 'g-fgll o7 ctr1d.r,ou.to?uo?! puD e-oell a! uzD'tuopD s! 6l acods aqu.r,treqJ (g1'g uraroaqa) 'uraroaql s.rallnurqtraJ

We postpone giving a proof of this theorem until Chapter 5, though we shall use the assertions freely in the remainder of this chapter.

Teichmiiller's theorem. (Theorem 5.15) The Fricke space Fg is a domain in R 6g-6 and homeomorphic to R 6g-6.

'.{1e,rr1rn1ur reql€r 1nq 'o3e aurrl 3uo1e pelrecuof, sehruaJoaql Euro,o11o; eq1 're,roarohtr'e-6etlJolasqnse sl tdr ecedse4crrgaql'flsnoue.rd pa1e1ssy '.{rlaruoa3crloqred,tq Sursn ,,lq 6g o1 saleurproo) Jo ad.{1raqloue acnpor}ur a,r. 'uotlces slt{l uI 'saceJrnsJo slepotu uelsqcnd Sursn ,{q 'eceds ueeprpng tg lasqns l€uorsueurp-(S-0g) learJo (eceds e{clq aq1 parueu) (Z ?) f snua3 ;o tg aceds rellnuqtlal aql peluasardar am,'6 reldeqS u1

In Chapter 2, we represented the Teichmiiller space T g of genus g (~ 2) as a subset F g (named the Fricke space) ofreal (6g-6)-dimensional Euclidean space, by using Fuchsian models of surfaces. In this section, we introduce another type of coordinates to T g by using hyperbolic geometry. As stated previously, the Fricke space F g is a subset of R 6g- 6 • Moreover, the following theorem was conceived a long time ago, but rather intuitively. * *

sa +B ur P r o o cu a sla r N{ a q r u a J' z' 8

3.2. Fenchel-Nielsen Coordinates '?'8'ttJ

Fig. 3.4. 3.2. Fenchel-Nielsen Coordinates

59

sat"urProoc uaslarNlaqruad'z'8

60 60

3. 3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-NielsenCoordinates Coordinates

grve a precise precise definition Now,. these coordinates, and verify verify that that they they Now,. we give definition of of these give a system of global coordinates on T To. g• point [R, For this purpose, first E] of first fix a point of T ?r. mutually disjoint disjoint 4 of mutually g • A set £[.R,.D] geodesics on R simple closed closed geodesics .R is termed maximal matirnal if if there is no set £' 4' which properly. We call a maximal maximal set £.C = {Lj mutually disjoint disjoint includes £4 properly. {fi}j!,If=l of mutually geodesics on R simple closed syslem of closed geodesics rt a system of decomposing decomposing curves, curaes, and the family family pants of all connected components of of R -- Uf=lLj the pants p consisting of the UltLi P = {Pd~l {PxW' decomposition of R corresponding corresponding to £-. decompositionof L.

Example. Emmple. When When 9g = 2, there are the two kinds of pants decompositions shown in Fig. 3.5. 3.5.

Fig. F i g . 33.5. .5. system of decomposing decomposing an arbitrary arbitrary system Proposition 3.6. Let £If=l bebe an Proposition L = {Lj {li}l=, genas 9C (~ (>-2), letPP = {Pkl~l 2), and let curves curaes on a closed closed Riemann surface R of genus Riemann surface lPxlf=t

be pants decomposition and N satisfy satisfy decomposition of of R corresponding concsponding to £-. Then M and be the lhe pants t,. Then N ==3 3g g - -3 3

and a n d M ==2 2g g - -2 .2.

Proof. Cut of connected element Ly1 of £-. n1 be the number of r? along an elementL L. Let n1 Cut R genera components be the sum of genera of all connected components connected components components of R .R-- L .t1, 1, and gl 91 of R -- L we have have tr1. 1 . Then we

gl 9 r-- nn1r ==( (g g - -1 )1)- 1- . 1.

Clearly, components of R ,R-- L -t1 Clearly, the number of boundary components 1 is two. Moreover, we can whenever we we add a cut along along a new can see see inductively that, whenever Moreover, we element increasesby two, and the components increases element of £-, 4, the number of boundary components connected sum of genera genera of all connected connected components components minus the number of connected we have components by one. one. Hence, Hence, we have components decreases decrea^ses _ _2N 2 N and gM = = ((gg _- 11)) _- NN, , 3M a n d 0 -_ M =

I9

u3,{'z't seleuProoc uaslarN-Iaqf,

3.2. Fenchel-Nielsen Coordina.tes

61

o

which imply the assertion.

'uorlrasse aql ,(1dun qcqaa

suorlcur\{ q1Eua1 crsapoaD 'Z'Z't

3.2.2. Geodesic Length Functions Fix a point [R, 17] of Tg , and a system £ = {L j }f=l of decomposing curves on R. For every t in the Fricke space F g , we denote by [Rt,17t ] the point in T g corresponding to t. Then, we can determine uniquely a system £t = {Lj (t)}f=,l of decomposing curves on Rto Namely, take a marking-preserving homeomorphism ft : R ---+ R t (cf. Theorem 1.4). For every Lj in £, let Lj(t) be the unique closed geodesic in the free homotopy class of the closed curve ft (Lj) on Rt. It is not difficult to show that Lj(t) is simple, and that Lj(t) and Ljl(t) are mutually disjoint when i :f; i'· Hence, £t {Lj(tnf=l is a system of decomposing curves on Rto Let r t be the Fuchsian model of R t represented by t. Note that Lj(t) is the projection of the axis of an element of rt which covers ft(L j ) for every i. Now, for every tin Fg and every i, we denote the hyperbolic length l(Lj(t)) of Lj(t) simply by lj(t). We consider lj(t) as a function on Fg (or equivalently, on T g ) and call it the geodesic length function for Lj. Then Proposition 3.3 implies the following lemma.

uo 'flluelerrrnbe.ro) 6g uo uorlcunJ € se (1)f7 raprsuoc eM'(ib fq fldrurs Ol7 p qfual cqoqrad{q eq} elouep eiu'f l(ra,repue td ur 3 itre,re ro;'aro11 ((l)!l)l 'f, fra,ra q (!1)tg sreloc qcq/rt ,J Jo luatuala u€ Jo slx€ eq1 ;o uotlcafo,rd aq1 sr (1)f7 leq? eloN'1fq paluesardarrgrJo lepow u"rsqcr\{ eqt aq tJ ta1 .rar uo salrnc Sursodruocap;o uralsfs e sr t--;f{11;t1\ = rj'eoue11 ',1 + | uaqar lurofsrp .r(11en1nur ere (7),17 pue (3)f7 l€rl? pue 'eldurrs sr (1)f7 ?€rll aorls ot llnclgrp tou '? uo (!7)r1 a^rne pasole arlt q ss€p fdolouroq eerJaql ur crsapoe3pesolc lI Jo enbrun aql eq (l)fZ f"t 'J ul !7 {ra,re rog '(y'1 uraroaq; 'Jc) tU A : tt 'tgr uo selrnf, Sutsodurooap ursrqdrouroauroq Eur,rrasard-Eur4retue alel tr(1arue11 I=J{(t) l?} = ,J urelsfs e ,tlanbrun aunurelap uec er$'t".II '? o1 Eurpuodserroc Jo t; ur "U] ,tq alouep e$'6,tr a*ds aqcrq4eqt ul t fra,ra rog '3. uo aqt lurod lt3' '6J selrnc Eursodurocepf" tlf{fZ} 7 ure1s.{se pue lo [g'g,] lurod e xld

=

serTdur ueqy t7 rc1uot1cun! 6'9uorlrsodord vtfuq ?*?po;fflttftTopH iU

Lemma 3.7. Every geodesic length function lj (t) is real-analytic on Fg •

'6d uo cr1fi1ouo1oa.r, st (l)h uo4cunt y76ua7nsapoa| fi.taag 'Z'g BuruxaT sralaruerBd Eullqar;'g'Z'g

3.2.3. Twisting Parameters alouep pu" '(e'I o1 fre11oro3 eql 'z'f4 leql II€lsU g" f7 Eul\eq d

Next, for every i, let Pj,l and Pj,2 be two pairs of pants in P having Lj as a boundary component. Here we allow the case where Pj,l = Pj,2' Recall that Pj,l and Pj,2 admit the reflection J l and J 2, respectively, by the Corollary to Theorem 3.5. Take a fixed point of h, on Lj for each Pj,k (k = 1,2), and denote it by Cj,k. Fix also an orientation on Lj (see Fig. 3.6). As before, let [R t , 17t ] be the point of Tg corresponding to t for every t in Fg • For every t and i, let Pj, 1 (t) and Pj ,2 (t) be the connected components of Rt Uf=,lLj(t) (which are pants of Rt) corresponding to Pj,l and Pj,2, respectively. Recall that each Cj,k (k = 1,2) is the end point on Lj of the geodesic Dj,k joining Lj and another boundary component, say Lj,k' in Pj,k' Let Lj,k(t) be the boundary component of Pj,k(t) corresponding to Lj,k' Denote by Dj,k(t) the geodesic joining Lj(t) and Lj,k(t) in Pj,k(t) with minimal length, and by Cj,k(t) the point of Dj,k(t) on Lj(t). Then each Cj,k(t) (k = 1,2) is a fixed point of the reflection of Pj,k(t). Let 1j(t) be the oriented arc on Lj(t) from Cj,l(t) to Cj,2(t). Since Lj(t) has the natural orientation determined from that of L j , we can define the signed hyperbolic length Tj(t) of 1j(t) (so that Tj(t) is positive or negative according to whether the orientation of 1j(t) is compatible with that of Lj(t) or not). Set

1ag'(1ou rc (7)17 Jo 1eql qlrar alqrleduroc s1 (3)fuJo uorteluerro eql raqleqa ol Eurproace a.,rrleEauro a,rrlrsod sg (3).r.r.1eq1 os) (l)l,l,l" (l)f, qt3n"t cqoqred,tq pau3rs eql eusap ust e^l 'lI lo l"ql uro+ peururalep uorleluarro l€rnl"u eql seq (l)fZ acurg'(3)z'fc o1 (3)t'fa urory (3)f7 uo f,trepatuerro eqt aq (t)fap1 .(ic'ta Jo uorlcagar aqlgo lurod pexss sl (U'I = t) (1)t'!c qcee uaql'(l)f7 uo Ot't1;o lurod aq1 (ic'ft fq pue '{t3ua1 prurunu qtyn (1)r'12,ur (7)t'!7 pue (1)f7 Eururof crsapoa3 aq1 (1)r'fc, i(q alouaq 't'17 ol Eurpuodsarro) (i't'!d;o luauodruoc frepunoq eq1 eq (7)t'!7 p"l'1'14 u1 tr'!7 fes 'luauoduroc f.repunoq raqloue pue !7 Sururof r'f6. crsepoe3 aq1 yo f7 uo lurod pua a{} sl (Z,I - q) t'fc qcee }€ql IIeceU 'f1a,rr1cadsa.r 'z'!4 pue r'14 o1 Surpuodsa.uo?(tU;o slued a.re qcrq,u,)(l)fZt=Jn - ? Jo sluauoduroc palcauuo? aql aq (iz'fa pue (1)t'f4' 1a1'f, pue l fra,re rog ''d.rl 'a.ro;aq ';;o sy I {rara.rog 1o1 Eurpuodserroc lurod aql aq l,3'rA) 1a1 '(g'g'3t.{ aas) f7 uo uol}sluerro us osls t1 "g'r'fa,tq = {) r't, qcea rc1 !7 uo Y Jo lurod paxg " e{"I 'g'g ureroaqJ ^q 'flarrrlcadse.r 'zf pue r/ uorlcager aql lrurpe z'!4 pue t'14 - r'.r2'a.raq,r,r es€l aql ,&roll"e^r areH'lueuoduroc drepunoq e ul slued;o s.rred oiu.1aq z'!4 pue r'fd 1a1'/ fra,re roy 'lxaN

Tj (t) Bj(t) = 211" lj(t)' (t)!t-"

.l.r!t!"^ -= (7)!6

62 62

3. Hyperbolic 3. Hyperbolic Geometry and Fenchel-Nielsen Fenchel-Nielsen Coordinates

Dt,z

D;,r

(P.;,1: Pi,z)

Fig. F i g . 3.6. 3.6.

Then Bj(t) Bj(t) the twisting parameler parameter with 0i(f) is well-defined well-defined modulo 271". hr. We call caII01(t) respect respect to Lj. L1. Lemma j, exp(iBj(t)) Lemma 3.8. For every eueryj, exp(id1(t)) is well-defined well-definedand and real-analytic real-analylicon on F Fn. g• Proof j. For every Proof. Fix Ftx 1. every tf in F .Fr, Fuchsian group represented represented by t. fi t be the Fuchsian g , let r Take which covers Lj(t), and denote it by 'Yj(t). Take an element element of r covers tr;(l), denote each 4t 71(t). Next, for each 1,2),Iet be the element of which k&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&(= (= 1,2), let 'Yj,k(t) be the element of which covers Lj,k(t) and satisfies that covers L1,x(t) and satisfes that [ t 7i,x(t) geodesicDj,k(t), Di,n(t), connecting connecting Aj(t) the geodesic ,41(f) and and Aj,k(t) A1,*Q) with the minimal length, length, is is projected onto Dj,k(t), Di,*(t), where where,4i(t) and Aj,k(t) A1,r(t) are are the axes axesof 'Yj projected Aj (t) and (t) and and 'Yj,k(t), 7i(l) T,x(t), (see Fig. respectively respectively(see Fig. 3.7). 3.7). points of 'Yj(t), we may assume Here, we assumethat the fixed fixed points Here, and 'Yj,2(t) move 7i(t), 'Yj,l(t), ti,t(t), and 7i,2(t) move • Hence, real-analytically on F when we take a conjugation of r by an element Hence, when we take aconjugation ,Q an element g t d. \(t) > j ( r ) .. zz((Aj(t) o f Aut(H) s o that g o e sto points of A u t ( H ) so t h a t 'Yj(t) t o 'Yj(t)(z) 1 ) , the t h e fixed f i x e d points ) 1), 7 1 ( t ) goes f u ( t ) ( z ) = . \Aj(t) corresponding to 'Yj,k(t) move also of 'Yj,k(t) also real-analytically on F Fo each k. #. g for each 7i,1(t) move fu,*(t) corresponding projection of the end c1,1(t) is the projection Now, Cj,k(t) end point Cj,k(t) Zi,r(t) of "t Dj,k(t) Now, to Aj(t). A1Q). fu,x(t) Ifence, if we show Hence, if we show that Cj,k(t) 6i,x(t) moves moves real-analytically on F Fr, assertionfollows follows g , the assertion by the definition definition of Tj(t) 11(t) and and Lemma 3.7. 3.7. p1 and P2 p2 be show this, fix k, /c, and let PI To show be the fixed points of 'Yj,k(t). Set ii,i(l). Set (v* > 0). c13(t) = iYk iv* (Yk 0). Since Since Cj,k(t)

r

+ (PI - P2) = + P2 ) , =(PI '7* (ry)' (o'to')' 2 2' 2

Yk

we see we seereal-analyticity real-analyticity of Cj,k(t). ci,r(l).

2

2

o

';c) ursrqd.rouroe$lpe sl g_ogtl x ,_rg(ag)

-

.([tqz] pue ,h-yl 1.rad1o44 6l : 'dgenlcy 'IrDur?A 4i

----+

(R+?g-3

X

R 3g- 3 is a diffeomorphism (cf. Abikoff

.Uo{lqy

[A-l), and Wolpert [251]). Remark. Actually, .p : F g

'(sa,r.rncEursodurocep;o uralsfs eql qtyr\ 'to) d uotltsodtuocap slued aql 7 rl?-rrA paler)osse t;;o selouxprooxueslery-Ieq)uef ^seler;rprooc asaql II€r elA

We call these coordinates.p Fenchel-Nielsen coordinates ofTg associated with the pants decomposition P (or, with the system I:- of decomposing curves).

.6a uo uI 'e-oellX nuaq puD'6,I uo sa?Durproo? 7oqo16 lo ue7sfrro saat64i'.to7nc4.r,od 'OI'8 tuaroaql 6g to rusttld;oruoeuoq e s? 4 6utddnu, s?ttJ e-re(+U) oTuo

Theorem 3.10. This mapping .p is a homeomorphism of F g onto (R+)3 g-3 xR3g- 3 . In particular, .p gives a system of global coordinates on F g , and hence on T g . 'le,roelotr41 'uaroaql 3uralo1o; eq1 erro.rd,r.ou ilBqs aru

Moreover, we shall now prove the following theorem.

'uf uo ctTfipuo-pat st

is real-analytic on F g . ( ( l ) " - u " a ' . . . , ( t ) r0 , ( t ) e - u e T. . . , ( t ) r t ) = ( t )r t

ueqJ '! tuaaa ut 6g uo (7)lg nTatuo.tod 6ut7stm7aqyto qcun"tqsnonurluo? panlna-e16utsD ottr '6'g eurtrrarl

Lemma 3.9. Fix a single-valued continuous branch of the twisting parameter OJ(t) on F g for every j. Then

:3ur,r.lo11o; eql e^€r{ e,lr snqtr '(uraroaql ftuo.rpouour eql) t/ uo r{ou€rq snonurluoc panp,r,-e13urs € s€rl (3){6 frarra',{13urp.roccy'uleurop paltauuof, .{ldurrs (pueq raqlo aqt uO e s-rtJ l"q1 sale?s(91'g ura.roaql) ura.roaqls(rellnuqclatr

Here we set R+ {x E R \ x > O} and Sl {z E C \ \z\ l}. On the other hand, Teichmiiller's theorem (Theorem 5.15) states that Fg is a simply connected domain. Accordingly, every OJ(t) has a single-valued continuous branch on Fg (the monodromy theorem). Thus we have the following:

'{f = Itl I C > t} = rS pue tO < t lu>

t} - +lI 13seA\ereg

=

=

=

' (((r)t-rtOl)dxa'. . .' ((1)t6r)dxa'(t)e-aq . .'' (t)rt) = (t)'t

tJt(t) = (£l(t), .. ·£3g_3(t),exp(iB 1 (t)), ... ,exp(i03g _ 3(t))).

:e-re(rS) x e-oe(+tI) -

6tr :

e Paugepeleq ax\ '.re;o5 4i Surddeu cr1.,{1eue-1ear

So far, we have defined a real-analytic mapping tJt : F g

----+

(R+)3 g-3

X

(51 )3 g-3;

3.2.4. Fenchel-Nielsen Coordinates uaslalN-IaqcuoJ'v'z'

salBurPlooc

I

't'8'ttd

Fig. 3.7. Q|'r,

(t)z''g

(t1z''t

U7t'rt

63

sal"urprooc uaslarN-leqf,uaJ'z't

3.2. Fenchel-Nielsen Coordinates

t9

64 64

Ilyperbolic Geometry 3.3. Hyperbolic Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates

Prool. First, First, we we show show that thatitrr injective. Suppose Proof ~ isis injective. ~(t2) for Supposethat that ~(td fr(tr) ==rir(t) for some some t1 and t2 in Fo. Let be the Ra, the Riemann Riemann surface surface represented t 1 and t 2 in Fg- Let Rt. be representedby by tit; (with (with the the natural marking), marking), and and Pi pa^ntsdecomposition natural b" the th" pants decomposition of of R A == {Pk(i)}~~;;3 {P}(i)};s=;" be &,t • correspondingtoP, for each eachif (= (= 1,2). 1,2).BV corresponding to P, for By Theorem Theorem 3.5 3.5 and and the the assumption, assumption, there is is aa conformal conformal mapping, mapping, say g&, of say gk, of Pk(l) P7,(1)onto there onto Pk(2) P1(2) which which respects respectsthe the boundary correspondence correspondencefor for every every k. /c.Moreover, proof of boundary Moreover,the the proof of Theorem rheorem 3.5 implies B.bimplies that that dsl== (gk)*(ds~). (si4.@sl). dsi = 1,2) Here,ds; dsl (i (f = 1,2) is is the the hyperbolic hyperbolicmetric metric on on Rt pa.rticular,every Here, •. In particular, every gk is aa &,.In 9r is hyperbolic isometry isometry of of the the closure closureof of Pk(l) Pi(l) onto hyperbolic onto the the closure closureof of Pk(2). Pp(2). Since Since |j(tr) = ?i(tr),

j = 1,...,39- 3,

g1 can all gk glued together can be be glued together into into aa marking-preserving all ma^rking-preservinghomeomorphism, homeomorphism, say say h, h, of R rt1, onto R -R1r. is holomorphic on Since hlr is of on R r?1, except for for aa finite finite number of of t2 . Since t1 except t1 onto analytic curves, curves, so so is is hh on on the entire entire Rt .R1, analytic by Painleve's theorem. Hence, h is aa Painlev6's theorem. Hence, h is 1 biholo-morphicmapping of R -R1,, which implies implies that t1 biholomorphic t1 = t2. lz. Thus we we have proved have proved t1 , which P is is injective. injective. that ~ we show show that ~ f is is surjective. surjective. For purpose, we Next, we For this purpose, we begin begin with fixing aa g -3 x R 3 g-3 arbitrarily. For every .. . ,a3g-3,0'1,'" (4a1, (R+;sc-a point (a1,'" R3s-3 arbitra.rily.For every ,eas-s,e1t. . . ,0'3g-3) ,ass-z) of (R+)3 P3 decomposition P of R, denote denote by {L P (C .C) the boundary k,j}1=1 (C,C) k in the pants decomposition {Il,i}i=, components of P Pp. there is components 1.5, there is aa unique unique pair of pants, pants, say say P~, Pto,such such that $1.5, k . From § lengths of the boundary components the triple of the hyperbolic lengths components of P~ Pf is is equal equal pil pi,2 given triple {ak,j}1=1' to the given Set pi = {pk}~~~3. As before, let Pj,l and Pj,2 be and be {ou,i}i=r.SetPt= {P;}ir=lt.As before, let pj,xbe elements of P neighboring j, neighboring each each other along the elements Lj for every j, and let PIl be along .Li every element of P/ correspondingto Pj,l Pip (f (l = 1,2). the element pi corresponding be the point on I,2). Let cj,l Ci,2be oii the th" j Pjp corresponding correspondingto ci,t for every boundary of Pj,l to Cj,l and f. every and l.Now, by gluing Pj,1 suitablV along Pj,l and P/,2 Pj,2 suitably L j for along curves curves corresponding corresponding to ,Li every j, we obtain a Riemann surface, surface, say every j, we R . We need to choose a suitable say r?'. need choose ' gluing (and aa-suitable suitable marking of .R') R ' ) so R ' corresponds so that that R/ corresponds to a point tt,' of F u c htthat h a t f~(t') ( t / ) iiss e i v e n((a1' q u a ltto o tthe he g a 1 ,... . . . ,,aa3g-3, 1 s _ B 0'1,' , d l , . .. . . , ,a0'3g-3)' ss_a).T Fg, ssuch equal given This h i s ccan an be achieved gluing by Pr{,1 and Pj,2so achieved Pj,l Pj,2 so that that the twisting twisting parameter becomes becomes the given ai 0' j for every j. j. We shall explain this procedure procedure more rigorously by using Fuchsian Fuchsian models. models. pi,z, In the proof, rest of this we consider In of consider only only the case case where where Pj,l Pil t'"I Pj,2, for the other case case can be considered considered similarly. Fix Fix j,j, and let 4,r rj,k be a Fuchsian Fuchsian model of of the Nielsen Nielsen extension extension Pj,k of fj,t Pj,k P|,r "t for each each &. k. Here we assume assume that that every 4.,r rj,k acts on the upper half-ptanl half-plane fH,, aird and that that the transformation transformation

=

AZ, l,(Z) (z)=\2 ,

=

)A- e xexpaj p a i ) L> 1

belongs li,z, and, and 7 covers belongs to to both both 4,r r j ,l and rj,2, covers the the boundary boundary component, say Lj,k' of of say Ll1r, Pj,r Pj,k corresponding to.Li to Lj for each each & k (& (k == 1,2). We We also assume assume that that the the nilural natural orientation orientation of of the the axis axis ,4 A == {z {z €E Hl, HI Z -= 'iu,y iy, y )> 0} O} of of 7, corresponds corresponds to to the the prescribed prescribed orientation orientation of of .ti, Lj, and and that that the the point point ii €E /lH lies lies over cj,k with with respect respect over cl,x to to li,* Ij ,k for for each each /c. k.

99

sal"urProoc uaslarN-Iaqf,uad'z't

3.2. Fenchel-Nielsen Coordinates

65

luauale eql raplsuoopue'(r,6f lofo)dxa = fp 1eg

Set di = exp(aiG:i/27r), and consider the element 0 < lp

'ztp - (z)g

of Aut(H). Identify every z on the axis A of'Y considered as an element of rj ,2 with b(z) on the axis A of'Y considered as an element of ri,l' (See Fig. 3.8.) ('g'g'q.{ aag) 't'1Jo tuetuelerrc se pareprsuoo,LJoy srxe aq} uo (")g qft^ z'!tr p lueutueleue se pereprsuoc,LJo y srx€ eql uo z f.tete,tg11uap1'(n)WV lo |;e

'nJ

Fig. 3.8.

'8'8'EtJ

'eroJeq "'fa r'{" 3urn13 * pue l€tll q?ns ,U oluo A p rt tusqd.rouroauoqe xrJ fq paurclqoec"Jrnsuueuerg eql sq ,A 1eI"tr;o dlrrrrlcaf.rns;oyoord er{l Jod ('r 'dtq3 '[tz-f] tl{se4 acu€tsulroJeas'sruaroaql uorl€urquo) .rog'(saaeg.rns om1Surlaauuor.rog)uuoeql uorlDurquocs(uNelNse ulrou{ flpcrsselc q slqtr) '!/A Jo uorsuelxeueslarNeql Jo Iapou u€rsqcr\{ e sr I'lJr {q paleraueEdnorSuersqf,ndaq1'sprornraqlo uI'{ rl"€a roJ ,-gz'!J9, l ' pue ! , tt'!4 uo '?"p ,It!^ {.raaaEuole'}'I$ ul s (a,unaf.repunoqEurureura.r lueprcuror r'la p uorsuelxe uaslarN aqlJo uorlrnrlsuor eql ur pasnse^rqf,rq^\'ureuropEurr elq"lrns e 3urqce11e fq lnt urorJpeurelqoec"Jrnsuusuarg eql ''e'l) fr14ureurop ?ql uorsuelxeueslerNeql uo f,rrlerucqoqred,(q"ql'(H)?nV 3 9 acurg 3ur11nsar Jo 'fo r'fa uorJ fp3o1 olnpou ot lenba sr z'!p oI {fua1 uor}elsusrlarlt }erll qcns t'l,l z'!4 pue t'fa p 3urn13e e^eqe^{ueqJ to uorlecgrluapleql ,(q) 1z'!,7pue

Then we have a gluing of Pj,l and Pj,2 (by the identification of Lj,l and Lj,2) such that the translation length from Cj,l to cj,2 is equal to log di modulo ai' Since b E A ut( H), the hyperbolic metric on the Nielsen extension of the resulting domain Wi (i.e., the Riemann surface obtained from Wi by attaching a suitable ring domain, which was used in the construction of the Nielsen extension of P;,k in §1.4, along every remaining boundary curve) is coincident with dS~/,, on Pj,k 1,

for each k. In other words, the Fuchsian group generated by ri,l and bri,2b-1 is a Fuchsian model of the Nielsen extension of Wi' (This is classically known as Klein's combination theorem (for connecting two surfaces). For combination theorems, see for instance Maskit [A-71], Chap. 7.) For the proof of surjectivity of .j" let R' be the Riemann surface obtained by gluing Pj,l and Pj,2 as before. Fix a homeomorphism h of R onto R' such that 'Z-6?,'...'I = {

"la =(qa)rl

h(Pk)=P£,

k=1,···,2g-2.

66 66

3. 3. Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates

point ofT Then Then [R',h.(E)] determinesaa point be the point of of To.Let l'be the corresponding correspondingpoint g • Let t/ [rR',h-(t)] determines preceding .Fo. • F From the preceding construction, construction, it is is clear clear that g

=

=

, " ' ,3g l i ( t ' ) = aj, a t , ji = I1,··· fj(t') , 3 9-- 33,, and that and ).t

0 1 ( t ' 1 : ^ : I " e d 1 p o 1 ) ( m o d2 r ) , uj

j = t , . . ' , 3 9- 3 .

preseni, we we cannot (Note that, at present, cannot say say that OJ(t/) 01(t') = aj, ai, because because the choice choice of 0i is is not unique.) unique.) branch of OJ ------R' Hence,letting 7j Ti; : R' ---+ R'be the Dehn Dehnlwist Hence, be the twist with respect respectto Lj, trl , the the curve curve on.R'corresponding we can on R' corresponding to to.ti, Lj, we can find find integers integersn1, ... ,n3g-3 such that ftrt.'. such ,fl3s-3

[R :,(T i 'o...orti :i" o h) .( t) ] corresponds correspondsto aa point, say say t", ttt, which satisfies satisfies , i r 1 t ,= , 1( (a1' o r , .... . . , ,a3g-3, a s s _ s ,a1, o t r... . . . ,a3g-3). ,osc_s). !it(t") we have have shown that that j, fr is is surjective. Thus we Here, the Dehn twist 7j with Here, with respect respect to to.Lj Lj is, is, by definition, aa homeomorphism homeomorphism Q of R' onto R' ,? corresponding to the following following surgery: cut R' along along Lj L! and a.ndreglue rotation of 211" (see Fig. 3.9). 2zr (see after aa rotation 3.9). Note that applying T Ti, we can can make make the j , we value of OJ value di increase increase by 211" 2r while every every other OJ j) remains unchanged. 0i (i/ remains unchanged. U'+j)

t

o0 o twist F Fig. i g . 33.9. .9.

g-3 xX R3r-3 -------+ we have have proved that fr j, : F (R+)3 R 3g- 3 is bijective. By Now, we Fo (R+)sr-a g ---+ j, is also also continuous. continuous. On the other hand, Teichmiiller's theorem Lemma 3.9, 3.9, tit 6 6 (Theorem 5.15) homeomorphic to R g- • Hence Hence the following states that F 5.15) states Fo R6c-0. g is homeomorphic theorem, Brouwer's theorem on invariance theorem, invariance of of domains, domains, implies that thatV!it is actually n a homeomorphism. 0 homeomorphism.

aldurrs (anbrun) aql aq lV m 'raqtrr\{ 'IV tq tr alouap pue '17 s])asre]ur qcrq^r fr14ur crsapoaEpasolc aldurs e xrg 'z'f4, n lI nt'!4 - lr14 ps'f fre,ra rog as"r eq} ^{olls e,lr'ure3y) 'luauoduroc,trepunoq e se lI Eur,teq ('7"{d = I'fd teqf '7 o1 Surpuodserroc 'J > lI {rarra rog dJo slueuele eq1 a'f4, pue I'ld' fq elouep U Jo uorlrsodurocapslued aql aq d +aI'U uo selrnc Sursodurocapgo 7 uelsds e pue (6 f' snua3 ersJrns uu€ualu e xg 'uorlces snor,rerd eql q sY l) Ur Jo '21'9 uorlrs 'uorl€zrJlerue.red;o pur{ raqloue arrrS leqs aira. -odo.r4 ur 'ra1e1 'suorlrunJ {lEua1 crsapoa3o^\} qll^r sa}eurproof,ueslerN-leqrueg paxg q relaurered 3ur1sral1qcea Surcelde.rfq '6g go slurod aleredas suoll?unJ 'a.re11 {}8ua1 esor{^\ scrsapoa3 pesolc aldurrs 6 - d6 Jo }es e lcn.rlsuoc aa,r '(FOt] IIBAToSpue eleddag ';c) elqrssodurr q slql '.rala,no11'oJuo selsurprooc pqo13 a,rrEsuorlcuny qfual asoql( srrsapoa3 posolt aldurs g - 69 Jo les € 6?ilrererll ;t elqsrrsap lsoru eq plnoa lI 'areJrns eql eururalep sqfual crloqradfq asoq^\ k ?) 0 snuaS;o af,eJrnsuuetuarg pasol, e uo scrsapoe3pesop eldurrs;o 1ese Surpug;o uralqo.rdeql replsuor e,n 'uotlaas qql uI

In this section, we consider the problem of finding a set of simple closed geodesics on a closed Riemann surface of genus 9 (~ 2) whose hyperbolic lengths determine the surface. It would be most desirable if there was a set of 6g - 6 simple closed geodesics whose length functions give global coordinates on Tg- However, this is impossible (cf. Seppala and Sorvali [194]). Here, we construct a set of 9g - 9 simple closed geodesics whose length functions separate points of T g , by replacing each twisting parameter in fixed Fenchel-Nielsen coordinates with two geodesic length functions. Later, in Proposition 6.17, we shall give another kind of parametrization. As in the previous section, fix a Riemann surface R of genus 9 (~ 2) and a system .c of decomposing curves on R. Let P be the pants decomposition of R corresponding to .c. For every Lj E .c, denote by Pj,l and Pj,2 the elements of P having Lj as a boundary component. (Again, we allow the case that Pj,l = Pj,2.) For every j, set Wj = Pj ,l U L j U P j ,2. Fix a simple closed geodesic in Wj which intersects L j , and denote it by ,1J. Further, let ,1} be the (unique) simple

3.3. Fricke-Klein Embedding tu rp p e q tu g u la lx- a { r l4 4 ' g ' g

e. ')

corresponds to a continuous curve in T g • The variation of a Riemann surface represented by such a curve is called a Fenchel-Nielsen deformation. We shall investigate the deformation of this type in Chapter 8 by using quasiconformal mappings. We shall also give a direct proof of continuity of ~-1 with respect to

o1 lcadsar {1a r_4;o flmurluor;o;oo.rd }cerrp € arlr3 osle lleqs a1ys3urddeu IeuroJuocrsenb Sursn fq g reldeq3 ur adfl $rll Jo uorleruroJep eq1 ele3rlsaaut Upqs a1yuotTotu.totap u?slerN-Ieycuadre pelpr sr elrnf, " qf,ns ,tq pelueselder ac€Jrns uu"tuerg e Jo uorlerJel eql 'tJ ur eAJnf,snonulluot e o1 spuodsauoc ( { U > I | ( € - 6 s o t. . . r r * l n , J ' t - ! n , . . . , I r 2 , 8 - 6 t D. ,. . , I p ) } ) r _ 4

ljf

- -1

({(a1,··· ,a3g-3,0'1,··· ,O'j-1,e,0'i+1,··· ,0'3g-3) leER})

eqt yo fg e3eurre.rdeql '0I'g ueroeqtr dg 'sralaure.red3ur1sr,ra1 e u oe { € l p u e ' n - r s ? I x e - n e ( + u ) ; o ( e - E e o ' . . . ' r D ' s - 6 8 D' . ' . ' 1 o ) l u r o d e x r g (}j

Fix a point (a1, ... , a3g-3, 0'1, ... , 0'3g-3) of (R+)3 g-3 x R 3g- 3 , and take one of the twisting parameters. By Theorem 3.10, the preimage

'uollJasse aql a^eq arrr "{rerlrq.re ! sr f erurg'IO uo snonurluocsl r-d pue'O;o lurod rorJelur ue sr f '.re1nct1.red uI'IO qlr^r luaprf,uroceq plnoqs d dq g Jo g lul rorrelur eqf Jo (g' 1u1)d e3eur 'pueq retllo eq} uO -,lI aq1 'acue11'pelrauuoc s\ u^roqs eq u"? ll ler{} fI '/s' Jo uleluop rolrelxe eql pue ureruop rorJalur eql 'fla^llredsar 'zO pue IO {q alouaq 'slueuoduroc pelcauuoc o,r.l s€q ,S - .rll (uraloeql s(u€pJof leuorsueurp-, "ql fq 'acue11 'rll ul 'fle.rrrlcedser'ereqds pcrSolodol e pu€ IIeq pesol) 1ecr3o1odo1 e erc (gg)dt - rS pue,g're1ncr1redu1 '(g)dt = ,g oluo €r go ursrqdrouroeuoq € s.r g uo o1yo uorlculsar aq1 'lceduot sr Br aculs 'r reluar qll/rl - o pue'f1t.re.r1tqre pesop e f1tre.r1tq.re xg'(f)r-d las 5' ileq '[ZI-y] sreg 'y3) '(qc1aqs y) too.r4 O q n lurod e xIJ ('[98-y] ueu^\eN pue

Proof (A sketch). (Cf. Bers [A-12], and Newman [A-85].) Fix a point y in D arbitrarily, and set x = ep-1(y). Fix arbitrarily a closed ball B with center x. Since B is compact, the restriction of ep on B is a homeomorphism of B onto B' ep( B). In particular, B' and S' ep( BB) are a topological closed ball and a topological sphere, respectively, in R n . Hence, by the n-dimensional Jordan's theorem, R n - S' has two connected components. Denote by D 1 and D 2 , respectively, the interior domain and the exterior domain of S'. On the other hand, it can be shown that R n - B' is connected. Hence, the image ep(Int B) of the interior Int B of B by ep should be coincident with D 1 . In particular, y is an interior point of D, and ep -1 is continuous on D 1 . Since y is 0 arbitrary, we have the assertion.

=

=

'o oluo 'utotuop o s.t ("g)dt ueqJ ' iy olu! uE O ;g to tustrliltoutoeuoU o st d) puo 'onl uDlI ssq lou ta,a\ut. uo to uo4catut snonur?uo?p eg uE {_ ull : 6 7a7 eq u pI (sureurop Jo acuBrJBlur uo uraroaql s6rar*no.rg) 'tt'B uraroatlJ

Theorem 3.11. (Brouwer's theorem on invariance of domains) Let n be an integer not less than two. Let ep : R n ---+ R n be a continuous injection of R n into R n . Then D = ep(Rn ) is a domain, and ep is a homeomorphism of Rn onto D. turppaqurg uralx-a{f,rrJ't't

3.3. Fricke-Klein Embedding

67

L9

68 68

3. Coordinates 3. Hyperbolic HyperbolicGeometry Geometryand and Fenchel-Nielsen Fenchel-Nielsen Coordinates

geodesicwhich is closed closed geodesic is freely freely homotopic to the simple simple curve curve obtained from L1J 4! (seeFig. by applying L j (see 3.10). applying the Dehn twist with respect respectto to,ti Fig.3.10).

(Pi.r: Pi,z)

Fig. F i g . 3.10. 3.10.

?0. For every every For Fc,let b" the corresponding correspondingpoint of T For every every t E e F t , L't] be g • For g , let [R [it1,&] geodesicon geodesicL on R, we closed geodesic closed we express express as as L(t) I(t) the corresponding corresponding closed closed geodesic R f(L(t)) the hyperbolic length of L(t). I(t). Set Set denote by l(L(t)) ft1, t , and denote

fj(t) = t(Lj(t)), f(Lj(t)), tuo_"+i(t) f 3g - 3+j (t) = t(al(t)), f(L1J(t)), ftao-a+i(t) f(L1}(t)) ti(t) 6g - 6+j(t) = t(Aj(t)) j, and set for every set every j,

I 1 t 1= ( r ( t ) , . . . , / g o - s ( r ) ) . We the following: following: Wehave havethe -s. Theorem inlo (R+)9 (n+;seg-9. mapping 1 L is it a o proper embedding embeddingof F Fo Theorem 3.12. 3.L2. The The mapping g into

(That is, preimage of any and the the preimage any onto the the image irnage l(F t(F),g ), and is, 1 I is a homeomorphism homeomorphism onto g-9 under compact (R+)ee-s L is compact.) compact.) sel in (R+)9 under 1 compactset To prove prove this theorem, we fix a point to ts of F Fs theorem, first we g arbitrarily, and write

3 c -33 . .Tr( = (( a1,'" ( R + ; a9 s-3 -t x R R 3go r , . . . ,a3g-3,lll,'" f r 1tot )o 1 € (R+)3 '£' -_ X . , e 3 s - J , e r t . . . ,ll3g-3 , o e g - s) )E t(s) Fix j, and for every s) of F every Ss E define a point t( Fix j, € R, define d g by --1

t ( s ) = f r - ' ((a1,··· o r , " ' ,a3g-3,lll,'" t(s)=1Jf ,asc-e). , a ! - r t o i + s ' o . i + r , " ' ,ll3g-3). , a 3 s - s , Q r t " ' ,llj_1,llj+S,llj+1,'"

Then we we have have the following Proposition.

alsq era acueg 'f1a,rr1cadse.r ',? pue og ol pue 0z spues pue '? slt<eeq? qll,r{ crloq.redfq e s rLoeL luetuele ,z uorltsoduror aql uaqf 'slurod paxg se'f1a^rrrleadse.r'0rn pue oz er.ie-q qcrrl^it,olrr1 raPro Jo {H)?nV Jo slueruala arldqla eql eq zl pue Il, }al 'pueq reqto aql uO

On the other hand, let /1 and /2 be the elliptic elements of Aut(H) of order two which have Zo and wo, respectively, as fixed points. Then the composition /2°/1 is a hyperbolic element with the axis L, and sends Zo and Zl to io and i', respectively. Hence we have '(,V'

(' ') +pw,z. (' ") pz,Z _pz,w ( I ")
(l)rt leql qcns ? a rnc pasolo alduns € eleq e^\ acuaH ',| 1e spua pue q}-I/'astlsls qcrq^r,Jr Jo J3el 3 Jo cr€qns e sr areql 'uraroaql ecuerrncer s