Geometry of Riemann surfaces and Teichmüller spaces

GEOMETRY OF RIEMANN SURFACES AND TEICHMULLER SPACES NORTH-HOLLAND MATHEMATICS STUDIES 169 (Continuation of the Notas ...

Author: M. Seppälä | T. Sorvali

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GEOMETRY OF RIEMANN SURFACES AND TEICHMULLER SPACES

NORTH-HOLLAND MATHEMATICS STUDIES 169 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

GEOMETRY OF RIEMANN SURFACES AND TEICHMULLER SPACES

Mika SEPPALA Academy of Finland Helsinki, Fin land

Tuomas SORVALI University of Joensuu Joensuu, Finland

1992

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.

Library of Congress Cataloging-in-Publication

Data

Seppala. Mika. G e o m e t r y o f R i e m a n n s u r f a c e s and T e i c h m u l l e r s p a c e s / M i k a Seppala, Tuomas Sorvali. p. c m . -- ( N o r t h - H o l l a n d Mathernalics s t u d i e s , 169) I n c l u o e s b i b l i o g r a p h i c a l r e f e r e n c e s a n d index. I S B N 0-444-88846-2 1. R i e m a n n surfaces. 2. T e i c h n u l l e r s p a c e s . I. S o r v a l i . T u o n a s . 1944. 11. T i t l e . 1 1 1 . S e r i e s . O A 3 3 3 . S42 1992 515 .223--dc20 9 1-34760 CIP

ISBN: 0 444 88846 2

0 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in The Netherlands

Preface This monograph grew out of a series of lectures held by the first author at the University of Regensburg in 1986 and in 1987 and by the second author at the University of Joensuu in 1990. This book would presumably not have been written without the initiative of Professor Leopoldo Nachbin. A large part of the present work has been carried out at the University of Regensburg and at the Mittag-Leffler Institute. We thank these both institutes for their warm hospitality. Finally we thank Ari Lehtonen for several figures, especially for his intriguing illustration of the Klein bottle.

In Helsinki and in Joensuu, Finland August 1991 Mika Seppala

Tuoinas Sorvali

1

This Page Intentionally Left Blank

Introduction The moduli problem is to describe the structure of the space of isomorphism classes of Riemann surfaces of a given topological type. This space is known as the moduli space. It has been in the center of pure mathematics for more than 100 years now. In spite of its age, this field still attracts lots of attention. The reason lies in the fact that smooth compact Riemann surfaces are simply complex projective algebraic curves. Therefore the moduli space of compact Riemann surfaces is also the moduli space of complex algebraic curves. This space lies in the intersection of many fields of mathematics and can, therefore, be studied from many different points of view. Our aim is to get information about the structure of the moduli space using as concrete and as elementary methods as possible. This monograph has been written in the classical spirit of Fricke and Klein ([31]) and in that of Lehner ([57]). Our main goal is to see how far the concrete computations based on uniformization take us. It turns out that this simple approach leads to a rich theory and opens a new way of treating the moduli problem. Or rather puts new life in the classical methods that were used in the study of moduli problems already in the 1920’s. Some results, like the Uniformization of Riemann surfaces, have to be presented here without proofs. They are, however, used almost exclusively to interpret the results derived by other means. Proofs are not really based on them. In all cases, where we do not present proofs, we furnish exact references. If one is willing to accept Uniformization and some related facts, then this monograph is self-contained and can be read without much prior knowledge about complex analysis. In Chapter 1 we develop an engine that will power other chapters. There we consider Mobius transformations and matrices. One of our aims in Chapter 1 is to understand thoroughly how commutators of Mobius transformations behave and how groups generated by Mobius transformations can be parametrized. All considerations here are elementary, but sometimes technically complicated. In Chapter 2 we present some basic results of the theory of quasiconfor3

4

INTRODUCTION

ma1 mappings. Everything there is presented without proofs, which can be found in the monograph of Lars V. Ahlfors [6] and in that of Olli Lehto and Kalle Virtanen [59]. Quasiconformal mappings have played an important role in the theory of Teichmiiller spaces. They provided the tools with which it was possible to develop the first rigorous treatment of the moduli problem. Today most of the results concerning Teichmiiller spaces and moduli spaces can be shown even without quasiconformal mappings. Quasiconforma1 mappings are only absolutely necessary to show that the moduli space of symmetric Riemann surfaces of a given topological type is connected (cf. Theorem 4.4.1 on page 147). In Chapter 3 we first review the Uniformization of Riemann surfaces without proofs. Then we show how considerations of Chapter 1 can be applied to study the geometry of Riemann surfaces. Our main concern in this Chapter is to study the geometry of hyperbolic metrics of Riemann surfaces of negative Euler characteristics. We derive many results concerning simple closed geodesics and sizes of collars around them. We pay special attention to the geometry of symmetric Riemann surfaces, i.e., to non-classical Klein surfaces. Everything here can be shown in detail using the results of Chapter 1. It is actually surprising how much information can be obtained from detailed analysis of the commutator of Mobius transformations. Considerations of Chapter 3 form a quite comprehensive treatment of certain aspects of the geometry of hyperbolic surfaces. So it may be of some interest for its own sake already. Main target is, however, to get information about the moduli problem using considerations of Chapter 1 alone. The beginning of Chapter 3 provides an environment in which considerations of Chapter 1 can be interpreted so that we get useful results for later applications. In Chapter 4 we introduce Teichmuller spaces and define its topology using quasiconformal mappings. Here we have to resort to the review presented without proofs in Chapter 2. We will, however, later derive an alternative way of parametrizing the Teichmuller space using the geodesic length functions. That is done in detail here (see page 161). Quasiconformal mappings provide a simple way to describe the complex structure of the Teichmuller space of classical Riemann surfaces (cf. page 148). We will take benefit of that description and indicate how our considerations lead to a real analytic theory of Teichmuller spaces. This also leads to a presentation of Teichmiiller spaces as a component of an affine real algebraic variety (Section 4.12). This affine structure is derived here in detail. This is a n important part in the theory of Teichmiiller spaces, albeit not central, because it opens new ways of compactifying the Teichmiiller space by using methods of real algebraic geometry (cf. 1641, [16], [73]). We will not consider these interesting approaches to the compactification problem here.

INTRODUCTION

5

Figure 0.1: T h e Mobius strip and the Klein bottle are two genus 1 real algebraic curves that are not homeomorphic to each other. In this monograph a new moduli space is constructed for the these non-classical Klein surfaces. T h e presentation of the affine structure of Teichmuller spaces is, however, partly motivated by these new applications of real algebraic geometry. Points of the moduli space of compact genus g Riemann surfaces are isomorphism classes of mutually homeomorphic genus g Riemann surfaces. Such Riemann surfaces are smooth projective complex algebraic curves. So the moduli space of genus g Riemann surfaces is the same thing as the moduli space of smooth genus g complex algebraic curves. In Chapter 5 we consider this moduli space and define a natural topology for it. The definition of the topology is based on the Fenchel-Nielsen coordinates. In that topology the moduli space is connected but not compact. Using the considerations of Chapter 3 we then consider degenerating sequences of Riemann surfaces. It turns out that by adding points corresponding t o so called stable Riemann surfaces it is possible t o compactify the moduli space of compact and smooth genus g Riemann surfaces. This is quite classical today and has first been shown by David Mumford and others using the methods of complex algebraic geometry. Smooth projective real algebraic curves have more structure than complex curves. They can be viewed as compact Riemann surfaces with symmetry. Equally well they can be viewed as compact non-classical Klein surfaces, i.e., surfaces that are obtained as the quotient of a smooth Riemann surface by the action of the symmetry. This fact was realized already by Felix Klein (cf. [46]). Therefore the inoduli spaces of non-classical compact Riemann surfaces are simply moduli spaces of real algebraic curves. A compact genus g surfaces has L(3g +4)/2J topologically different orientation reversing symmetries. It follows, especially, that real algebraic curves of the same genus need not be homeomorphic t o each other. This implies that, in any reasonable topology, the moduli space of smooth genus g real

6

INTRODUCTION

algebraic curves has several connected components. The situation changes completely when we consider the natural compactification of the moduli space of real algebraic curves. That space is obtained by adding points corresponding to stable genus g real algebraic curves. We show, in Chapter 5 , that this moduli space of stable real algebraic curves of a given genus g , g > 1, is a connected and compact Hausdorff space. This fact was already conjectured by Felix Klein in [48, Page 81. We start Chapter 1 with the assumption that the reader is familiar with elementary properties of Mobius transformations. For the sake of completeness we have included also an appendix in which we develop the elementary and classical theory of Mobius transformations. A proof for the so called Nielsen Criterium for discreteness of Mobius groups acting in a disk is also included. Basic properties of the hyperbolic geometry are considered in an appendix as well. Resorting to the appendices, if necessary, this monograph can be read with only basic knowledge of complex analysis.

Contents 1 Geometry of Mobius transformations 1.1 Introduction t o Chapter 1 . . . . . . . . . . . . . . . . . . 1.2 Mobius transformations . . . . . . . . . . . . . . . . . . . . . 1.3 Multiplier preserving isomorphisms . . . . . . . . . . . . . . 1.4 Parametrization problem and classes H . P and & . . . . . . 1.5 Geometrical properties of the classes P and 'H . . . . . . . . 1.6 Parametrization of principal-circle groups by multipliers . . 1.7 Orthogonal decompositions and twist parameters . . . . . . 2

3

Quasiconformal mappings 2.1 Introduction t o Chapter 2 . . . . . . . . . . . . . . . . . 2.2 Conformal invariants . . . . . . . . . . . . . . . . . . . . . . . 2.3 Definitions for quasiconformal mappings . . . . . . . . . . 2.4 Complex dilatation . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

.

11 11 12 17 23 29

40 46

59

.. ..

59 59 61 62

Geometry of Riemann surfaces 69 3.1 Introduction to Chapter 3 . . . . . . . . . . . . . . . . . . . 69 3.2 Riemann and Klein surfaces . . . . . . . . . . . . . . . . . . . 69 3.3 Elementary surfaces . . . . . . . . . . . . . . . . . . . . . . . 71 3.4 Topological classification of surfaces . . . . . . . . . . . . . . 72 3.5 Discrete groups of Mobius transformations . . . . . . . . . . . 80 3.6 Uniforinization . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.7 Models for symmetric surfaces . . . . . . . . . . . . . . . . . . 93 3.8 Hyperbolic metric of Rieinann surfaces . . . . . . . . . . . . . 95 97 3.9 Hurwitz Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.10 Horocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Nielsen's criterium for discontinuity . . . . . . . . . . . . . . . 104 3.12 Classification of Fuchsian groups . . . . . . . . . . . . . . . .107 3.13 Short closed curves . . . . . . . . . . . . . . . . . . . . . . . . 108 3.14 Collars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.15 Length spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 115

7

CONTENTS

8

3.16 Pants decompositions of compact surfaces . . . . . . . . . . . 117 3.17 Shortest curves on a hyperbolic Riemann surface with a symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.18 Selection of additional simple closed curves on a hyperbolic Riemann surface with a symmetry . . . . . . . . . . . . . . . 124 3.19 Numerical estimate . . . . . . . . . . . . . . . . . . . . . . . . 129 3.20 Groups of Mobius transformations and matrix groups . . . . 131 3.21 Traces of commutators . . . . . . . . . . . . . . . . . . . . . . 132 3.22 Liftings of Fuchsian groups . . . . . . . . . . . . . . . . . . . 135 4 Moduli problems and Teichrnuller spaces

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14

137 Introduction to Chapter 4 . . . . . . . . . . . . . . . . . . . . 137 Quasiconformal mappings of Riemann surfaces . . . . . . . . 141 Teichiniiller spaces of Klein surfaces . . . . . . . . . . . . . . 144 Teichiniiller spaces of Beltrami differentials . . . . . . . . . . 147 Non-classical Klein surfaces . . . . . . . . . . . . . . . . . . . 148 Teichinuller spaces of genus 1 surfaces . . . . . . . . . . . . . 150 Teichinuller spaces of reflection groups . . . . . . . . . . . . . 152 Parametrization of Teichinuller spaces . . . . . . . . . . . . . 155 158 Geodesic length functions . . . . . . . . . . . . . . . . . . . . Discontinuity of the action of the modular group . . . . . . . 162 Representations of groups . . . . . . . . . . . . . . . . . . . . 164 170 The algebraic structure . . . . . . . . . . . . . . . . . . . . . Reduction of parameters . . . . . . . . . . . . . . . . . . . . . 172 Extension to non-classical surfaces . . . . . . . . . . . . . . . 174

5 Moduli spaces 177 5.1 Introduction to Chapter 5 . . . . . . . . . . . . . . . . . . . 177 5.2 Moduli spaces of smooth Riemann surfaces . . . . . . . . . . 178 5.3 Moduli spaces of genus 1 surfaces . . . . . . . . . . . . . . . . 180 185 5.4 Stable Rieinann surfaces . . . . . . . . . . . . . . . . . . . . . 5.5 Fenchel-Nielseii coordinates . . . . . . . . . . . . . . . . . . . 190 5.6 Topology for the inoduli space of stable Riemann surfaces . 192 5.7 Compactness theorem . . . . . . . . . . . . . . . . . . . . . . 194 5.8 Syinmetric Rieinann surfaces and real algebraic curves . . . . 195 5.9 Connectedness of the real moduli space . . . . . . . . . . . . 197 5.10 Compactness of the real moduli space . . . . . . . . . . . . . 201 5.11 Review on results concerning the analytic structure of moduli spaces of compact Rieinann surfaces . . . . . . . . . . . . . . 204

.

CONTENTS

9

A Hyperbolic metric and MSbius groups 209 A.l Length and area elements . . . . . . . . . . . . . . . . . . . . 209 A.2 Isometries of the hyperbolic metric . . . . . . . . . . . . . . . 211 A.3 Geometry of the hyperbolic metric . . . . . . . . . . . . . . . 211 A.4 Matrixgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 218 A.5 Representation of groups . . . . . . . . . . . . . . . . . . . . . A.6 Complex Mobius transformations . . . . . . . . . . . . . . . . 223 A.7 Abelian groups of Mobius transformations . . . . . . . . . . . 231 A.8 Discrete groups of Mobius transformations . . . . . . . . . . . 238

B Traces of matrices B.l Trace functions

..........................

245 245

Bibliography

249

Subject Index

258


Chapter 1

Geometry of Mobius transformations 1.1

Introduction to Chapter 1

Our main interest lies in parametrizing groups generated.by Mobius transformations. By the Uniformization such a parametrization can be used t o study surfaces and their complex structures. That is the theme of chapters 3 , 4 and 5. In this chapter we derive the necessary preliminary results which then power the rest of the book (excluding Chapter 2). First we recall briefly the classification of Mobius-transformations. That is explained in more detail in Appendix B in which we review the classical theory of Mobius transformations and groups of Mobius transformations. Natural question is t o find ways of parametrizing Mobius groups up to a conjugation by a Mobius transformation. Therefore we need parameters which remain invariant under such conjugations. Natural candidates for such parameters are the multipliers of Mobius transformations. Equivalently one may use also traces of the corresponding matrices. In this chapter we will mainly use multipliers. They are more natural since they are uniquely determined while the trace of the corresponding matrix is determined only up to the sign by the Mobius transformation. This difference may a t first sight appear only as a small technical complication, but it turns out to be one of the main difficulties. The problem lies in the fact that the sign the trace of a product of t w o matrices is not determined by the signs of traces of the matrices in question. To deal with that difficulty we introduce, in Section 1.4, a classification of pairs of hyperbolic Mobius transformations. We use that classification to find natural parameters which determine a group (generated by a finite number of hyperbolic Mobius transformations) up to a conjugation by a MGbius transformation. 11

CHAPTER 1. MOBIUS TRANSFORMATIONS

12

The main problem of this section is to find a minimal set of parameters for Mobius groups generated by a finite number of hyperbolic Mobius transformations. That is also important for later applications. In the general case these groups are not freely generated. The generators usually satisfy a relation which typically says that the product of certain commutators is the identity. Such a relation is difficult to deal with. In the general case the problem of finding a minimal parametrization using only multipliers of elements of the group remains open. In Theorem 1.6.4 we summarize our results concerning this problem. It is our conjecture that the parametrization given by Theorem 1.6.4 is actually minimal. Recent investigations by Chen Min support this conjecture, but it has not been shown yet. A minimal parametrization can be constructed if we use also certain twist parameters or gluing angles. That is done in Section 1.7. The twist parameters presented there are closely related t o the usual Fenchel-Nielsen gluing angles1, but they are not the same.

1.2

Mijbius transformations

We shall consider groups of conformal automorphisms of the extended complex plane C. Directly conformal automorphisms of C are orientation preserving Mobius trunsformations Z H -

az+ b cz d'

+

a d - b c = 1,

whereas indirectly conformal ones are orientation reversing Mobius transformations a?+ Z H -

L

cZ+ d'

ad

- bc = -1.

We shall concentrate on groups which act in the upper half-plane U . It is a well-known exercise in complex analysis to show that transformations (1.1) and (1.2) fix U if and only if the coefficients a , b, c and d are real. Rays or half-circles in U perpendicular t o the real axis R are called nonEuclidean lines. A Mobius transformation fixing U maps a non-Euclidean line onto a non-Euclidean line. There are the following types of Mobius transformations fixing U : 0

the identity transformation,

0

hyperbolic transformations,

0

parabolic transform at ions,

'For the usual definition of the Fenchel-Nielsen gluing angles see Section 5.5

1.2. MOBIUS TRANSFORMATIONS a

elliptic transformations,

a

reflections,

13

glide-reflections. The first four types are orientation preserving whereas the last two ones are orientation reversing. It will turn out that hyperbolic transformations are the most essential ones. Geometrically, the hyperbolic transformation

is determined by the following three parameters: a

the attracting fixed point a ( g ) = limn+m g n ( z )

a

the repelling fixed point r ( g ) = limn-,oog - n ( z )

a

the multiplier

where z is any point in C not fixed by g . Especially, k(g) > 1. The fixed points are real if and only if g ( U ) = U . In this case, the non-Euclidean line through a ( g ) and T ( g ) , the axis of g , is denoted by a z ( g ) . It has natural orientation by r ( g ) + a ( g ) . Denoting k = k ( g ) , z = a ( g ) and y = r ( g ) we obtain, from the crossratio defining k ( g ) , the following representations for g : g(z) =

(kx - y ) z - z y ( k - 1) if 3 : # c Q # Y , ( k - 1). 2 - k y

+

g(z)

= k z - y ( k - 1) if x = 00,

g(z)

= -+z(1

t

k

-

1 -) k

if

y =

(1.4)

00.

Conversely, if k > 0, x and y , x f y , are given, then the Mobius transformation g defined by formulae (1.4) has the following properties: a

if k = 1, then g = id,

a

if k

a

if k < 1, then g is hyperbolic, k(g) = l/k, a ( g ) = y and r ( g ) = z.

> 1, then g is hyperbolic, k ( g ) = k , a ( g ) = z and r ( g ) = y,

CHAPTER 1 . MOBIUS TRANSFORMATIONS

14

Since the coefficients in (1.3) are determined up to the sign, it follows by the formulae (1.4) that

+

Hence la d( > 2. Consider conjugate Mobius transformations g and g' = h o g o h-' where h : C c--) C is a Mobius transformation. Then g and g' are of the same type. Suppose that g is hyperbolic and fixes U . Then 0

g' fixes h ( U ) ,

0

4s') = h(a(g))and 4 g ' ) = h ( T ( d ) ,

0

a49') = h ( 4 7 ) ) ,

0

h maps the non-Euclidean lines of U onto the non-Euclidean lines of h(U).

The conjugacy class of a hyperbolic transformation is determined by its multiplier. In fact, for any hyperbolic transformations

g(z)= and g'(z)

az + b

a'z = c'z ~

+ b' +d '

ad - bc = 1,

a'd' - b'c' = 1,

the following conditions are equivalent: 0

g and g' are conjugate,

0

qY) =W),

0

la + dl = la'

+ d'l.

Let us consider the other types of Mobius transformations fixing U. A parabolic transformation has one fixed point only. For (1.1) this occurs if and only if ( a dl = 2. If we set k ( g ) = 1 in the parabolic case, then (1.7) remains valid. Elliptic transformations are conjugate t o rotations

+

2 H P.2

(1.8)

15

1.2. MO3ZUS TRANSFORMATIONS of the complex plane. If gl and g2 are conjugate to

z

H

ei'l z

and

z

H

eiB2z,

+

respectively, then g1 and g2 are conjugate if and only if 91 = f92 n27r. Therefore, we may define the multiplier k(g) of an elliptic g conjugate to (1.8) by setting k(g) = e", where 0 < .9 5 K.

If we denote by x and y the fixed points of g, then either = (g(z),z , x , Y>

2' or

= ( ! l ( z ) ,z , 2,9)

,-i19

for all z # x, y. Hence formulae (1.4) hold also in the elliptic case either with k = ei' or with k = e-i'. Moreover, since 05

2'

+ e-i' + 2 < 4

for all 6 E R, it follows that (1.7) is valid and .1 d ] < 2. An elliptic transformation fixes U if and only if its fixed points are complex conjugates. The composition of two orientation preserving transformations

+

g(4 =

az

+b

5

7

ad - bc

# 0,

and

is obtained by multiplying the corresponding matrices: ua+

(: :)(; ! ) = ( c a t dby7 Hence g(w)

(aa = (ca

@+b6 @+d6

(1.9)

+ b7)z + ap + b6

+ d y ) z + cy + d6'

For orientation reversing transformations this is not true in general. But restricting ourselves to transformations with real coefficients, the coefficients of the composite transformation are obtained by (1.9) regardless of whether any of the transformations is orientation reversing. The axis az(o) = { z I u ( z )= z }

C H A P T E R 1. MOBIUS TRANSFORMATIONS

16

of a reflection 5 fixing U is a circle or line perpendicular t o R, i.e., a non-Euclidean line in U. Denote by z and y the real fixed points of U . Then

)(.

= +(y)

+

where is the elliptic transformation defined by k(+) = -1, +(z) = z and $(y) = y. Hence, inserting k = -1 in formulae (1.4), we obtain the following representations: (1.10)

( ~ ( z )= - S - t 2y if z = 00

(1.11)

Especially, (T and '1c, agree on the real axis. A glide-reflection s fixing U is of the form

where 0

'1c, is a hyperbolic transformakion fixing U ,

a 5 0

is a reflection fixing U ,

a x ( $ ) = az(a).

+.

Hence the glide-reflection s is uniquely determined by Moreover, since s2 = $ 2 , also the hyperbolic transformation s2 defines s uniquely. Especially, s and $ have the same fixed points and the same axis. ) k = k ( s ) = -k($). Formulae (1.4) and Denote x = a(s), y = ~ ( s ant1 (1.10) then yield the following representations:

(kz - y)z - xy(k - 1 ) if z#..#y, ( k - 1)zt 2 - Icy s ( z ) = Icz- y(k - 1) if z = 00, 1 z tx(1if y = 00. s(z) = s(z)

=

x)

(1.12)

(1.13) (1.14)

If we define k(a) = -1 for a reflection 0 , then formulae (1.10) are obtained from (1.12) and (1.13) as special cases. It follows that la

+ 4 = m--J==Wl 1

for all transformations (1.2) fixing U . For reflections we have la whereas ( a -t dl is positive for all glide-reflections.

+ dl = 0

1.3. MULTIPLIER PRESERVING ISOMORPHISMS

17

A transformation (1.4) with k < -1 is loxodromic. A loxodromic transformation g has well-defined fixed points a ( g ) and r ( g ) and hence also a well-defined multiplier k ( g ) . By formulae (1.12), a glidereflection s fixing U admits also a representation

4.) = II,(z)

where II,is the loxodromic transformation defined by k($) = k ( s ) , u(+) = a ( s ) and T ( + ) = T ( s ) . On the real axis, s and h?, agree. Note that II,maps U onto the lower half-plane and that is decreasing on the real axis.

+

1.3

Multiplier preserving isomorphisms

In this section, we consider groups G of Mobius transformations acting in U. We are interested in developing conformally invariant systems of identification for the groups G. Our final goal will be to define a minimal set of identification numbers. In fact, in the next sections, we will give a "social security vector" to every group with a certain normalization (Theorem 1.5.6). For precise formulation of the results we consider isomorphisms of the groups G. The next leinnia shows that the Mobius group G is in most cases determined by its hyperbolic elements.

Lemma 1.3.1 Letg and h be Mobius transformationsfixing U , h hyperbolic and g ( a ( h ) ) # r ( h ) . Then g o h" is hyperbolic or a glide-reflection for suficiently large values of n. Proof. We may suppose that u ( h ) = 00 and r ( h ) = 0. Let k = k ( h ) and consider the representation (1.1) or (1.2) of g . Then

Since g ( o 0 ) # 0, we have a

It follows that g

o

# 0.

Hence

h" is either hyperbolic or a glide-reflection for n 2 no.

If, in the above lemma, g o 1%" is a glide-reflection, then it is determined by the hyperbolic transformation ( g o h n ) 2 . Hence g is determined by the hyperbolic transformations h and ( g o hn)2 for any n 2 no.


18

We show next that, under quite general assumptions, the group G is in fact determined up to conjugation already by the multipliers of its hyperbolic elements. To that end we need some technical lemmas. The function

f(k)=

-

/*

is well-defined and non-negative for both k

= f(k2)

f@l)

fl

kl = k,

t 1

=

> 0 and k = ea8. We have

9

f(k) >_ f(1) = 2 for k > 0, 0 = f(-l) 5 f(k) 5 f(1) = 2 for k = e”,

0

f(k) 4 00

if and only if

max(k, l / k )

+ 00.

is defined for all orientation preserving transformations

fixing a disk or a half-plane and

f(s)= .I

+ dl7

> 2 e g hyperbolic,

0

f(g)

0

f(g) = 2

0

f(g)

e g parabolic or the identity,

< 2 e g elliptic,

f(gi) = f(g2) formations.

W

k(gl) = k(g2)

* g1 and g2 are conjugate trans-

Let (g,h ) be a pair of hyperbolic transformations fixing the upper hdfplane U. Suppose that g and h have no common fixed points and denote 0

t = (r(g),T ( W 7 a

’ kl = 0

w

7

k2

= k(h),

k3

= k(g

0

h).

7

.(g))7

1.3. MULTIPLIER PRESERVING ISOMORPHISMS

19

In order t o derive an expression for f ( k 3 ) in terms o f t , Icl and 1 2 we normalize by conjugation such that r ( h ) = 1, a ( h ) = 0 and a ( g ) = 00. Then t = r ( g ) and we have by formulae (1.4)

g ( z ) = Iclz - t(k1 - I),

It follows that

L e m m a 1.3.2 The multipliers k l , k2, k3 and kq = k(g2 o h ) determine t and hence also the conjugacy class of ( 9 , h ) uniquely. If only k l , k2 and kg are fixed, then t has two possible alternatives.

Proof. Retaining the above normalization we have by (1.15) either

or

Then t - t' > 0. By eliminating f ( k 3 ) we get

Similarly, replacing g by g2 we get two values t and satisfying- 7 k; k2 t+t =-2 ( k ; - 1)(k2 - 1 ) '

for r ( g ) = r ( g 2 )

+

+

Since k2 > 1, the function k H ( k k 2 ) / ( k - l ) ( k 2 - 1) is strictly decreasing f o r k > 1,and we have t+t' < t+tl'.Therefore, the sets { t , t ' } and { i,? } can share at most one point. On the other hand, r ( g ) belongs to the intersection of the sets { b , t ' } and { t, Z'}. Hence t = r ( g ) is uniquely determined.

L e m m a 1.3.3

Ic(gmi

o hni)+ 00 whenever min(rn;,ni) + 00.

Proof. By (1.15)

+

f(gm8o h n a )= f ( k ~ ' k ~ t' ) ( 1 - t )

kldmi + kTnl 1 + k y m lk;nl

I-

CHAPTER I. MOBIUS TRANSFORMATIONS

20

Since g and h have no common fixed points, we have t follows. a

# 0, and the assertion

It follows from Lemma 1.3.3 that g" o h" is hyperbolic for sufficiently large values of n.

Lemma 1.3.4 a(g"

o

h") + a ( g ) and r(g* o h " )

-+

r ( h ) as n + 00.

Proof. We may suppose that the fixed points of g and h are finite. Choose disjoint closed intervals 11 c R and 12 c R containing a ( g ) and a(h) as interior points, respectively, but not containing r ( g ) or r(h). Choose no such that g"(12) C 11 and h"(11) c 1 2 for n 2 no. Then g"(h"(11)) c 11 and it follows that a(g"oh") E 11 for n 2 no. Since r ( g n o h n ) = a(h-"og-*), it follows similarly that r ( g n o h") + r ( h ) as n + 00. a

In the next lemma we suppose that g and h share at least one fixed point.

Lemma 1.3.5 I f r ( g ) = a ( h ) , then there are indices m;+ 00 and ni such that k ( g m ; o h"*) stays bounded as i -+ 00.

--* 00

Proof. If a ( g ) # r f h ) , then (1.15) remains valid with t = 0, and we have

f ( g m i o h"') = f(k;"'/k;'>.

(1.16)

If a ( g ) = r ( h ) then gmi o h"* is conjugate to z I+ ( k r i / k g i ) zand (1.16) holds also in this case. Since kl > 1 and k2 > 1, the assertion follows. In the following theorems, we consider groups G and GI of Mobius transformations acting in the upper half-plane U . An isomorphisms j : G + GI is induced by a Mobius transformation 1c, if j ( g ) = 1c, o g o 1c,-l for all g E G. Note that is not uniquely determined by j. In fact 1c, and $ both induce j. Hence can always be chosen such that + ( U ) = U . On the other hand, an orientation preserving 1c, inducing j may map U onto the lower half-plane. The isomorphism j is type-preserving if g and j ( g ) are of the same type for all g E G.

+

+

Theorem 1.3.6 Suppose that G is generated by a finite or countably infinite set E = ( g l , g 2 , . . .} of hyperbolic transformations. Suppose that gn and g , share no fixed points for any n # m. Let j : G + GI be a typepreserving isomorphisms. I f k ( g ) = k ( j ( g ) )for every hyperbolic g E G, then j is induced by a Mobius transformation.

1.3. MULTIPLIER PRESERVING IS 0M 0R P HISMS

21

Proof. If E contains only one element, then there is nothing to prove. Let 2 1 , 22,23, z 4 be distinct fixed points of E . Suppose that 2 1 = ~ ( h l ) , 22 = ~ ( h 2 )z3 , = u(h3) and 2 4 = u(h4) where either hi E E or hr' E E , i = 1,2,3,4. To show that the points y1 = ~ ( j ( h l ) y2 ) , = ~ ( j ( h 2 ) y3 ) , = a(j(h3)) and y4 = u ( j ( h 4 ) ) are distinct, suppose, e.g., that y2 = 93. Then by Lemma 1.3.5, there exist indices mi -, 00 and ni +. 00 such that k(j(h2)m: oj(h3)n')5 M

< 00

(1.17)

as i + 00. On the other hand, the points 2 2 = ~ ( h 2and ) 2 3 = u(h3) are distinct, If we had h2 = h3, then the transformation j ( h 2 ) = j ( h 3 ) would have only one fixed point and j could not be type-preserving. Hence h2 and h3 share no fixed points. Then by Lemma 1.3.3,

which contradicts (1.17). We show next that (21,22,z3,54)

To that end, let

gln

(yl,y2,y3,y4)-

(1.18)

= h: o hy and gan = h? o h;. By Lemma 1.3.4,

T(gin) a(gin>

+

xi,

r(j(gin))

+

yi,

a ( j ( g i n ) ) yii-2, i = 1,2. Consider the pair ( 9 1 n 7 g 2 n ) . By Lemma 1.3.3, k ( g l n ) + 00 and k ( g 2 n ) -+ CQ. Since the cross-ratio of the fixed points of g l n and g2,, is bounded away from 1 , 0 and 00, it follows by 1.15 that gln0gZn and g f n o g 2 n are hyperbolic for sufficiently large values of n. Then by Lemma 1.3.2, the pairs (gin, g2n) and ( j ( g l n ) , j ( g z n ) ) are conjugate, i.e., +

5ii-21

+

from which (1.18) follows by letting n + 00. Finally, keep the points 5 2 , 5 3 and 2 4 fixed and let $ be the orientation preserving Mobius transformation defined by $ ( z ; ) = y i , i = 2 , 3 , 4 . Let 2 1 = u ( h i ) be any point distinct from x 2 , z 3 and z4 such that either hi E E or (hi)-' E E . Then by (1.18), $ ( X I ) = u ( j ( h ; ) ) ,and we have for all g ; E E

= u(j(gi)>, $(T(gi)) = T G b i ) ) , k(gi) = k(j(gi)).

$(u(gi))

22


Hence j( g i ) = $ogio7/h-'

for all generators g; E E , and the assertion follows.

0

For a Mobius group G acting in a disk or a half-plane, let G+ denote the subgroup of the orientation preserving elements of G. For a moment, we restrict ourselves to countable groups G satisfying the following condition: For every z E C, there exits g z E G+ such that { g E G+ I g ( z ) = z} = { g :

I n = O , f l , f 2 ,...}.

(1.19)

For example, all Fuchsian groups satisfy these conditions. The last condition states that the stabilizer { g E G+ 1 g ( z ) = z } of z E C is a maximal cyclic subgroup of G+ whenever g z # id. Theorem 1.3.7 Let G and G' be countable M6bius groups acting in U. Suppose that G satisfies (1.19) and has at least four distinct hyperbolic fixed points. Let j : G + G' be a type-preserving isomorphism. If k ( g ) = k ( j ( g ) ) for every hyperbolic g E G , then j is induced by a M6bius transformation.

Proof. Let the set E = { g 1 , g 2 , . . .} contain exactly one generator of every maximal hyperbolic cyclic subgroup of G. Let Go be the group generated by E . Then, by Theorem 1.3.6, j I Go is induced by a Mobius transformation $, i.e., j ( g ) = $ o g o $-' for all g E Go. If g E G \ Go, then g is either parabolic, elliptic, a reflection or a glidereflection. For a glide-reflection g E G, g 2 is hyperbolic. Hence Since a glide-reflection is uniquely determined by its square, we have j ( g ) = 7/h 0 g 0 $-1. Suppose that g E G \ Go is not a glide-reflection. Choose a hyperbolic h E G such that g (a (h ))# r ( h ) (such an h exists by Lemma 1.3.4). Then, by Lemma 1.3.1, g o h" is hyperbolic or a glide-reflection for sufficiently large values of n. Then

j ( g ) o j( h " ) = j ( g o hn) = 1 ~ ,o g o hn o $-' = $ 0 9 0 $-l o $ o h" o$-' = $ago$-' o j (h " ) and hence j ( g ) = II, o g o $-l.

Corollary 1.3.8 Let G and GI be countable M6bius groups acting in U. Suppose that G satisfies (1.19) and has at least four distinct hyperbolic fixed points. Let j : G + G' be a type-preserving isomorphism. If j ( g ) = g for all g E G+, then G = G' and j = id.

1.4. PARAMETRTZATION PROBLEM

23

It follows from the Corollary 1.3.8 that it suffices in many cases to consider Mobius groups containing only orientation preserving elements. It is an interesting exercise to enumerate all groups having at most two hyperbolic fixed points.

1.4

Parametrization problem and classes 7 f , P and I

The motivation of considering isomorphisms of Mobius groups has its roots in the theory of Teichmuller spaces. However, in the Teichmiiller theory, only isomorphisms j : G1 -+ G2 induced by quasiconformal mappings are considered. This gives rise to the following definition: For k = 1, 2, let Gk be a Mobius group acting in a disk or half-plane Dk. Denote by Kk the boundary of Dk. An isomorphism j : G1 -+ G2 is geometric (on K1) if 0

j is type-preserving,

0

there exists a homeomorphism 11, : K1

-+

K2 inducing j on K1.

It follows that 11, maps the hyperbolic or parabolic fixed points of GI onto the respective fixed points of G z . Hence j preserves the cyclic order of the fixed points of G I on K1. In the following, we consider only principul-circle groups, i.e., Mobius groups acting in a disk or half-plane. Fix a principal-circle group Go and denote by J(G0) the set of all pairs ( j , G ) where G is a principal-group and j : Go + G is a geometric isomorphism. Parametrization problem. Find a set A c Go having the following property: If ( j l , GI) E J(G0) and ( j z , G z ) E J(Go), then j2 o j,' : G I -+ G2 is induced by a Mobius transformation if and only if

for every g E A. The set A pummetrites J(G0) and the numbers k(j(g)), g E A , are coordinates of (j,G) E J(G0). In view of Theorem 1.3.7, we restrict ourselves to principal-circle groups generated by hyperbolic elements. If, firstly, the group Go is generated by one hyperbolic transformation g , then A = { g } parametrizes J(G0). Secondly, let Go be generated by a pair ( 9 ,h ) of hyperbolic transformations sharing no fixed points. By Lemma 1.3.2, the set

parametrizes J(G0). However, the fourth coordinate k> = k ( j ( g z o h)) of ( j , G ) is in a special position: As soon as the first three coordinates


24

Figure 1.1:

ki = k(j(g)), kt = k ( j ( h ) ) and k$ = k(j(g o h ) ) are fixed, ki has only two possible alternatives. We shall prove in Theorem 1.4.3 that k> is an almost superfluous coordinate for a geometric isomorphism j . We call (9,h ) a principal-circle pair if it satisfies the following conditions: 0

g and h are hyperbolic transformations sharing no fixed points,

0

the cross-ratio t = ( T ( g ) , r( h ) ,a(h),a ( g ) ) is real.

Groups generated by principal-circle pairs (g, h ) are building blocks of all principal-circle groups. To our purposes it is necessary to limit ourselves to cases where kl = k(g), k2 = k ( h ) and k3 = k(g o h ) determine k4 = k(g2 o h ) uniquely. To that end, it suffices that t = (r(g),r(h),a(h),u(g))will be uniquely determined. This in turn will be achieved by dividing the pairs (9,h ) into three disjoint classes ‘H, P and E (cf. formula (1.15)): ( g , h ) E 7-i

(g,h)E E

*

f ( g 0 h) = t f ( W 2 )

+ (1 - t ) f ( k l / k 2 )1 2

t2 < t < t l .

Here ‘H stands for “handle”, P for “pants” and E for “elliptic”. Since ( 9 , h) is a principal-circle pair, g o h is either hyperbolic, parabolic or elliptic. Note that g o h is elliptic if and only if ( 9 , h ) E 8. Normalize the pair (g, h ) by conjugation such that a(h) = 0 a(g)

= 00

and ~ ( h=) 1, and r(g) = t .

1.4. PARAMETRIZATION PROBLEM

25

In Figure 1.1, the axes of g and h are drawn in the same z = t with the graph of

4 t ) = f ( g 0 h ) = Itf(klk2)

+ (1 - t ) f ( f l / k 2 ) l .

+ izl plane (1.20)

Then f ( g o h ) is the ordinate of the intersection point of (1.20) and the axis of g whereas the abscissa t of the intersection point gives the class of (g,h). In the class H,t = r ( g ) omits the points 0 = ~ ( hand ) 1 = r ( h ) . Therefore, f ( g o h ) omits the values f ( k l k 2 ) and f ( k l l k 2 ) = f ( k 2 / k l ) in H. Since f maps the interval [I, oa]injectively onto [2,00], we have, by Figure 1.1, the following lemma:

Lemma 1.4.1 For any k1 > 1, k2 > 1 and conjugation unique pair ( 9 , h ) E P such that k ( g ) = k1,

k ( h ) = k2,

k3

2 1 there exists an up to

k(g o h ) = k3.

(1.21)

For a pair ( g , h ) E ( k 3 - k I k 2 ) ( k 3 - k l / k 2 ) ( k 3 - k 2 / k l ) # 0. If kl > 1, k2 > 1 and k3 2 1 satisfy this condition, then there exists an up to conjugation unique pair ( g , h )E IH sutisfying (1.21). ' H j

In subsequent applications of the above lemma, we shall exclude the class & and consider only principal-circle pairs (9,h ) with non-elliptic g o h . If we then know that the class of (9,I t ) is fixed, then k l , k2 and k3 determine the conjugacy class of ( 9 ,h). Hence we need not know whether ( 9 ,h ) is actually in P or in H,it suffices to know the invariance of the class only. We show next that the invariance of the classes P and 'H is pertinent to the parametrization problem. We consider first the classes H ,P and & in detail. Considering the cross-ratios

and 1 - t = ( r ( g - ' ) , r ( h ) , u ( h ) , u ( g - 1 ) )= ( m , l , O , t ) associated with the pairs (9,h ) and (g-', h ) , respectively, we see that 0

the pairs (9,h ) , ( h ,g ) , ( g - ' , h-') and (h-', g - l ) are in the same class,

a

the pairs (g-l, k ) , (Ii,g-'), ( g , h-') and ( h - l , g ) are in the same class.

For a moment, we normalize by conjugation such that g and h map the unit disk D onto itself. Then g and h have well-defined isometric circles I ( g ) and I ( h ) . In general, I(g) is defined by the following properties (Figure 1.2):

26

CHAPTER I. MOBIUSTRANSFORMATIONS

Figure 1.2: 0

I(g) is perpendicular to az(g) and to the unit circle

0

g(I(g)) and I(g) have the same radius.

K,

It follows that g(I(g)) = I ( q - ' ) and g maps the inside of I(g) onto the outside of I(g-’). Since g and h share no fixed points, we may suppose that

a(h) = 1 and r ( h ) = -1, a ( g ) = ei’ and r(g) = -efi',

0 < 19 < R .

In Figure 1.3, three different alternatives 3.1-3.3 for the cyclic order of the fixed points of g and h are represented. Here 3.1 contains the whole classes P and & and possibly a part of H . The rest of the class 'H is contained in 3.2 and 3.3, cf. Figure 1.1. The case 3.1 has five different subcases 3.1.1-3.1.5 which are found by drawing the isometric circles of y, h, 9-l and h-l. 3.1.1. t 0 , k # 1, be the hyperbolic transformation with the real fixed points a and r and with the multiplier m a x ( k , l / k ) . If there exist real numbers y1 and yz such that y2+(y1) = y l + ( y 2 ) , then, by direct calculation, ( 1.30) If we insert a = 1, r = x < 0 , y1 < x and impossible. Hence = id and y = y’. o

+

y2

< x, we obtain k < 0 which is

Theorem 1.6.2 Suppose that the principal-circle group Go is generated by a set EO = { g l , g z , . . .} of hyperbolic transformations satisfying (1.28). Suppose that the trunsformations g, o g1 and g; o g2 are non-elliptic f o r i = 3,4,. ... Then A = {gi,gi+l 0 g',y;+z o g2 1 i = 1,2,. . .) parametrizes J(G0). Proof. Choose ( j k , G k ) E J ( G o ) , k = 1,2. Since j , : Go + GI and j , : GO + Gz preserve the cyclic order of the real fixed points of Go, the sets


42

jl(E0) and jz(E0) are conjugate to sets El and E2, respectively, for which conditions of the type (1.28) are valid. Suppose that

for i = 1 , 2 , . . . . Since the pairs (gi,gl) and (gi,gZ) are in P U 3-1 for i = 3 , 4 , . . ., we have, by Theorem 1.4.2 and Lemma 1.6.1, El = E2, and the assertion follows. Retaining the assumptions of Theorem 1.6.2, suppose that

is a finite set. Then the parametrizing set A contains the following 3s - 3 elements 0

g;, i = 1, ..., s,

a

g i o g 1 , i = 2 ,...,s ,

a g ; o g 2 , i = 3 ,..., s.

The result of Theorem 1.6.2 is best possible if Go is generated freely by Eo. Hence, if we want to reduce the number of the elements of the parametrizing set A , we have t o consider groups with defining relations. If follows from (1.28) that g1 and 92 have intersecting axes. This assumption was needed in proving that k in (1.30) is negative. If the axes of g1 and g2 do not intersect, then k(gi), k(gi o g l ) and k(g; o 92) do not determine g; uniquely. This fact will make trouble in the proof of Lemma 1.6.3. The uniformization theory constitutes the connecting link between surfaces and principal-circle groups. Surfaces are represented as quotients of a disk or a half-plane by principal-circle groups, and principal-circle groups corresponding t o a surface are isomorphic t o its fundamental group. We shall see later that Theorem 1.6.2 gives minimal parametrizing sets for principal-circle groups corresponding t o non-orientable or non-compact surfaces. It remains the rather complicated case of the compact and orientable surfaces. The fundamental group of a compact and orientable surface of genus p is generated by 2 p elements 71, . . .,~2~ with the defining relation (1.31)

43

1.6. PARAMETRIZATION B Y MULTIPLIERS

Figure 1.14: If, in the usual notation, gk is the Mobius transformation corresponding to Y k , then gk o g; corresponds to ’)’;Yk. For our purposes it is noteworthy in relation (1.31) that it gives a representation for the commutator -1 -1 Y 2 p - l Y 2 p YZp-lY2P

in terms of 71,. .. , y z P - 2 . In order to find a minimal parametrization by multipliers, it suffices t o know that principal-circle groups corresponding to compact and orientable surfaces can be characterized as follows. Consider a set E = {gl, . . .,gzP} of hyperbolic transformations fixing the upper half-plane U. Suppose that the following conditions are satisfied: The transformations g; o yl, i = 3 , . . ., 2 p - 2, and g; o g2, i = 3 , . . ., 2 p are non-elliptic. The coinmutator c p = [g2p-l,g2p] is hyperbolic and it has a given representation in the group generated by {gl, . . .,g2p-2}. 0

E is normalized by

a(g1) = 1, a ( g z ) =

00,

~ ( g 2= ) 0 and

T(g1)

=x

< 0.

For any g;, i = 3 , . . . , 2 p - 2, the cyclic order of the fixed points is given by Figure 1.14.

A set E satisfying the above conditions is said to be of compact type. A group G is of c o m p a c t type if it is generated by a set conjugate to E . The number p is called the genus of E or G. Hence we allow, for a moment, some


44

ambiguity in the definition of the genus of a group. However, if a compact and orientable surface of genus p is a represented as a quotient U / G , then p is the smallest number satisfying the above definition. In the following lemma, we give a minimal set of multipliers determining E uniquely. Lemma 1.6.3 Let E = { 91,. . .,gzP} be a set of compact type. Suppose that the classes of the pairs (gi,gl), i = 3 , . . . , 2 p - 2, and (g;,g2), i = 3 , . . ., 2 p , are fixed in P U H . Then the following 6 p - 4 numbers 0

k(g;), i = 1,. . . , 2 p ,

0

k(g; o gl), i = 2 , . . . , 2 p - 2 ,

0

k(g; o 9 2 ) , i = 3 , . . ., 2 p ,

0

4 2 p - 1 og2p)

determine E uniquely. This set is minimal. Proof. Firstly, give the following Gp - 9 multipliers: 0

k(y&

0

k ( g ; 0 g1), i = 2 , . . . , 2 p - 2 ,

0

k(g; 0 g2), i = 3 , . . ., 2 p - 2 .

i = 1,. . .,2 p - 2 ,

Then, by Lemma 1.6.1, the set (91,. . .,g2p-2} and hence also the commutator cp = [g2p-1, gzP]are determined. Assume that 0

qgzp-1)

and

%2p)

are given. Then k.(g2p-l o g z P ) has (at most) two possible values by Theorem 1.6.2. Hence, by Lemma 1.4.1, also the cross-ratio

has at most two possible values. To choose one of these we have to spend one parameter, i.e., we have to suppose that also 0

Q2p-10

Y2p)

is given. Since (g2p-l,g2p) E H , the pair conjugation. Give the multiplier

(g2p-1,g2p)

is determined up to

1.6. P A R A M E T R I Z A T I O N B Y MULTIPLIERS

45

Suppose that ( g i p - l , g & , ) is another candidate for the pair ( g ~ ~ - - l , g 2Then ~). there exists a Mobius transformation 11, fixing U such that = 11, og2p-1

g:p-1

= 11, og2p

s:,

0

$5

0 $-I.

Since ( g ~ ~ - - l , gand 2 ~ )( g i p - l , g i p ) have the same commutator c p , we have c p = 1c, 0 c p 0 +-I.

Hence 11, fixes the axis of c p . By the normalization of Figure 1.14, II,maps the interval [T, a] increasingly onto itself. Hence 11, is hyperbolic or the identity transformation. Since also (g2p,92) and (9:,,92) are conjugate, we have by formulae (1.29) and (1.30) two alternatives: Either 1c, = id or

‘(’)

=

( a k - T ) Z - a r ( k - 1) ( k - 1). - ~r a

+

with

k=

T(Y1

- “XY2 -

“1

“(Yl

- .)(Y2

4‘

-

Hence, to distinguish between these two alternatives, we have t o spend one more parameter. Suppose, therefore, that also Ng2p-1 o g 2 )

is given. To show that then only the case 11, = id remains, denote 01 = ~ ( g 2 ~ - and 1 ) 02 = a(gzP-1). Then, similarly as for g2p, we get an expression for k: k = T ( V 1 - .)(% u(v1 - T ) ( V 2

Hence

yl-T v1 -.^__--.01-T

y1-a

-

.>

-T)’

y2-av2-r y 2 - T v2-a.

But this is impossible since ( 0 1 , y l , a , ~ > ) 1 and ( y 2 , 0 2 , a , ~ < ) 1. Hence 11, = i d . We have seen that 6 p - 4 multipliers determine E uniquely. If we drop ) k(gaP-1 0 9 2 ) and keep the remaining 6 p - 6 the multipliers k(gzP-1 0 g 2 ~ and multipliers fixed, then the set (gl, . . ,,g2p-2} is fixed but the pair ( g 2 p - 1 ,g2p) has at most four alternatives. The remaining 6 p - 6 multipliers are all essential: If one of them is dropped, then there are uncountably infinite set of alternatives with the same 6 p - 7 remaining parameters.


46

Suppose that the essential 6p - 6 multipliers are given. Then fixing of the value of k(gi?p-l o g2p) reduces the number of the alternatives from four to two and finally k(g2p-10g2) makes the choice between the remaining two cases. o In the next sections, we shall investigate the geometric meaning of the inessential” parameters l ~ ( g 2 ~ -o1gzP) and k ( g ~ o~ g2). - ~

LC.

Theorem 1.6.4 Suppose that the reflection group Go is generated by a set Eo = {gl, ...,gzP} of compact type. Then the following 6 p - 4 elements gi, i = 1, ...,2p,

g i O g l , i = 2 ,..., 2 p - 2 , 0

g;og2,i=3

’ QZp-1

, ” . ) 2p,

0 g2p

constitute a niinimal parametrizing set A of J(G0). Proof. Since the pairs (gi,gl), i = 3 , . . .,2p - 2, and (g;,g2), i = 3 , . . . , 2 p , are in P tl ‘H, Theorem 1.4.2 and Lemma 1.6.3 can be applied, and the assertions follow similarly as in the proof of Theorem 1.6.2. Theorem 1.6.2 gives a parametrizing set for J(G0) containing 6p - 3 elements. Hence the relation of Go derived from (1.31) reduces the number of elements of a minimal parametrizing set only by one. In the next sections we shall show that it is possible to parametrize J(G0) by 6p - 6 numbers if two of them are not multipliers but functions of multipliers. It is open question whether all minimal parametrizing sets of J(G0) contain (at least) 6p - 4 elements.

Orthogonal decompositions and twist parameters

1.7

In this section, we derive a parametrization by 6 p - 6 parameters for compact type groups of genus p. To that end, we continue the study of the coinmutator c = [g,h] of two hyperbolic transformations. Consider two Mobius transformations @ I and @ 2 fixing e.g. the upper half-plane U ,and let L be a non-Euclidean line in U. We say that h = @2 o 9 1 is an orthogonal decomposition with respect to L if either 91 and L

are hyperbolic, their axes are perpendicular and a z ( @ l )=

47

1.7. ORTHOGONAL DECOMPOSITIONS

or 0

@I

= i d , 4 9 is hyperbolic and az(iP2) and L are perpendicular.

Before considering the existence and uniqueness of orthogonal decompositions we prove the following lemma:

Lemma 1.7.1 Let h ( z ) = ( a z b ) / ( c z d ) , ad - bc = 1, be a hyperbolic transformation fixing U . Then abcd # 0 i f and only i f h(O) # 0,w and h(m) # 0,w. I n this case, we may choose a > 0 , b > 0 , c > 0 and d > 0 if and only i f r ( h ) < 0 < a(Jz).

+

+

Proof. Since h(0) = b / d and h(w) = a / c , the first assertion holds. Suppose, that abcd # 0. The fixed points X I = a ( h ) and 2 2 = r ( h ) satisfy the equation c x 2 - (U - d ) z - b = 0. ( 1.32) Since 2 1 x 2 = -b/c, we have ~ 1 x < 2 0 if and only if bc > 0. Suppose that r ( h ) < 0 < a ( h ) . Then h ( 0 ) = b / d > 0 and bc > 0. Choose b > 0. Then c > 0 and d > 0. From ad - bc = 1 it then follows that a > 0. Suppose conversely that a > 0, b > 0 , c > 0 and d > 0. Then a ( h ) r ( h )< 0. Since h(0) = b/d > 0 , we have r ( h ) < 0 < a ( h ) . Theorem 1.7.2 Let L be a non-Euclidean line in U and let h be a hyperbolic transformation fixing U . If ax( h ) and L intersect, then there exists a unique orthogonal decomposition h = o @ 1 with respect to L .

Proof. We may suppose without loss of generality that L is the positive imaginary axis and a ( h ) > 0. Since a z ( h ) and L intersect, we have ~ ( h 0, b > 0 , c > 0 and d > 0. Let @ 2 0 @ 1 be an orthogonal decomposition with respect t o L. Let x > 0 and -x be the fixed points of @ 2 . Then by formulae (1.4)-(1.6), @l(Z)

= k'z,

@ 2 ( 4

=

We have * 2 ( @ 1 ( z ) )=

k'

> 0,

( k +- 1)zz + x 2 ( k - 1 ) , k>O. ( k - 1). ( k 1).

+ +

+

+

( k 1)k'xz + x 2 ( k - 1) -a z b cz + d ( k - 1)k'z + ( k + 1)"

48


if and only if a = c

=

(k

+ 1)k‘

x(k - 1) 2 m ’ d=- k + l 2 m ’

b=

2422 ’ ( k - 1)k’ 2 2 m

’

i.e., if and only if

Theorem 1.7.3 Let ( @ I , @ z ) be a principal-circle pair with intersecting axes, i e . , ( @ 1 , + 2 ) E H R . Then h = @*

0

(1.33)

@1

is an orthogonal decomposition i f and only if

Proof. B y Figure 1 .l, (1.33) is orthogonal if and only if

t = (r(@1),T(@z), Since

(@I, @2)

.(@2),

1 a ( % ) ) = 2.

E H R , this occurs if and only if

f(W =

1 s[f ( k ( @ l P ( @ 2 )+) M @ l ) / k ( @ 2 ) ) 1 .

The assertion follows by simple calculation. Since f ( i d ) = 2 , the formula (1.34) holds for all orthogonal decompositions h = @ 2 o $1. Let L be an oriented non-Euclidean line in U ,i.e., denote one of the ideal end points of L by r ( L ) and the other by a ( L ) and let the positive direction on L be defined by r ( L ) + a ( L ) . The axis of a hyperbolic transformation h has a natural orientation given by ~ ( hand ) a ( h ) . Then h translates the points of a x ( h ) towards a ( h ) , i.e., to the positive direction. Suppose that L and a x ( h ) intersect. Let a ( L ,h ) be the angle between L and a z ( h ) determined by the positive orientations of L and a z ( h ) . In Figure 1.15, we have the same normalization as in the proof of Theorem 1.7.2.

1.7. ORTHOGONAL DECOMPOSITIONS

-X

49

0

Figure 1.15:

Theorem 1.7.4 Let h = @ 2 o $1 be u n orthogonal decomposition with respect to L . Then k ( h ) k ( Q 2 ) . Moreover

>

--

a ( L , h ) = n / 2 I$1 = id a k ( h ) = k ( @ 2 ) , 0

cw(L,h) < n / 2

0

a( L , h ) > 7r/2

= a@), u( @ I ) =

r( L ) .

2 2, we have by (1.34) f ( h ) 1 f ( Q 2 ) . Then k ( h ) 2 the function k H f(k) is strictly increasing for I; 1. The equality holds if and only if f(@,) = 2, i.e., $1 = id. In this case, h = @ 2 and a ( L , h ) = n/2. Normalize by a ( L ) = 00, r ( L ) = 0, X I = a ( h ) > 0 and 2 2 = ~ ( h 0, 6 > 0, c > 0 and d > 0 (cf. Leiiinia 1.7.1). Then by (1.32) X I + 2 2 = ( a - d ) / c . On the other hand, by the proof of Theorem 1.7.2, Proof. Since

f(@l)

>

k ( @ 2 ) because

+

+

U

@&) = -z. d

Hence


50

Consider a pair ( 9 , h ) E

XR whose

commutator

c = [ g , h ]= hog-’ o h - 1 o g

is hyperbolic. Let h = t o L = a z ( g ) . Then 9:'

o 91 be the orthogonal decomposition with respect and g f l commute, and it follows that c

= 19, hl = [g, 9 2 1 .

Hence only the component 9 2 of h whose axis is perpendicular to a z ( g ) contributes to the coinmutator of g and h. By Theorem 1.7.4,

k(h) L

W2).

Moreover, by Theorem 1.5.3 and its Corollary 1.5.4,

Hence k(c) and k ( g ) determine k(92)uniquely. Let k1 = k ( g ) and k d = k(c). Define ko > 1 by the formula

On the other hand, define

0 1

if if if

-1

=id, ~ ( 9 1=) u ( g ) , u(@pl) = r ( g ) . 91

Hence ko 5 k ( h ) and Lo = k ( h ) if and only if t ( g , h ) = 0 . Consider the number k(h) In=

c(gh)

[XI

Let a ( g , h ) be the angle determined by the positive directions of the axes of g and h. Then, by Theorem 1.7.4, a ( g , h )= 7r/2

J

t(g,h) = 0

0

a ( g ,h ) < 7r/2 I t(9, h ) = 1

0

a ( g , h )> 7r/2

t ( g , h )= -1

m = 1, m

> 1,

e m < 1.

I. 7. ORTHOGONAL DECOMPOSITIONS

51

Theorem 1.7.5 Let ( 9 ,h ) E H R be a pair with hyperbolic c = [ g , h ] . If the multipliers kl = k ( g ) and k4 = k ( c ) and the number

are known, then the pair ( 9 , h ) is determined up to conjugation. Proof. Since k1 and k4 determine ko, the number m determines then both kz = k ( h ) and c ( g , h ) . By Corollary 1.5.4 of Theorem 1.5.3, the multipliers k1, k2 and k4 determine the acute angle between the axes of g and h. On the other hand, ( ( g , h ) tells which one of the adjacent angles formed by the axes of g and h is acute. Hence t = ( r ( g ) ,r ( h ) ,a ( h ) ,a ( g ) ) is uniquely determined and the assertion follows.

Consider a set E = { g l , . . . ,g z P } of compact type. The first 6 p - 9 multipliers given in the proof of Lemma 1.6.3 determine the commutator cp = [gzp-l, g2pl.

Then by Theorem 1.7.5, the numbers k ’ ( g l p - l ) and m ( g 2 p - l , gZp) determine the pair ( g ~ ~ - l , g up 2 ~ t)o conjugation. The only freedom left is conjugation by hyperbolic transformations fixing the axis of c p . Geometrically, the “handle” determined by ( g z p - l , g z p ) is then “rotated around” the commutator c p . This rotation can be determined by one real parameter, so called twist parameter, as follows. Let L be a non-Euclidean line in U and let C ( L ) denote the group generated by the hyperbolic transformations : U + U with a x ( + ) = L . A family F of pairs ( g , h ) of hyperbolic transformations fixing U is L-invariant if the following conditions are satisfied:

+

0

If (g,h ) E 3- and $ E C ( L ) then $ ( g , h)$-’ = ( + o g o $ - l , +oho+-’)

3.

E

If ( g i , h i ) E 3- and ( g 2 , h l ) E F ,then there exists a uniquely determined € C ( L ) for which (p2, h 2 ) = +(gl, h1)+-l.

+

Fix ( g o , ho) E 3 and suppose that 3 is L-invariant. The mapping (Y,

+

if

( 9 ,h ) = +(go, ho)+-’

(1.35)

is a bijection F -+ C ( L ) . Choose an orientation r ( L ) + a ( L ) on L . For (9,h ) = $(go, / L O ) $ - ’ , call the number (1.36)

52


the twist parameter of (g,h) E F with (g0,ho) as the origin. Hence, by (1.35), every (go,ho) E F defines a bijection (go,ho)) : 3 + R.

~ ( 0 ,

We can now choose the right element from an L-invariant family 3 as soon as we fix an element (go, ho) E F and give the twist parameter of the required element. Let c : U 4 U be a given hyperbolic transformation with r ( c ) < U ( C ) < 0. Consider all pairs (g,h) E ' ? ffor ~ which r(c> < r(g) < r ( h ) < a(g)

E The last assertion follows now by Theorem 1.7.6 and its Corollary 1.7.7. , P o

0

Corollary 1.7.9 For the fumily 3 in Theorem 1.7.8, choose (go, ho) minimizing d U ( 4 l t > , ax(g2)) as the origin of the twist parameter (1.36). If (hO,g2) E P , then the multiplier k(h o 92) is a convex function of the twist parameter 2

= 4 ( 9 ,I t ) , (90, ho))

and as a function of x , k ( h o 9 2 ) attains its minimum in R at x = 0. Proof. T h e convexity of the function x H k(hog2) follows from the convexity of the function (1.39). By Theorem 1.7.8, (go, ho) minimizes also k ( h o g 2 ) . 0

Suppose that (92, c ) E Int P . Then by Theorem 1.7.6 (ho,92) E P . For ( g , h ) E 3 define E(l1)

=

{

1 -1

if 4 - A 14, (go, 1x0)) L 0, if w((st, (go, ho)) < 0.

14,

Then we have, by the above corollary, the following result:


56

Theorem 1.7.10 The function

is a bijection 3 --f R.

In order to apply the preceding considerations to the parametrization of compact type groups, fix the genus p and consider sets E = (91,.. .,g z P } of compact type for which ( 9 2 , c p ) E Int P, Give first the following 6 p - 8 multipliers: 0

k ( g ; ) , i = 1 , . . .,2 p - 1,

0

k(g; 0 gl), i = 2 , . . . 2 p - 2,

0

k(ga o 9 2 ) , i = 3 , . .., 2 p - 2.

Then the set (91, . . . ,g2p-2} and hence also cp = [g2p-1, gzP]are determined. Consider all sets E of genus 1) with these 6 p - 8 parameters fixed. Let ~2~ = @ 2 o @ I be the orthogonal decomposition with respect to a z ( g z p - l ) . Then the numbers k ( c p ) and k(gZp-1) determine k(i92) and we have Ic(gZp) 2 k ( @ 2 ) . Hence, if we consider all possible pairs ( g , h ) = (g2p-1,g2p), there exists a minimum Lo = min k ( g z p ) . E

Fix the number

Then the az(c,)-invariant family F containing all pairs ( 9 , h ) = ( g 2 p - l , g 2 p ) with the given values of k ( g 2 p - ~ and ) g 2 p ) is determined. We may apply Theorem 1.7.6 to 3. Let k , = niin k(gzP o g 2 ) . E

Then, finally, the number

I determines the pair ( Y ~ ~ -92,). Fix a compact type set EO = { h l , . . .,hzp} of genus p . Denote 7p = [h2p-1,hZp]and suppose that (hz,yp) E Int P . Let Go be the principal-circle group generated by Eo. Define in J(G0) an equivalence relation by setting N

1.7. ORTHOGONAL DECOMPOSlTIONS

57

( j l ,G I ) ( j 2 , Gz) if j 2 o j,' : GI -+ G2 is induced by a Mobius transformation. Then every equivalence class [ j ,G] contains a unique representative (j,G) such that the set j ( & ) = { g l , . . .,g2p} with g; = j ( h ; ) ,i = 1 , . . .,2 p , is of compact type. Moreover ( g z , c p ) E Int P by Theorem 1.4.2. We have proved the following result: N

Theorem 1.7.11 T h e mapping J(Go)/ a

-

-+

R6P-6 defined by

z; = log k ( g i ) , i = 1 , . . . , 2 p - 1 ,

a ~ 2 ~ - 1 += ;

log k(g; o g l ) , i = 2 , . . ., 2 p - 2,

a ~ ~ ~ - -= 4 + log i

k ( g i o g2), i = 3 , . . ., 2 p - 2,

is injective. o We shall see later that the coordinates xi, i = 1,. . . ,Gp - 6 , have rather simple geometrical interpretations on a compact and orientable surface of genus p .


Chapter 2

Quasiconformal mappings 2.1

Introduction to Chapter 2

It is our ultimate goal to give parameters which define Riemann surfaces up to conformal isomorphisms. This is the famous moduli problem. The first solution was based on the theory of quasiconformal mappings. They give a way to deform one conformal type of Rieinann surfaces t o another conformal type. In this Chapter we will review the necessary part of the theory of quasiconformal mappings. Everything here belongs t o the classical foundations of this theory. Consequently, we will, in this Chapter, give only a quick review of quasiconformal mappings omitting all the proofs. We try to keep the presentations as readable as possible giving exact references for all the omitted proofs. Our main references for quasiconformal mappings are the monograph of Lars Ahlfors [6] and that of Olli Lelito and I 0). A homeomorphism f : A + f ( A ) is called dianalytic if either f itself or the complex conjugate o f f is holomorphic in each component of the set A . Let z : U + z ( U ) and w : V -+ w(V) be two local variables such that U n V # 8. Then we may form the mapping z o w-l : w ( V n U ) + z ( V n U ) . This mapping is called the coordinate transition function.

Definition 3.2.1 A collection U = {(Ui,zi)liE I } of coordinate charts of C is an atlas of the surface C i f C = UiCrU;. A n atlas U is: orientable if each coordinate transition junction zi o zJT1is an orientation preserving homeoinorphism. dianalytic if all the coordinate transition functions are dianalytic homeom o rphisms. complex analytic i f all the coordinate transition functions are holomorphic homeomorpliisms. Two complex analytic atlases U and V are called equivalent if U U V is a complex atlas as well. An equivalence relation is introduced in the same manner for the other types of atlases. An equivalence class of complex atlases is called a complez structure of the surface C. Dianalytic, and orientable structures are defined in the same way as equivalence classes of the respective structures. A surface C which has an orientable structure is called orientable. An orientable structure of an orientable surface determines an orientation.

Definition 3.2.2 A Riemann surface is a topological surface C together with a complex structure X . A Klcin surface is a topological surface, possibly with boundary, C together with a dianalytic structure Y. Observe that Riemann surfaces, as defined above, do not have boundary points and that each Riemann surface automatically is a Klein surface as well. A Klein surface ( C , X ) that cannot be made into a Riemann surface

3.3. ELEMENTARY SURFACES

71

is called non-classical while orientable Klein surfaces with empty boundary are called classical. A classical Klein surface always carries two Riemann surface structures which are complex conjugates, or mirror images, of each other. It is well known that every surface with boundary can be made into a Klein surface, i.e., every such surface carries dianalytic structures. It is also well known that every orientable surface can be made into a Riemann surface. These results follow from the topological fact that every surface with or without boundary can be represented as a branched covering of the Riemann sphere or the unit disk (for a proof in the case of compact surfaces see e.g. [8, Theorem 1.7.2, page 491). We will not prove this result here because we do not use it anywhere. The problem that we are concerned about is t o parametrize the set of all analytic or dianalytic structures of a given surface in some reasonable way. This is also the famous moduli problem.

3.3

Elementary surfaces

Simpliest compact topological surface is the sphere S2 in R3

s2= ((2,y, z ) I x 2 t y2 t t2= 1).

Identifying the antipodal points (z,y, z ) and (-z, -9, -t) on the sphere S 2 one obtains the real projective plane P'(R). This surface is not anymore orientable in the sense that it has only one side. It is not possible to embedd the real projective plane in R3. The torus T is the quotient surface

T = C / ( z I-+ z

+ 1, z

H

z

+ i).

Here ( z H z+ 1, t H z+ i) is the group generated by the elements z and J H z t i. Let R be the strip

H

z+ 1

The torus can be obtained from a rectangle identifying its opposite sides. If we identify only one pair of opposite sides we get an annulus

A = R/(zH z

-+ 1).

The identification of a pair of opposite sides can also be done changing the orientation. T h a t is achieved by the mapping z I+ Re z 1 (1 - Im z)i. The quotient surface

+ +

M = R / ( z H Re z

+ 1 t (1 - Im z ) i )

72

CHAPTER 3. GEOMETRY OF RIEMANN SURJ-ACES

Figure 3.1: The sphere, the torus and the real projective plane. The figure shows the real projective plane cut open along an arc in such a way that the cutting yields a half-sphere. The straight lines indicate how antipodal boundary points have to be identified in order to get a 'real' real projective plane. is the well known MGbius strip. It is the only one-sided, i.e., a nonorientable, surface that can be embedded in an euclidean 3-space. Figure 3.2 shows the Mobius strip in the upper left hand corner. The Klein bottle is another famous non-orientable surface. The usual way to picture the Klein bottle is to consider first an ordinary bottle from which a small open disk is deleted from the bottom. This is actually an annulus. To get the Klein bottle, identify the two boundary components of the bottle (i.e. the annulus) we started with in such a way that the orientation gets reversed in the process. In this way one gets a one-sided bottle. Another way to picture the Klein bottle is shown in figures 3.2, 3.3 and 3.4. All these illustrations are due to Ari Lehtonen. In these figures the Klein bottle is formed by taking two copies of the Mobius strip and identifying the boundary points. Figure 3.2 illustrates this. The resulting surface is some kind of a twisted product of a figure-8 curve and the unit circle. Figure 3.3 illustrates this.

3.4

Topological classification of surfaces

Section 3.3 gives a rather concrete picture about some elementary nonorientable surfaces. We can make this a little bit more precise by considering curves on surfaces. A curve (Y on C is the image of the closed unit interval I = [0,13 under a continuous mapping (Y : I + C. We often use the same notation (Y for a mapping I + C and for its image in C. The points ( ~ ( 0 ) and ( ~ ( 1are ) the end-points of a. Observe that the orientation of the unit

3.4. TOPOLOGY O F SURFACES

73

Figure 3.2: Simpliest non-orient able surface with boundary components is the famous Mobius strip. It can be embedded in a 3-space. Gluing two Mobius strips together along the boundary curves yields the Klein bottle, which cannot be embedded into a 3-space. This illustration shows first the Mobius strip which is then gradually enlarged t o a projection of the Klein bottle in a 3-space. The projection necessarily cuts itself. We thank Ari Lehtonen for this picture.

74

CHAPTER 3. GEOMETRY OF RIEMANN SURFACES

Figure 3.3: The Klein bottle can be viewed as a twisted product of afigure-8 curve and the unit circle.

3.4. TOPOLOGY OF SURFACES

Figure 3.4: The Klein Bottle.

75

76

C H A P T E R 3. G E O M E T R Y O F RIEMANN SURFACES

Figure 3.5: A surface of genus g is a sphere with g handles.

interval [O,1] induces an orientation for each curve a : [ O , l ] + C. A curve a is closed if its end-points agree. A closed curve a is called simple if the mapping a is injective when restricted to the half-open interval [0,1[. A simple closed curve cr is two-sided if there exists an open and connected set U such that a c U and U \ a has two components. A curve a is one-sided if it is not two sided. Another characterization for orientable surfaces can be obtained by means of simple closed curves: A surface C is orientable if and only if every simple closed curve in (the interior of) C is two-sided. We will not prove this result here and we do not need it in our applications. The Euler cliaructeristic x(C) of a compact surface C is defined (for more details see e.g. (91, p. 113 and p. 1721 or [3]) in terms of finite triangulations of C. If n j is the number of j-faces of a finite triangulation of C, j = 0,1,2, then x ( C ) = no - n1+ n2.

For an orientable compact surfaces C, the Euler characteristic is an even number and the number g = g ( C ) = 1 - x ( C ) / 2 is called the genus of the orient able surface C. The genus g of an orientable surface C is always non-negative. A compact surface of genus 0 is homeomorphic to the Riemann sphere C. A compact orientable surface of genus g , g > 0, can be thought of as a sphere with g handles.

3.4. TOPOLOGY OF SURFACES

77

Let now C be a surface, orientable or not. Let p E C. We proceed and define next the orientable covering C" of the surface C as follows. An atlas U = { ( U i , zi)li E I} of C is maximal if the following condition is satisfied: Let V be an atlas of C such that U U V is also an atlas of C. Then U

c V.

Assume now that U = {(Ui, z;)li E I} is a maximal atlas of C . Form first the disjoint union

Let ( U i , z i ) and ( U j , z j ) be twocharts such that U i n U j # 8. Let p E U i n U j . Being a disjoint union the set S has two points which both correspond to the point p E C, namely the point p E Ui and the same point p E U j . To make a distinction between these points, call the latter one p'. Next we identify the points p and p' if the corresponding coordinate transition function zi o zlT' is orientable. This gives us the set C". We still have to define a topology for the set C". That is done via the natural projection K : C" + C. A topology for C" is defined requiring a : C" -+ C be locally homeomorphic. Then C" is clearly an orientable surface. It is connected if and only if C is non-orientable. For an orientable C, Co has two components which are both homeomorphic t o C. If dC # 0, then C" has two points lying over each boundary point of C. Identifying these two points gives us a surface Cc which is called the complex double of the surface C. The covering R : C" + C induces a mapping a : C" -+ C which is a ramified double covering of C. It is a local homeomorphism a t all points p E C" for which ~ ( p #) dC. At points lying over the boundary of C, the projection is a folding similar to the mapping z i y H 3: ilyl at the real axis. For more details about this mapping see [8, 1.61. In this way we form the complex double C" of a surface with boundary C. If C is orientable and dC = 8, then C" has two components which are both homeomorphic to C. For all other surfaces C, C" is a connected orientable surface without boundary. Observe that the covering group of the branched covering 7r : C" + C is generated by an orientation reversing involution (T : C" + C". (An involution is a mapping whose square is the identity.) Also C = C C / ( a ) ,where (a)is the group generated by (T.Here CC/(a) is the quotient surface obtained by identifying p with a ( p ) for all p E C". Above we defined the genus of a classical compact surface. The genus, or, more precisely, the arithmetic genus of a non-classical compact surface C is defined as the genus of the complex double Cc of C.

+

+

C H A P T E R 3. G E O M E T R Y OF RIEMANN SURFACES

78

When speaking of the genus of non-classical surfaces we always mean this arithmetic genus. Observe that in some other text books the genus of a non-classical surface has another meaning. Since the Euler characteristic of a circle vanishes, we have x ( C " ) = 2 x(C) for all non-classical compact surfaces C. Therefore we conclude that for nonclassical compact surfaces x( C) = 1 - g . Recall that for classical surfaces C, x( C) = 2 - 2g. Compact topological surfaces are classified topologically by the following parameters: 0

the genus g = g ( C ) of C,

0

the number n = n ( C ) of components of a C ,

0

the index of orientability k = k ( C ) which is defined setting k = 0 for orientable surfaces C and k = 1 for non-orientable surfaces C.

It is clear that the above parameters g , n and k are topological invariants in the sense that for surfaces, that are homeomorphic to each other, the genus, the number of boundary components and the index of orientability agree. These parameters are also related to each other as described in the following theorem [96]. Theorem 3.4.1 The index of orientability k and the number n of boundary components of a compact genus g surface C with boundary components

satisfy: I f k = 0 then n = g + 1 (mod 2) and n 0

> 0.

If k = 1 then 0 5 n 5 g .

These are the only restrictions und all possible configurations of g , n and

k satisfying these conditions appear as invariants of some compact genus g surface with boundary components. Corollary 3.4.2 There are

39+4

l,-l

topologically diflerent non-classical compact genus g surfaces with boundary components. These are classical results of Weichhold. They can also be found in the works of Klein ([45],[47], [49]). We will not prove this result here. In Section 3.7 we will, however, construct a partial proof for Theorem 3.4.1. In that

3.4. TOPOLOGY O F SURFACES

79

section we show a way to construct all the surfaces of Theorem 3.4.1. What remains not shown in this monograph is that there are no other surfaces. That can be found in the works referred above. Observe that classifying non-classical topological compact genus g surfaces is the same thing as classifying all orientation reversing involutions u of a classical compact genus g surface. This follows from the above constructions. The complex double Cc of a non-classical genus g surface C is a branched covering of C and the cover group is generated by a single orientation reversing involution CT : Cc + C". Then C = C'/((T). Conversely, given a classical compact surface C' of genus g and an orientation reversing involution u : C' + C', then C'/(o) is a compact nonclassical surface of genus g .

Definition 3.4.1 A symmetric surface ( C , a ) is a classical surface C together with an Orientation reversing involution u : C -+ C . The involution u is the symmetry of the symmetric surface C. For a symmetry u : C + C define T L ( ( T ) as the number of the components of the fixed-point set C, of u . Define also k ( a ) setting k ( n ) = 2 - number of components of C \ C,.

It is immediate that the invariants n(u)and k(cr) are simply the corresponding invariants n and k of the quotient surface C / ( a ) . We say that two symmetries (T and T of a topological surface C are conjugate to each other if there exists a homeomorphism f : C + C such that u o f = f o T . This is equivalent to the condition that the surfaces C / ( u ) and C / ( T ) are homeomorphic to each other. Corollary 3.4.2 implies then that the number of different conjugacy classes of symmetries of a classical genus g surface is given by formula (3.1). For a later application it is useful t o be able to have a concrete understanding of the topology of a compact surface. To that end we will now describe a n explicit way of building a compact surface starting with certain elementary surfaces. We use the three elementary surfaces, the sphere S 2 , the torus T and the real projective plane P'(R), to construct concrete models of more complicated surfaces. Let p and k be non-negative integers, and let S;+k denote the complement of the union of n+ k open disks on S2whose closures are disjoint. Let TI denote the complement of an open disk on T . Likewise P1(R)l is the complement of an open disk on P'(R). The Euler characteristics are:

80


Let A and B be two compact subsurfaces with boundary of a topological surface. The Euler characteristics satisfy

Build a surface E; by taking s i + k and gluing to it p copies of the surface TI to it along the boundary components and k copies of the surface P1(R)l also along the boundary components. In this way one obtains a compact surface CP, without boundary components. Using (3.2) and the fact that the Euler characteristic of a circle vanishes we compute that

x ( q ) = 2 - p - k t px(T1) = 2 - 2 p - k. In this way one can build all compact surfaces without boundary components starting from the sphere, the torus and the real projective plane. We say that the surface C! is the direct sum of the sphere, p torii and k real projective planes. These real projective planes are also called cross caps of the surface CP,. Any compact surface with boundary components can then be obtained from a surface of type CP, by deleting a suitable number of open disks with disjoint closures. Let CP,', be a compact surface with p handles (i.e. torii), k cross caps and n boundary components. Using the above computation we easily get the formula x(CP,'") = 2 - 2 p - k - n for the Euler characteristic of the surface

3.5

Xi',.

Discrete groups of Mobius transformations

A topologicul group G is a group G together with a topology for which the inverse G -, G, g H g-' and the group operation G x G -+ G are continuous mappings. Let G now be a group of Mobius-transformations. Elements of G are of the form

3.5. DISCRETE GROUPS

+

81

+

+

+ ... Figure 3.6: Any compact topological surface can be built combining a finite number of simple structures. This figure shows a few first steps of such a construction. The holes shown in the figure may be boundary components of the resulting surface or may be used to add cross caps or may be used t o add more structure. We thank Ari Lehtonen for this illustration.


82

We can define a topology on G by saying that a sequence (gk, k = 1 , 2 , . . .) converges to a,z b, gm(z) = ccoz t d , if and only if we may choose representations of the type (3.3) for the elements gk in such a way that ak --* a m , b k + b,, ck -+ c , and dk + d , as k + 00. This is the usual definition. Together with this topology G is a topological group. Recall that a topological space is called discrete if all of its subsets are open. Likewise we say that a topological group G is discrete if all of its subsets are open. Observe especially that for a discrete topological group all subsets consisting of one point only are open. In the applications of the methods presented in Chapter 1, discrete groups of MSbius transformations play an important role. A related concept to discreteness is discontinuity. We say that a group G whose elements are Mobius transformations acts discontinuously at a point x E C if the following holds:

+

w

The stabilizer of G at x, G, = { g E G I g(2) = is finite.

w

XI

There is an open set U c C, x E U , such that g ( U ) = U for all g E G, and g ( U ) n U = 0 for all other elements of G.

The set of points where G acts discontinuously is called the set of discontinuity, or regular set, and is usually denoted by R = R(G). It follows from the definition that 52 is an open G-invariant subset of the extended complex plane. A group of Mobius transforinations that acts discontinuously in some domain D is sometimes said t o be properly discontinuous. Definition 3.5.1 A group G of Mobius transformations is a Kleinian group ifR(G) # 0. L e m m a 3.5.1 A li'leinian group is either finite or countable. Proof, Choose a point Then

tE

R(G) such that the stabilizer of z , G,, is trivial.

G(z.1 = {dz) 19 E GI is a discrete set. Therefore G(z) is either finite or countable. Observe finally, that the cardinality of the set G ( z ) is the same as the cardinality of the group G.

3.5. DISCRETE GROUPS

a3

Definition 3.5.2 A Kleinian group G is a Fuchsian group if the following holds: there is a disk (or a half-plane) D of the extended complex plane C such that each element of the group G maps D onto itself and D c Q(G). The following result is immediate by definitions:

Lemma 3.5.2 Every Kleinian group is discrete. The converse is not true. The Picard group

P=(zH-

az + b I ad cz d

+

-

bc = 1 and a , b, c, d E Z [ i ] }

is clearly discrete but not discontinuous because it, can be shown that for any z E C, the set { g ( z ) I g E P } is dense in C.

Theorem 3.5.3 Let G be a group of Mobius transformations mapping the unit disk D onto itself. Then G is discontinuous if and only i f it is discrete. Proof. As we have already observed, a discontinuous group is clearly also discrete. It suffices, therefore, to show that a discrete group of Mobius transformations mapping the unit disk onto itself is also discontinuous. To prove this assume that the group G is not discontinuous at some point zo E D . This means that we can find infinite sequences 2 1 , z2,. . . E D and g 1 , g 2 , . . . E G such that z, -+ t o as n --+ 00 and gn(zn) = zo for each n. Set now and

C, = A,+1

o

g i l l o gn o A;’,

n = 1 , 2 , . . ..

Since C,(O) = 0, we conclude by Schwarz’s lemma that

C n ( t )= A ~ z ,

IXnI = 1.

Thus, by passing to a subsequence if necessary, we may assume that A, A0 as n --+ 00. This means that the sequence C, converges to Co as n 4 00. The points z, are assumed to be distinct. Therefore also the elements g , of the group G are distinct. The mapping h, = g;:l o gn maps z, onto z,+1 for each n. Assume that infinitely many of the mappings h, agree. Then we may as well suppose that h, = h for all n. We conclude that --f

20

= lim z, = lini h(z,) n+m

n+m

(3.4)

is a fixed point of the mapping h, which is, therefore, an elliptic element of the group G. But since h is elliptic, (3.4) can happen if and only if z, = zo


84

for large enough values of n . But this is not possible, since we assumed that all the points z, are distinct. This implies that the mappings h, = g;il ogn are distinct elements of the group G. On the other hand, the above considerations imply that

h, = A,+I

o

C,

o

A;'

-+

A;'

o Coo A0 as n + 00.

This is not possible since the group G is discrete. Observe that in the above proof the contradiction was derived from the assumption that there is one point 20 E D at which the group G does not act discontinuously. Above proof actually implies the following stronger statement: Theorem 3.5.4 Assume that the Mobius group G leaves the unit disk invariant. The following conditions are equivalent:

1. D

c R(G).

2. D fl R(G)

# 0.

3. G is discrete.

In this monograph we are concerned with Mobius groups that act either in the unit disk or in the upper half-plane. By Theorem 3.5.3 discontinuity and discreteness are equivalent properties for such groups. Let G be a discrete Mobius group that acts in the upper half-plane U. Assume that the stabilizer of z E U in G is trivial. Form the set

D,(G) = (2’ E U I d ( z ' , z ) < d ( g ( z ' ) , z ) V g E G , g

#

l}.

It is clearly an open set and it has the following properties: 1. No two points of D,(G) are equivalent under the action of the group

2. For every point w E U there exists an gw E G such that gw(w) E

LUG).

3. The relative boundary of D,(G) in U consists of piecewise analytic arcs.

4. For every arc a c dD,(G) t,here is an arc a' c aD,(G) and an element g E G such that g ( a ) = a'.

3.6. UNIFORMIZATION

85

We will not show here that D,(G) satisfies these properties. A detailed proof can be found in the monograph of Alan F. Beardon [lo, §9.4.,pp. 226 - 2341.

Definition 3.5.3 By a fundamental domain a Fuchsian group G acting in the upper half-plane U we mean an open subset D(G) of U satisfying the above conditions 1 - 4. The above defined fundamental domain D,(G), for a point I E U that is not a fixed-point of a non-identity element of G, is called the Dirichlet or the Poincar6 polygon for G.

3.6

Uniformization

For any surface C we may form the universal covering surface 2 which is simply connected and admits a projection A : 2 -+ C that is a local homeomorphism. Furthermore each point p E C has a neighborhood W, such the restriction of the projection K to each component of .-'(Up) is a homeomorphism between the component and U p . Homeomorphic self-mappings g of % satisfying A o g = A form the cover group G of the universal covering A : 9 + C. Since A is a local homeomorphism, the group G acts discontinuously on 2. The action is also free in the sense that no non-identity element of G has fixed-points in 2. The cover group G of the universal covering of C has the property that if ~ ( p=) n ( q ) , for p , q E 9, then there exists an g E G such that g ( p ) = q . It follows that C = g/G. This is, of course, quite standard. More details concerning the universal cover can be found, for instance, in [3]. Let X be a dianalytic structure of the surface C. Then, by requiring the mapping A be locally analytic, we may lift the dianalytic structure of C t o a dianalytic structure 2 to 2. Next observe that, since 2 is simply connected, it is orientable. Therefore the dianalytic structure X of % is induced by a some complex structure Y . Hence we may suppose that 2 is a complex structure such that A : (2,k)+ (C, X ) is dianalytic. There are actually two possible complex structures d satisfying this condition. They are complex conjugates of each other. We conclude that any Klein surface ( C , X ) has a Riemann surface as its universal covering surface. From the equation A o g = A and from the fact that x : (2,z)-+ (C, X ) is dianalytic and a local homeomorphism, it follows that each element g of the cover group G is a dianalytic self mapping of ( 2 , d ) . The following result is the famous Rieinann mapping theorem:


86

Theorem 3.6.1 A simply connected Riemann surface without boundary is either the extended complex plane, the finite complex plane or the upper half plane U. We will not prove this result here. A proof can be found, for instance, in the monograph of Farkas and Kra [29, Theorem IV.4.4, p. 1821. By this theorem we may suppose that, for any Klein surface ( C , X ) , the interior of the universal covering is one of the standard Riemann surfaces of Theorem 3.6.1. Then, by the preceding observation, we conclude that elements of a cover group G corresponding t o a Klein surface ( C , X ) are all Mobius transformations. If the Klein surface ( C , X ) is not orientable, then the group G necessarily contains orientation reversing Mobius transformations. Analyzing all possible groups that act properly discontinuously on the Riemann sphere or on the finite complex plane we conclude that:

(E,z)

0

(%,%) is the Riemann sphere if and only if ( C , X ) is either the Riemann sphere itself or the real projective plane. In this case the Euler characteristic of C is positive.

0

(c,%)

is the finite complex plane if and only if ( C , X ) is one of the following surfaces: -

finite complex plane,

-

torus,

- infinite cylinder, -

Klein bottle.

In this case the Euler characteristic of C vanishes. If the Euler characteristic of C is negative, then the universal covering ( C , X ) is always the upper half-plane together with certain intervals on the real axis. These intervals correspond t o the boundary of the Klein surface (C, X). Then the upper half-plane itself is the universal covering of the interior of the Klein surface (C, X ) . It is sometimes technically easier to consider the interior of a surface instead of the whole surface with boundary. By abuse of language, we may later speak of the upper halfplane as the universal covering of a Klein surface which may have boundary components. Let Q E C be a point and a , p closed curves on C with end-points at Q. We say that a and p are homotopic if there exists a continuous mapping h : [0,1] x [0,1] + C such that h ( 0 , s ) = h(1,t) = Q and h(s,O) = ( ~ ( s ) ,

( 2 , R ) of

3.6. UNIFORMIZATION

87

h ( t, 1) = P ( t ) , for all s,t E [0,1]. We use the notation (Y x p to indicate that a and p are homotopic to each other. It is obvious that x is a equivalence relation in the set of closed curves with end-points a t Q. The corresponding set of equivalence classes, or homotopy classes, of closed curves at Q is denoted by nl(Z,Q). If a and p are closed curves as above, then their product ap is defined setting

It is rather straightforward to verify that this multiplication determines a multiplication in r l ( C , Q ) and that nI(C,Q) is then a group. It is called the fundamental group or the first homotopy group of the surface C a t the base point Q. The definition of the fundamental group r l ( C , Q) depends on the choice of the base-point Q. The choice of this base-point is, however, irrelevant. Standard arguments show that if Q’ is another point of C then the groups rl(E,Q) and .rrl(C,Q’) are isomorphic to each other. This isomorphism can be constructed in the following way. Let first y be a curve such that y(0) = Q and y(1) = Q’. Then define

This is a well defined mapping and an isomorphism. The isomorphism (3.5) depends, of course, on the choice of the connecting curve y. An other choice of y changes the isomorphism (3.5) by an inner automorphism of r l ( Z , Q ‘ ) . Let G be the covering group of the universal cover of the surface C. There is an almost canonical morphism

i : r l ( C , Q ) + G,

(3.6)

which is defined in the following way. Choose first a point Q E U (or E C or E C)lying over the point Q E C. Every closed curve 7 with end points at Q can be lifted to the universal covering space U (or C or C) of ( C , X ) . The lifting becomes unique when we require that its starting point is the be the end-point of this lifting. Then previously fixed point 0. Let also Q I is a point over Q and hence there is an element gr of G such that gy(Q) = The element gr defined by this condition is unique because non-identity elements the group G do not have fixed-points in D.

91

01.

Lemma 3.6.2 The Mobius transformation gr E G depends only on the homotopy class of 7 .

88


Proof. This is a standard result in topology and holds even in a more general setting. The result follows from the ‘homotopy lifting property’ of the universal cover and the discontinuity of the action of the cover group G. For a proof we refer to [91, Corollary 8 on page 881. By Lemma 3.6.2 [7]+ gr is a well-defined mapping K I ( C )4 G. The following result gives us the inverse of this morphism and thus proves that it is actually an isomorphism bet,ween n l ( C , Q ) and G.

Lemma 3.6.3 Assume that (C, X ) is a Klein surface such that the universal cover of the interior of ( C , X ) is the unit disk D with the cover group G. Let a be a closed curve representing a point of nl(C,Q). Let ga = i ( [ a ] ) be the element fo the group G which corresponds to [a]E r l ( X , Q ) . Then each curve in D with end-points z and ga(z), z E D ,projects to a closed curve on C that is homotopic to the curve a. Proof. This is also quite standard and follows from rather general topological arguments. To prove the result we have to construct a homotopy between the projected curve and the original curve a. That can be done using hyperbolic geometry. Choose now a point z E D and a lifting of the curve a to a curve ti in D. Then 5 is a continuous mapping I -+ D satisfying a = K o 6 , where K : D + C is the projection. Let p : I + D be a curve in D such that p(0) = z and p( 1) = ga(.z). We form first a continuous mapping F : I x I + D in the following way. Let ( t , s ) E I x I . Define F ( t , s ) as the point on the hyperbolic geodesic between p ( t ) and 5 ( t ) which divides that geodesic in the ratio s : (1 s). Then clearly F ( t , O ) = p(t), F ( t , 1 ) = & ( t ) for all t E I . We have, furthermore ga(F(O,s) = F(1,s) for all s E I. This implies that K o F : I x I + C is a homotopy between the closed curves a and K o p proving the lemma. By the above result we conclude that r l ( C , Q ) -+ G, [a]H ga, is an isomorphism. It depends on the choice of the point Q E D lying over the base-point Q. Another choice of Q changes the corresponding isomorphism by an inner automorphism of G.

Definition 3.6.1 We say that the transformation ga E G of Lemma 3.6.3 covers the (homotopy class of the) curve a. Next we recall a result (cf. e.g. [91, statement 12 on page 1491) describing the fundamental group of a surface. Assume that C is compact surface with n boundary components, p handles and k cross-caps.

3.6. UNIFORMIZATION

89

Theorem 3.6.4 The fundamental group xl(C,Q) of the surface C is genemted b y the elements a l , 01,.. .,a,, Pp (which correspond to the handles of C ) , 71, .. .,-yn (which correspond to the boundary components) and 61, . . .,6 k (which correspond to cross-caps) satisfying the relation k

V

n

[c.j,pj] is the commutator of aj and

Recall that, in the above theorem, Pj-

Now let X be a dianalytic structure on C and assume that (the interior of) the Klein surface (C, X ) is DIG for a reflection group G. The group G is isomorphic to xl(C,Q). Let i : x l ( X , Q ) + G be an isomorphism and let gj = i(aj),hj = i(,f?j),d j = i(yj) and sj = i ( S j ) be the elements of G corresponding to the generators of x l ( X , Q). Then the set

= { g i , h l , . . .,gp,hp,dl,...,dn,51,...,sk} generates G and satisfies

l-I k

1)

nrgj,hj] j=1

n

dj

s;

i=l

= 1.

1=1

Definition 3.6.2 Generators gj, h j , d; and s1 of the group G satisfying the relation (3.8) are called the standard generators for G. Observe that if X is a compact classical Riemann surface of genus p , then there are no generators of type d j or si. The standard generators for such a group are Mobius transformations 91, h l , . . ., g,, h, satisfying the single relation P

n r g j ,hj] = 1. j= 1

(3.9)

This is also the most complicated case since, froin the relation (3.9), it is not possible t o solve any one of the generators in terms of the other generators. But if n > 0, then the group G is actually freely generated, since, in this case, one can solve one of the elements d j by the relation (3.8). If 7t = 0 but k > 0, then we have minor technicalities to take care of. In this case we can express, by the relation (3.8), one of the Mobius transformations sj” in terms of the other generators. By the construction, sj’s are now orientation reversing Mobius transformations mapping the unit disk onto itself. Such a Mobius transformation is a glide reflection and its square is a hyperbolic Mobius transformation.

90

C H A P T E R 3. GEOMETRY OF RIEMANN SURFACES

Recall then (cf. considerations on page 16) that the hyperbolic Mobius transformation s! alone determines the glide reflection s, uniquely. Therefore, even if the group G in this case is not freely generated, we know everything about the generator s1 when all the other generators are given. The classical case of compact and oriented Riemann surfaces remains the most complicated one. Let X = ( C , X ) and Y = ( C , Y )be Klein surfaces. We need t o consider continuous mappings between X and Y . Assume that X = DIG and Y = DIG', where G and G' are Fuchsian groups. Let f : X --f Y be a continuous mapping. It can then be lifted to a continuous mapping F : D -+ D satisfying For'= f or (3.10) where x : D + X and x' : D -+ Y are the projections. If F is one such lifting, then also g'o F 0 9 , g E G, g' E GI, is a lifting o f f , i.e., a continuous mapping satisfying (3.10). All continuous liftings of a continuous mapping f are obtained from one lifting in this manner. The equation (3.10) implies to the following observation:

For each z E D and for each g E G there exists an g: E G' such that (3.11)

Lemma 3.6.5 The Mobius trarisformation gc in equation (3.11) does not depend on the point z E D . Proof. Let to and z1 be two points of D and let a : I -+ D be a curve such that a ( 0 ) = 20 and a(1) = 21. Denote a ( t ) = zt. Then we may define the element g:, for each t E I by the equation (3.11). Recall that since the group G' acts freely in D the element g:, is uniquely defined by (3.11). It suffices to show that g;, = gi,. To that end, let I' = {t E I I g i , = gi,. Clearly I' # 0, since 0 E Z’. By the discontinuity of the group G’ we then conclude that both I' and its complement are open in I . This implies then that I' = I proving the lemma. The above result follows also from the discreteness of the group G. To see this, consider the above defined elements gzt E G. They give a continuous mapping I t G, t I+ g z r . But since G is discrete and I connected, such a continuous mapping is necessarily constant. The difficulty in this reasoning is to show the continuity of the mapping t H gtt. By Lemma 3.6.5 a continuous lifting F : D -+ D of a continuous mapping f : D I G -+ DIG' defines a morphism f # : G + G', setting f # ( g ) = g l for any z E D. This inorphism does not depend on the choice of the point z E D.

3.6. UNIFORMIZATION

91

Definition 3.6.3 The homomorphism f# : G to be induced by the mapping f.

+

G' defined above is said

Observe that one continuous mapping induces many homomorphisms between the corresponding group. An induced morphism depends, of course, on the choice of the lifting F. Another choice F' induces a homomorphism (f’)# which is obtained from f # by composing it with an inner automorphism of GI.

Definition 3.6.4 Homomorphisms i : G + G' and j : G + G' are called equivalent if there exists an element gh E G' such that i ( g ) = gh o j ( g ) o (g;)-', holds for all g E G , i.e., if i is obtained composing j with an inner automorphism of G'. We conclude now that all continuous liftings of a continuous mapping f : DIG +- DIG' induce equivalent homomorphisms G G'. In our applications we are mainly interested in homeomorphisms f : DIG + DIG'. Assume now that f is a homeomorphism. The above construction that was done for the mapping f can just as well be applied to the mapping f - I . If F : D + D is a lifting of f : DIG 4 DIG', then F-' is a lifting of f-'. Using these liftings in the constructions for f # and ( f - l ) # we conclude easily that (j-')# = (j#)-'. --f

This means that for a homeomorphism f : DIG homomorphism f # is an isomorphism.

--f

DIG' the induced

L e m m a 3.6.6 Assume that f : D fG + DIG' and g : D fG + DIG' are homotopic homeomorphisms. Then they induce equivalent isomorphisms f# : G + G' and g# : G +- G'.

Proof. Let H : ( D I G ) x I + DIG' be a continuous mapping such that H ( p , 0) = f ( p ) and H ( p , 1 ) = g ( p ) for all p E DIG. This homotopy between the mappings f and g can be lifted to a continuous mapping

H:DXI+D

a(-,

such that #(-,O) is a lifting of the mapping f and 1) is that of g . The mapping B is, furthermore, compatible with the action of the groups G and G' in the sense that for each t E I and for each g E G there exists an gi E G' for which f i ( g ( z ) t, ) = g i (H (z,2 ) ) for each z E D. It suffices to show that this element gl does not depend on t. Then it follows that 0) and I?(-, 1) induce the same isomorphism. Repeating the reasoning of the proof of Lemma 3.G.5 we see that this follows from the discontinuity of the action of the group GI.

a(.,

92


Lemma 3.6.7 Assume that f : DIG + DIG' and g : DIG + DIG’ are two homeomorphisms such that the induced isomorphisms f# and g# are equivalent. Then the mappings f and g are homotopic to each other.

Proof. Assume that the isomorphism f # is defined by the lifting F of the mapping f and g# by the lifting G of g . The assumption that the isomorphisms f # and g # are equivalent means that there exists a Mobius transformation g; E G' such that

f#(s>= s:,0 S # ( d

0

(g:,)-l

(3.12)

for all g E G. Since G is a lifting of g, then such is G' = g; o G as well. Let us see which isomorphism is induced by this lifting. To that end write

to conclude, by (3.12), that the lifting G' = g& o G of g induces the same isomorphism than the lifting F of the mapping f . We may, therefore, assume that F and G are liftings of f and g, respectively, inducing the same isomorphism f # : G + GI. Next we construct a homotopy H : D x I -+ D between the mappings F and G defining f i ( z , t ) as that point on the geodesic arc from F ( z ) to G ( z ) which divides this arc in ratio t : (1 - t ) . Then I? is a continuous mapping and H ( - , O ) = F ( . ) , H ( . , 1) = G(.). Using the fact that F and G define the same isomorphism f # : G + G' and the fact that this isomorphism is an isometry of the hyperbolic metric, we then conclude that H ( g ( 4 , t ) = f#(s)0 H ( Z , t )

(3.13)

for all z E D and for all t E I . Equation (3.13) implies that fi induces a homotopy between the mappings f : DIG --f DIG' and g : DIG + DIG'. Lemma 3.6.7 has the following corollary that will be applied later. Lemma 3.6.8 Assume that G is a Fuchsian group acting in the upper halfplane U and that hyperbolic non-identity elements of G have at least three fixedpoints on R U { m}. Then if f : U / G -+ U/G and g : U/G + U / G are homotopic to each other and holomorphic, then f = g .

3.7. M O D E L S F O R S Y M M E T R I C SURFACES

93

Proof. Under the assumptions of the lemma, h = fog-’ is holomorphic and homotopic to the identity. Let H : U U be a lifting of h. Since H is a holomorphic homeomorphism, it is a Mobius transformation and can be immediately extended t o the whole infinite complex plane. Let us do that. Observe that the identity mapping U + U is, of course, a lifting of the identity mapping U / G + U/G. By Lemma 3.6.7 H and the identity mapping induce equivalent isomorphisms G + G. Repeating a part of argument of the proof of Lemma 3.6.7 we may suppose that H induces the same isomorphism G + G as the identity mapping, i.e., that H induces the G. identity G This means that for all g E G, z f 7 i and n E N we have --f

--f

H(g'"z)) = g n ( H ( z ) ) .

(3.14)

Assume that g is hyperbolic and fix a point z E U . Consider the equation (3.14) for various values of n. Recall that lirn g n ( z ) = a ( g ) and

n++m

lim g n ( z ) = T ( g )

n+-w

where a ( g ) is the attracting fixed-point of g and r ( g ) is the repelling fixedpoint. From equation (3.14) it then follows that H ( a ( g ) )= a ( g ) and H ( r ( g ) ) = T ( g ) . This applies to all hyperbolic elements of G. Since hyperbolic elements of G have a t least three different fixed-points. We conclude that the orientation preserving Mobius transformation I1 fixes three points. It is, therefore, the identity mapping.

3.7

Models for symmetric surfaces

Even though we did not prove Theorem 3.4.1 we will, in this section, give a concrete construction that shows u s the existence of all the symmetric surfaces of Theorem 3.4.1. Consider first involutions o with index of orientability k ( o ) = 0. Let n be an integer with g - (78 - 1) = g + 1 - 71 even. Take a Riemann surface of genus (g 1 - 72)/2. Delete 71 open disks from it. Assume that the disks are chosen in such a manner that their closures are disjoint. Then one gets a Riemann surface Y of genus (g 1 - n ) / 2 with n boundary components. Let P denote the Riemann surface obtained from Y by replacing the complex structure of Y with its conjugate structure, i.e. by replacing all local variables z with their complex conjugates z.Y is simply the mirror image of Y. Glue the Riemann surfaces Y and P together identifying the

+

+

94

C H A P T E R 3. GEOMETRY OF RIEMANN SURFACES

boundary points. In that way one gets a compact Riemann surface X of genus g . The identity mapping Y + P induces an antiholomorphic involution u : X + X such that the curves of X corresponding to the boundary curves of Y remain point-wise fixed. Therefore the parameters of Theorem 3.4.1 satisfy n(u) = n and k(u) = 0 for this involution. This is how one can construct topologically all symmetries u of a genus g Riemann surface X satisfying k ( o ) = 0 and n ( u ) = g l(mod 2). Let a be a closed curve left point-fixed under the above involution u. Let A be a tubular neighborhood of a. Then the universal covering of A is the strip A = { z E CI - 1 < I m z < 1).

+

Furthermore we may suppose that u maps A onto itself and that the complex conjugation is a lifting of u : A + A onto i. Then the real axis covers the curve a. Everything here is only topological. So assuming that the covering group of + A is generated by z H z 2 we do not restrict the generality. Define the function H : -+ setting H ( x iy) = (x 1 - y iy). Then the complex conjugation T ( Z iy) = x - iy and H o 7 are both selfmappings of A. Both of them map the real axis onto itself but only the complex conjugation keeps it point-wise fixed. Let fa : X + X be defined setting fa@)-= p for p E X \ A . In A define fa as the mapping induced by H : A --t A . The mapping fa : X + X defined in this way is clearly a homeomorphism.

+

+

+

Definition 3.7.1 The mapping the curve a.

fa :

X

+

+ X is the Dehn twist of

+

X along

It is easy to check that fa o u is also an involution of X. For this involution we have k(fa o u) = 1 and .(fa o ~7)= n(u) - 1. Figure 3.7 illustrates how the involution fa o u maps a curve that intersects the curve a. Repeating this procedure for each component of the fixed-point set of u we can clearly in this way construct topologically any symmetric surface for which n < n(u) and k = 1. This how one can construct topological models for all symmetric Riemann surfaces. Dehn twist was here used to construct topological models of symmetric surfaces. It is an important concept and has many applications. The rather formal definition given above can be replaced by the following shorter definition which relies partly on reader’s geometric intuition. To give this definition for the Dehn twist, orient first the simple closed curve a in some way. You have two choices, they both lead to the same deformation.

3.8. HYPERBOLIC METRIC OF RIEMANN SURFACES

95

Figure 3.7: The twisted involution f a o a does not keep the curve a pointwise fixed. The orientation of the curve a tells us the positive direction along a , and, since X is assumed t o be oriented, it also separates the left hand side of Q from the right hand side. Now cut the surface X open along a. You obtain in this way a new surface X ’ which has two boundary components a’ and a” corresponding to the curve a. These boundary components can be thought of as corresponding to the left hand side of a , call that one a’, and to right hand side of a. Turn the left hand side of a , i.e., the curve a’ full turn around in the positive direction of a and glue it back to the curve a“. In this way we obtain the topology of the surface X is not changed. But curves crossing cr get replaced by curves crossing a and going once around ‘the handle’ corresponding to a. This is also called the left Dehn twist. It is easy to check that the definition of the left Dehn twist does not depend on the orientation of the curve a .

3.8

Hyperbolic metric of Riemann surfaces

By the Uniforniization Theorem, every Riemann surface W (or, more generally, every Klein surface) with negative Euler characteristic can be expressed as W = D I G , where G is a discontinuous group consisting of Mobius transformations mapping the hyperbolic unit disk onto itself. We equip the the unit disk D by the hyperbolic metric of constant curva-

96


ture -1. For the definition and basic properties of this metric see Appendix A. By Theorem A.2.1 in Appendix A, the elements of G are isometries of the hyperbolic metric of D. We then conclude that the hyperbolic metric of the unit disk D projects to a metric of the surface W. We call this metric the hyperbolic metric of W. The hyperbolic metric of a Riemann surface is a complete metric of constant curvature -1 (cf. (A.4)). Let W now be a compact hyperbolic Riemann (or Klein) surface with boundary curves c q , . . .,aP.On W we often use the intrinsic hyperbolic metric in which the boundary components are geodesic curves of a finite length (see e.g. [ l , page 451). This metric can be obtained in the following way. First form the Schottky double of W by gluing W and its mirror image W together along the boundary components. The intrinsic metric of W is the restriction of the usual hypcrbolic metric of the Schottky double of W to W itself. Let W be a compact Riemann or Klein surface. The well-known GauilBonnet formula (cf. [34, Page 2281) gives the following expression for the hyperbolic area A ( W ) of the surface W

A ( W )= - 2 ~ x ( W ) .

(3.15)

We will not prove this result here. It will, nevertheless, play an important role in our considerations. The area formula (3.15) implies that the area of a compact Riemann or Klein surface depends only on the topological type of the surface and not on the complex or dianalytic structure.

A Riemann surface is usually thought of as being a sphere with handles. The hyperbolic metric then tells us how thick and how long the handles are. If the surface degenerates in such a way that some handles become very long, then they necessarily become thin at the same time. This is a loose observation can be understood by the invariance of the hyperbolic area. We will later give a completely different proof for this fact using the considerations of Chapter 1. From now on we assume that every compact Riemann or Klein surface without boundary is equipped with the hyperbolic metric. Surfaces with boundary components are - unless otherwise stated - assumed to be equipped with the intrinsic hyperbolic metric.

A main goal of this chapter, and also of this monograph, is to provide tools that allow us to make precise the above loose remarks concerning the degeneration of Riemann surfaces.

3.9. H U R W I T Z T H E O R E M

3.9

97

Hurwitz Theorem

For a later application we discuss here a classical construction which leads to an estimate concerning the order of the automorphism group of a compact classical Riemann surface. This result is known as the Hurwitz Theorem (see [42]). Assume that X and Y are classical Riemann surfaces and that f : X + Y is a non-constant holomorphic function. For any point p E X we may choose local variables z at p and w at f(p) such that z(p) = w ( f ( p ) ) = 0. Then w o f o t-’ is a holomorphic function defined in a neighborhood of the origin where it has series expansion of the form k=n

where n is a positive integer and a,, # 0. A closer analysis then reveals that we may actually choose the local variables z and w in such a way that

w0f

0 .-I((')

= 0, # 1. T h e set of the elements gn(z) = h-" o g o h"(z) = t bk-", n E Z, contains then a sequence of distinct elements of G converging t o the identity. This is not possible. Therefore h can not b e hyperbolic. Assume next that IL is parabolic. By conjugation we may assume again that both parabolic elements r~ and h have co as fixed point. Then they are of the form g ( z ) = z 6 and h ( z ) = z 6'. T h e statement of the Lemma 3.10.1 (equation (3.20)) is equivalent t o saying that b/b' is rational. But if this ratio were irrational, then

+

+

+

+

(7nb

+ nb'

I m, n E Z }

would be dense in R and one could easily build a sequence of distinct elements of G converging t o the identity. T h a t is not possible since G is assumed t o be discrete. L e m m a 3.10.2 Suppose that G is a Fuchsian group acting in the upper half-plane U and containing the parabolic transformation g ( z ) = z 1 as a primitive element. Let H = { z I I m z > 1) and assume that h E G. Then either h ( H ) n H = 0 or 12 = g n for some n E Z . H is the largest hag-plane having this property.

+

102


Proof. Assume that

+

az b h ( x ) = - ad c z + d’

-

be = 1 , and h

# gn for any n E Z.

If c = 0, then h has 00 as fixed point. By our present assumptions and by Lemma 3.10.1 this is not possible. Therefore we conclude that 1.1 > 0. We have to show that h ( H ) n H = 8. Let hl = h o g o h-’ and define inductively h k + l = h k o g 0 hi’. Write

Then

ck+l

= 1 -akck = a: = -Ci

dk+i

=

ak+l bk+l

1 -kakCk.

If 1c1< I , then Ck = - c -+~ 0. ~ This implies that U k -+ 1, bk + 1 and dk 1 as k + 00. Therefore we conclude that in this case h k + g. Since IcI > 0, elements h k are also distinct. This is not possible since G was assumed to be discrete. Hence IcI 2 1. Let z E H . Then Im h ( z ) = (Im z ) / l c z dI2. Provided that h is not a power of g , dH’ = h ( a H ) is a circle tangent t o R at a / c . We compute: .--f

+

diameter of H’

=

I

a(z+i)+ b a sup XER c ( x i) d -

+ +

el

-

which implies the first statement of the lemma. The elliptic modular group S L 2 ( Z ) provides an example which shows that H is the largest haJf-plane for which the first statement of the lemma holds. For the basic properties of the elliptic modular group we refer t o the discussion in Chapter 5. A fundamental domain for this group is described on page 182. This remark completes the proof of the lemma.

3.10. HOROCYCLES

103

Definition 3.10.2 Using the notation of Lemma 3.10.2, we say that H is a horocycle at the fixed point of g . We can improve the above result if we assume that the group G does not contain elliptic elements. Here we follow an argument of 1811 and [go]. Assume that this is the case, i.e., that G is a Fuchsian group which acts freely in the upper half-plane U . Let g E G, g ( o 0 ) # 00, and let I ( g ) denote the isometric circle of g . For the definition of the isometric circle see Section 1.4. The center of I ( g ) lies on the real axis, g(I(g)) = I(g-’), and I ( g ) and I ( g - ' ) have the same radius. If g is parabolic, then I ( g ) and I(g-') are tangent to each other at the the fixed point of g , otherwise I(g) n I ( g - l ) = 0. Lemma 3.10.3 The digerence g ( z ) - z is real for a point z in the upper half-plane if and only if t E I(g).

This lemma follows directly from the geometry of the action of the Mijbius transformation g and the definition of the isometric circle. Proof is left to the reader. Lemma 3.10.4 Let g E G be such that g(m) # 00. If G contains the translation gw : z H z w , then ( g ( z ) - z ) / w is not an integer for any 2

E

u.

+

Proof. Suppose that g ( z ) - z = nw for some integer n and z E U. Then o g," fixes z. It follows that 9-l o gc is elliptic, which is not possible by our assumptions.

g-'

Lemma 3.10.5 Suppose that gw : z H z t w is in G. I f g E G does not fix 00, then the radius rg of I(g) sutisfies rg 5 w/4.

Proof. By geometry (3.21) Now ( g ( z ) - z)/wis real on I(g) by Lemma 3.10.3. If the total variation of ( g ( z ) - z ) / w along I(g) were more than 1, then ( g ( z ) - z ) / w would necessarily take an integer value at some point in I(y). By Lemma 3.10.4 this is not possible. We conclude, therefore, that (3.22) Inequality (3.22) together with equation (3.21) implies now the lemma.


104

Theorem 3.10.6 Assume that the group G contains, besides the identity, only hyperbolic and parabolic M6bius transformations mapping the upper half-plane onto itself. Assume further that g l ( z ) = z w is a primitive element of the group G. Let g E G be such that g ( o 0 ) # 00. If I m z > w / 4 , then Im g ( z ) < w/4. The number w/4 is the smallest possible.

+

Proof. The first part of the statement follows directly from Lemma 3.10.5.

To prove that w/4 is the smallest number with this property, assume that w = 1 and let go be the transformation z H 2/(42 + 1) and G1 be the group generated by go and gl. Then G1 is a Fuchsian group without elliptic elements and g ( ( - 1 + i)/4) = (1 + 4 / 4 . Theorem 3.10.6 implies now immediately the following result: Theorem 3.10.7 Let X be a hyperbolic Riemann surface with punctures. Each puncture of X has a horocyclic neighborhood of area 4. The inner boundary curve of this horocycle has length 4.

Observe that the horocyclic neighborhoods of Theorem 3.10.6 at disjoint punctures are not necessarily disjoint. An example of this situation is provided by the group GI. Area 4 horocycle associated to the element go is the euclidean disk of radius with center at f . Area 4 horocycle associated to the element g l ( z ) = z 1 is the half-plane Im z > They overlap. Let g be any parabolic Mobius transformation mapping the upper halfplane onto itself. Then we may always assume that g is conjugate either to the transformation z --i z $- 1 or to z + z - 1. If g is conjugate to z -+ z 1, then g-' is conjugate to z -+ z - 1. The orientation reversing Mobius transformation z H -7 conjugates z H z 1 to z H z - 1. Assume now that G is any Fuchsian group acting in the upper half-plane U and let g be a primitive parabolic element in G. The above observation together with Lemma 3.10.2 implies that there is always a hyperbolic disk (or a half-plane) D, E U such that dD, is tangent to dU at the fixed-point of g and if h(D,) n D, # 8, then h E (9). In accordance with Definition 3.10.2 we say that the disk D, is a horocycle of the parabolic transformation 9.

+

+

3.11

a

a.

+

Nielsen's crit erium for discontinuity

Theorem 3.10.6 allows us to give a fairly general criterium that guarantees the discontinuity of a Mobius group acting in the upper half-plane. Theorem 3.11.1 Assume that G is a group of Mobius transformations mapping the upper half-plane I I onto itself and containing the translation

3.1 1. NIELSEN’S CRITERIUM FOR DISCONTINUITY

105

Figure 3.8: A horocyclic neighborhood of a puncture on a hyperbolic Riemann surface. This surface is the surface of revolution of the tracktrix curve which is characterized by the property that its tangent line meets the x-axis at unit distance from the point of tangency. This surface has curvature -1.

106


gl(z) = z -t 1. If G does not contain elliptic elements, then G is either properly discontinuous or contains only elements jxing 00.

Proof. Suppose that G contains elements not fixing the infinity. In view of Theorem 3.5.4 (on page 84) it suffices t o show that the regular set Q(G) of G contains a point in the upper half-plane. If g E G does not fix the infinity and gw(z) = z w is in G, then, by Lemma 3.10.5, w 2 4r,. (3.23)

+

From (3.23) it follows that the subgroup G, of the translations of G is cyclic. We show next that the group G does not contain any hyperbolic elements fixing the infinity. Assume, on the contrary, that h E G is hyperbolic and h(00) = 00. Then we may suppose that h ( z ) = kz for some k > 1. Observe that rg is not changed if g is conjugated by a translation. Now, 11-" o gw o h" E G, and

(h-" o gw o h " ) ( z ) = z

+ k-".

This contradicts (3.23). Let g w ( z ) = z w , w > 0, be a generator for G,. Choose zo E U such that Imz0 > By Theorem 3.10.6, the disk D ( z o , w / 2 ) = { z I Iz-zol < w / 2 } does not contain points z , z # zo, that are equivalent t o zo under G. We conclude that zo E S2(G).

+ :++.

Theorem 3.11.1 is a special case of the following more general result.

Theorem 3.11.2 Assume that G is a group of MObius transformations mapping the upper half-plane onto itself. If the group G does not contain elliptic elements, then it is discontinuous. In view of Theorem 3.1 1.1 we have to shown only that purely hyperbolic Mobius groups fixing the upper half-plane is discontinuous. Proof for this can be found in Appendix B, Theorem A.8.5 (on page 240). The above result is a special case of a more general result stating that a MZibius group mapping the upper half-plane onto itself is discontinuous if it does not contain infinitesimal elliptic elements. This has been first shown by C. L. Siegel in [ 8 5 ] , where Siegel calls this theorem 'a result of Jakob Nielsen'.

3.12. CLASSlFlCATlON OF FUCHSIAN GROUPS

3.12

107

Classification of Fuchsian groups

Let D, be a horocycle associated t o a parabolic Mobius transformation g. It is immediate that Dg/(g) is conformally equivalent t o the punctured unit disk D* = { z I 0 < 121 < l}. Assuming that g is a primitive element, we conclude, by Lemma 3.10.2, that Dg/(g) is conformally homeomorphic t o an open subset of the Rieinann surface U / G . Using this non-compact subset of U / G it is easy t o construct an open covering of U/G which does not have a finite subcovering of U/G. This implies that U/G is not compact. We have, therefore, the following result. Theorem 3.12.1 Let G be a Fuchsian group acting in the upper half-plane U. If U / G is a compact Riemann surface, then the group G does not contain parabolic elements.

By Theorem 3.12.1 we conclude that all non-identity elements of a Fuchsian group corresponding to a compact Riemann surface are hyperbolic Mobius transformations. Let now 7 be a closed curve on a Riemann surface UIG. Let g = gr be a Mobius transformation corresponding to the homotopy class of the curve 7 as explained in Lemma 3.6.2 and in Lemma 3.6.3. Recall that for a hyperbolic Mobius transformation infEEU& ( z , g(t)) is obtained for any z E u x ( g ) . Here a x ( g ) denotes the axis of the hyperbolic transformation g and du is the hyperbolic metric of the upper half-plane U. We have, furthermore, infzEUd u ( z , g ( z ) ) = log k(g), where b(g) > 1 is the multiplier of g. We conclude therefore that if the transformation gr is hyperbolic, then the homotopy class of the curve y contains a geodesic curve which is the projection of the geodesic arc from z to yy(z) for any point z E az(g,). Furthermore we conclude that if gr is parabolic, then the homotopy class of the curve 7 does not contain any geodesic curves. In this case we say that 7 is a curve going around a puncture of U / G . Let G be a Fuchsian group acting in the upper half-plane.

Definition 3.12.1 Let G ( z )= { g ( z ) I g E G } denote the orbit of a pont z under the action of G . A point x E R U {cs} i s a limit point o j G if there exists a point z E U such that x belongs to the closure of G ( t ) . The set L(G) ofG.

Definition 3.12.2 A Fuchsiun group G acting in the upper half-plane is said to be of the first kind if L ( G ) = R U {m}. Groups that are not of the first kind are said to be of the second kind.

108


This terminology is standard today. The reader should observe, however, that, in old literature, groups of the first kind were sometimes called groups of the second kind (and vica versa). For instance in a paper of Burnside ([ls])this terminology was used but the meaning was the contrary. Lemma 3.12.2 Assume that the Riemann surface U / G is compact. Then the Fuchsian group G is of the first kind. Proof. This argument is due to Pekka Tukia. Let r E R be an arbitrary point in the real line. I t suffices to show that the closer of the orbit of the point i E U contains the whole real line. Let 6 > 0 be arbitrary. Let

ZI,(r) = { z E

c 1 ) z - 7-1 < E }

denote the euclidean disk of radius c and center at r E R. By the properties of the hyperbolic metric, U C ( r fl ) U contains hyperbolic disks D R of arbitrarily large radius R. Now the image of any hyperbolic disk of radius

R > diameter of UIG = sup{dv/c(p, q ) I p , q E U/G} under the projection T : U +. U / G necessarily covers the whole Riemann surface U/G. Therefore any such disk contains a point z such that ~ ( i=) ~ ( 2 ) .This point z belongs to the orbit { g ( i ) [ g E G} of i . We conclude, therefore, that I- E R belongs t o the closure of the orbit of i. Repeating this argument we could prove the above observation stating that i can be replaced by an arbitrary point z in the upper half-plane. Likewise one can show that the limit set of a Fuchsian group is the closer of the set fixed-points of non-identity elements of that group.

3.13

Short closed curves

Theorem 3.12.1 allows us to apply considerations of Chapter 1 to estimate lengths of intersecting closed geodesic curves on a hyperbolic Riemann surface. We will next show that if two closed geodesic curves intersect, then they cannot both be short. Lemma 3.13.1 There exists a universal constant q, q > 0 , such that for any compact Riemann surface X (which may have boundary components) the following is true: Let (Y and /3 be closed geodesic curves on W with lengths < 7. Then either (Y = j3 (as set of points) or the curves a and p do not intersect.

3.14. COLLARS

109

Proof. This result follows directly from inequality (1.25) in Corollary 1.5.5 on page 33. Assume first that Q and fi are closed and intersecting geodesic curves such that Q # p (as sets of points). Write W = U / G for a Fuchsian group G. Let g and h be a hyperbolic transformations, g covering a and h covering ,f3 (cf. Definition 3.6.1). Provided that g and h are obtained using the same isomorphism i of Lemma 3.6.3, g and h are now hyperbolic Mobius transformations with intersecting axes. Applying the arguments of Theorem 3.12.1 we can conclude that the non-identity elements of the group G are hyperbolic Mobius transformations. This implies that the coininutator c = [ g , h] of the hyperbolic transformations g and h is hyperbolic as well. Therefore the assumptions of Corollary 1.5.5 are satisfied. Let La and Lo be the lengths of the geodesic curves Q and p, respectively. Then the multiplier kl of g satisfies kl = eea and that of h is k2 = eep. Inequality (1.25) gives then the following inequality for the lengths of the curves Q and /3 : deep (eea (3.24) < (eep - 1 ) 2 4eea ' Constant q is then the positive solution of the equation

(3.25) A numerical estimate for the positive solution of equation (3.25) is 7 x 1.33254. Another inequality of the same type as inequality (3.24) is given in [ l , Lemma 1 on page 941. It is not quite as strong as the one presented here. It yields the numeric estimate 1.01859 for the constant v. It is interesting t o observe that the above inequality was obtained using only the facts that the Mobius transformations 9 , h and their commutator c = [g, h] are hyperbolic. The discontinuity of the action of the Fuchsian group G does not play any role here.

3.14

Collars

Definition 3.14.1 Let A C X be a non-empty subset of a (hyperbolic) Riemann surface X and let 6 be a positive number. The €-distance neighborhood of A is the set

C H A P T E R 3. G E O M E T R Y OF R I E M A N N SURFACES

110

Let X now be a hyperbolic Riemann surface, p E X a point and a a simple closed geodesic curve on X. For an E > 0 we say that N , ( { p } ) is a disk if N , ( { p } ) is homeomorphic to a disk in the complex plane. Likewise we say that N , ( { p } ) is a collar if it is homeomorphic t o an annulus in the complex plane. We say further that a collar N E ( a )has width 6.

Let (Y be a simple closed geodesic curve in the interior of X . A collar N,({p}) of area 2p is called a p-collar at a. If a is a boundary component of X , then a collar N , ( c Y )of area p is called a p-collar. The area and the width of a collar a t a geodesic curve of length e are related t o each other in the following way. Lemma 3.14.1 Assume that cr is a simple closed geodesic curve in the interior of a hyperbolic Rieinann surface X and that N , ( a ) is a p-collar at a . Then we have p = e, sinh E

where

e,

denotes the length of cr on X .

The proof of this result is an elementary computation in hyperbolic geometry. We will leave the details to the reader. Arguments are similar t o the ones of the proof of Lemma 3.14.7. Next we use considerations of Chapter 1 t o estimate the distance dx(a,P) between closed geodesic curves a and on a Riemann surface X = DIG. By the considerations based on Theorem 3.12.1 we can associate t o any closed geodesic curves a and /? two hyperbolic Mobius transformations g E G and h E G such that the axis of g and h project onto a and p, respectively. Furthermore, if Icl = k ( g ) and kl = k ( h ) are the multipliers of g and h, then C, = log k1 and t?p = log kl. Let dx(a,P) = logk. If cr and ,d do not intersect each other, then we can deduce, by Theorem 1.5.7, that

(3.26) where f ( t ) = & +l/J? for t > 0. Estimate (3.26) gives us iinmcdiately an estimate for the distance d x ( a ,P ) in the following way. The right hand side of (3.26) can be rewritten so that one has (fit*)(&+*) k> (3.27)

(fi-*)(w2-&.

Estimate 3.27 implies the following result.

3.14. COLLARS

111

Lemma 3.14.2 Let a and p be closed geodesic curves of length C, and l p , respectively, o n a hyperbolic Riemann surface. Then either d X ( a , P ) = 0 ( i n which case a and p intersect) or d X ( f f ,p )

> log coth(CJ4)

+ log coth(Cp/4)

(3.28)

Estimate (3.28) means that short non-intersecting curves on any hyperbolic Riemann surface are rather far apart from each other. Estimate (3.28) has important applications for us. Let

6(a)= inf{dx(a,P) 1 P a closed geodesic curve ,f3 r l a = 8).

(3.29)

For a fixed value of C,, the right hand side of (3.28) is a decreasing function of C p tending t o the limit log coth(!,/4) as C p -+ 00. This shows the following result:

Lemma 3.14.3 For a simple closed geodesic curve a ,

We will now use the above considerations to estimate the size of the maximal collar a t a simple closed geodesic curve. To that end observe the following result. Lemma 3.14.4 A simple closed geodesic curve cy on a hy~erboZicRiemann surface X has always a collar N 6 ( a ) ( a )of width 6(a).

Proof. It is clearly enough t o show the following: Lemma 3.14.5 If the closure N , ( a ) of a €-distance neighborhood N , ( a ) of a simple closed geodesic curve a on a hyperbolic Riemann surface X is not homeomorphic to a x [ - E , E ] , then there exists a closed geodesic curve ,B that does not intersect a such that p n N , ( a ) # 0.

Proof. It is enough to consider the case where N , ( a ) z a x ( - E , € ) and N E ( a )$ a x [ - € , E ] . Then N , ( a ) has either one or two boundary components. Let us assume that N , ( a ) has two boundary components. T h e case of only one boundary component can be treated in the same way as the present case and will be left t o the reader. Figure 3.9 illustrates this case. Let a' and a'' denote the boundary components of N , ( a ) . At least one of them,


112

Figure 3.9: N,(cr) has two boundary components. say a', is not anymore a simple closed curve. Then a' can be expressed as a finite union of simple closed curves 71,. . .,yn, n 2 2, such that yi and y;+1 intersect at one point pi. Consider the curves y1 and y2 intersecting a t the point pl. Assume that both curves have the positive boundary orientation (as parts of the boundary of N , ( a ) ) . Let y be the geodesic curve freely homotopic to y1-y;'. Now y is a geodesic curve which is not homotopic to a. y is furthermore homotopic to a closed curve that does not intersect a. We deduce, therefore, that a n y = 0. For topological reasons it is, on the other hand, clear that y n N , ( ( Y )# 0.0 Lemmata 3.14.4 and 3.14.3 imply now the following result:

Theorem 3.14.6 Let

(Y be a simple closed geodesic curve of length la on a hyperbolic Riemann surface X . The curve a has afwuys a p-collar N,(a) for p = p ( & ) = l,sinh (logcoth(l,/4)). The width of this collar is 6 =

log coth e,/4. Observe that the above function p is a positive decreasing function and that lirn p ( l ) = 2. (3.30) e+o+

3.14. COLLARS

113

In order to apply the above results it is necessary to take a closer look at the geometry of collars. We are mainly interested in collars at boundary components of hyperbolic Riemann surfaces. We will, therefore, assume that a is a boundary geodesic of a hyperbolic Riemann surface X. What is said below is, nevertheless, true also for general simple closed geodesic curves. Let c > 0. Assume that the N j . Let now NO = max(N1,. . ., N6g--4). Then, for n > NO,l a j f,*(X)) ( is independent of n for all j. Theorem 4.8.2 implies then that all the points !,*([XI) E T g agree which is a contradiction.o

Theorem 4.10.5 Assume that g > 1 . The action 0 f P on T discontinuous.

g

is properly

Proof. It clearly suffices to show that each point [XI E Tg has an open neighborhood U X such that:

f * ( U x )n Ux # group P.

0 for

at most finitely many elements f* of the modular

CHAPTER 4 . TEICHMULLER SPACES

164

To construct such a neighborhood

is a positive number. Here

T

Ux observe first that by Lemma 4.10.4

is the Teichmuller metric. Let

Since all elements of the modular group are isometries of the Teichmiiller metric, the definition implies that f * ( T x n ) Bx = 8 for all f* E r g not fixing the point [XI. By Lemma 4.10.1 and by Tlleorem 3.9.3 (page 99), there are at most S 4 ( g - 1) elements of r g fixing the point [XI proving the theorem. Here we have considered Teichmuller spaces of compact classical surfaces of genus > 1 simply for technical convenience. It is not hard t o see that the same holds for actually all compact surfaces. The case of classical genus 1 surfaces will be dealt with separately. Same arguments can be applied also in the case of non-classical compact surfaces. Theorem 4.10.5 was first shown by S. Kravetz [54]. The arguments presented here follow the lines of the presentation of F. P. Gardiner 132, Section 8.51.

4.11

Representations of groups

Teichiiiiiller spaces of Riemann surfaces can be studied also in a more abstract setting using representations of groups. That leads to an interesting parametrization of the Teichmiiller space of a classical Riemann surface as a coriiponent of un uflize reul ulgebruic variety. This approach is interesting also because it leads to new ways of compactifying the Teichmiiller space ([73], [64], [16]). Within this monograph we cannot present these compactifications. We can, however, show that the Teichmuller space is a component of an affine variety. This important result has been shown independently by several authors. To our knowledge the first one to do this was Heinz Helling ([39]). We follow his constructions here. At this point it is necessary to review results from the theory of deformations of representations of groups. We cannot prove everything here. For more details w e refer to [83], [97], [98] and [39]. The following notation and definition is related to Definition 4.7.1 (on page 153) but it is not exactly the same. Definition 4.11.1 Let I? be a group und F a topological group. The deformation space K ( r , F ) of r in F consists of all homomorphisms r + F .

4.11. REPRESENTATIONS OF G R O U P S

165

We endow R(r,F ) with the topology of point-wise convergence. In this topology a sequence formed of the homomorphisms Bj : r 4 F, j = 1,2,. ., converges to a homomorphism 0 : r F if and only if for each 7 E I' the sequence &(7),&(7),... converges to 6(y) in the topological group F . For a topological group F , Aut(F) is the group of all continuous automorphisms of F . We denote by Auto(F) the group of inner automorphisms of F. The group Aut(F) acts on R ( I ' , F ) in the natural way: an element f E Aut(F) induces the mapping ’f :R ( r ,F ) R(r,F ) defined by setting f*(0) = f o 6 for every B E R ( r ,F ) .

.

--f

-+

Definition 4.11.2 The quotient

7(r,F ) = R(r, F ) / A u t o ( q is called the Teichiniiller space of representations of ’I in F .

For a locally compact group F , let

Ro(r,F ) = (0

E

R(r, F

) I ~ injective, qr) discrete, F/o(r)compact).

The subspace R0(r,F ) is clearly invariant under the action of Aut(F). We use the notation I o ( r F , ) for the image of Ro(I',F ) in l ( r ,F ) under the projection R + 7. Definition 4.11.3 This space P ( r ,F ) is called the Teichmiiller space of discrete representations of r in F .

In certain special cases the space 'To(I?, F ) is closely related to the usual Teichmiiller space of a surface. To see the connection consider the fundamental group r (at soiiie implicit base point) of a compact and oriented surface C without boundary. Let X be a complex structure on C. By the uniforinization theorem we get a presentation X = U / G where U is the upper half-plane and G C PSL2(R) is a discrete subgroup, i.e., a Fuchsian group. This, in turn, gives rise to an isomorphism (3.6) (see page

87)

8 : I?

--f

G C PSL2(R).

If G' is another group for which X = U/G' and 8' : r + G' is an isomorphism then Bo8-l : G -, G' is the restriction to G of an inner automorphism of P SL2(R). The isomorphism B : r -+ G is characterized by the following property: Let [a]E l?, (Y a closed curve, and let g = B([a]).Any curve joining a point z . E U to g ( z ) in the upper half plane projects to a curve /3 on U / G = X which is freely homotopic to the curve a.

a

CHAPTER 4. TEICHMULLER SPACES

166

It follows that - even thought there is some ambiguity in the choice of the group G and the isomorphism 0 - the point [e : r -, PSL2(R)] E p(r,PSL2(R)) depends only on X . A similar argument shows that the point [8 : r --f PSL2(R)] E 'TO(r,PSL2(R))depends only on [XI E T(C). We conclude therefore that p : T ( C )-+ p(I',PSL2(R)),[XI H [8 :

r -,PSL2(R)]

is a well defined mapping. With the help of the uniformization theorem, one can easily show that p is injective. The next thing that we should observe is that

The reason is that when defining the Teichmiiller space T ( C )we started with only those complex structures of C that agree with the given orientation of C. The complex conjugates of such complex structures form a mirror image of T ( C ) which can be mapped to 7'(r,PSLz(R)) as well. It follows that the Teichmiiller space P ( r ,PSL2(R))has two connected components which are both models for the Teichmiiller space T ( C ) . We denote these PSL2(R)). , components of P ( r ,PSL2(R)) by e ( r ,PSL2(R)) and ’Z?(I’ For more details we refer to [25]. For our purposes it is better to study representations in SL(2,R)instead of the representations in PSL(2, R). By Theorem 3.22.1 (on page 136), in the case of the fundamental group r of an oriented compact surface without boundary, every (faithful) representation 8 : r + PSL2(R) can be lifted to a (faithful) representation 8 : r + SL2(R). On the other hand, every (faithful) SL2(R) projects to a (faithful) representation B : representation 8 : r r -, PSL2(R) because the center of the fundamental group of a compact and oriented surface of genus > 1 is trivial. It follows, therefore, that the projection --f

is surjective. This projection is, of course, continuous and open. It is also obvious that Ro(l',SL(2, R)) projects to R0(r,PSL(2,R)). Since p ( r ,P W 2 , R)) has two connected components,

4.11. REPRESENTATIONS OF GROUPS

167

has only finitely many components. The part of 7"(I', SL(2, R))projecting to T ( I ' , P S L ( 2 , R ) ) will be denoted by T ( I ' , S L ( 2 , R ) ) . Since SL(2,R) is connected, the projection

R0(r,SL(2,R))

-, p ( I ' , S L ( 2 , R ) )

defines a bijective correspondence between the components of

RO(LSL(2, R ) ) and those of P ( r ,SL(2, R ) ) . Let G, be the group generated by the rotation of the upper half-plane around the point i by the rational angle m , / n in the positive direction. Assume that m, and n are relatively prime integers and that the sequence m,/n converges to an irrational number s. Then the generators gn of the groups G, form a converging sequence and limn+mgn = gs, which is a rotation by the angle s. So, in some sense, the discrete groups G, 'converge' to the group (gs) which is not discrete. This is, nevertheless, possible only if we allow G, change as a group. Above group G, is a cyclic group of order TZ.For different values of n these groups are not isomorphic to each other. In the case we are considering the situation is different. The following has been shown in [39].

Theorem 4.11.1 Connected components of Ro(I',SL(2, R ) ) are connected components of R ( r ,SL(2, R ) ) . Proof. We have to show that discrete faithful represtations of I’ in SL2(R) form an open and closed set in the space of all represtations. The fact that discrete representations form an open set is an important result of A. Weil who showed in [97, $11 that

Tt0(r,SL(2, R)) is open in R(r,SL(2, R ) ) . We will not reproduce his proof here. It remains to show that

is also closed in R(r,SL(2, R ) ) . In the case that we are considering, namely that of a fundamental group r of a compact surface, this fact follows immediately from Theorem 3.11.2 (on page 106). Assume that 8, : r + SL2(R), n = 1 , 2 , . . ., is a sequence of faithful representations such that for each n, U / O , ( r ) is a compact Riemann surare face. Then by Theorem 3.12.1, all non-identity elements of each @,(I?)

CHAPTER 4 . T E I C H M ~ L L E RSPACES

168

hyperbolic Mobius transformations. If On -+ 8 as n + 00, then, by the definition of the topology, 8,(7) + B(7) for each y E I?. The matrices On(7),7 E l? \ {Id}, correspond to hyperbolic Mobius transformations. A sequence of hyperbolic Mobius transformations can converge only to one of the following Mobius transformations: a

a hyperbolic Mobius transformation,

a

a parabolic Mobius transformation,

a

the identity.

It follows, especially, that the group 8(r)cannot contain matrices corresponding to elliptic Mobius transformations. Therefore it follows, by Theorem 3.11.2, that the Mobius group corresponding to 8(r)is discrete. The projection R ( r ,SL(2, R)) + 7(r,SL(2, R)) is open. Therefore the components of p ( r ,SL(2, R)) and those of SL(2, R)) are also components of l ( r ,SL(2, R)). The following lemma characterizes components of q(I', SL(2, R)):

q(r,

Lemma 4.11.2 Assume that the points corresponding to the representations 6 : r + SL2(R) and 8' : r + SL2(R) belong to the same component o f q ( r , SL(2, R)). Then, for any [a]E r, truces of the matrices 8([a])and B'([a])E SL2(R) have the same sign. Proof. Assume the contrary. Then there exists an a I? and representations 8 : I' SL2(R) and 8' : r + SL2(R) which belong to the same component of q ( r ,SL(2,R)) and for which the traces of the matrices @(a)and O'(a) have opposite signs. Since 8 and 8' belong to the same component of 7$(l?, SL(2, R)) we have a continuous mapping

[o, 11 + G ( ~ , S L R)), ( ~t ,

C*

such that 8, = 8 and Then the mapping

( 8 , :r

+

SL~(R))

= 8'.

is continuous as well. Here x(Bt(a))is the trace of the matrix @,(a)E SL2(R). Because of the assumptions concerning a , 8 and 8' this mapping changes sign on [0,1]. Therefore we can find an s E [0,1] such that x(e,(a))= 0.

4.11. REPRESENTATIONS OF G R O U P S

169

Since, for every t E [0,1], we have

every matrix &(a),a E I', corresponds to a Mlibius transformation in the covering group of a compact and oriented surface of genus > 1. Every such Mobius transformation is hyperbolic, i.e., the trace of such a matrix has to have absolute value 2 2. We have therefore reached a contradiction proving the lemma. Let u s now take a closer look at the projection 7O(FlSL(2,R)) + We make first the following observation: Representations 8 j : I’ + SL(2,R), j = 1 , 2 , project to the same representation I' + P S L ( 2 , R ) if and only if there exists a function ( on I', taking the values f l , such that & ( y ) = ((y)&(y) for all y E .’I Using this remark and the reasoning of the above lemma we prove:

P ( r ,PSL(2, R ) ) .

Theorem 4.11.3 Let C be a compact and oriented surface of genus > 1. Assume that I? C S L ( 2 , R ) is isomorphic to the fundamental group of the surface C with some base point. The space 7,"(I', SL(2, R ) ) has finitely many components each of which is homeomorphic to the Teichmziller space of C . Proof. The natural projection

Il : q ( r ,SL(2, R ) )

-+

q(I', PSL(2, R ) )

is continuous and open. It is clear that I$(I',SL(2,R)) has only finitely many components. It suffices to show that no components of

q(LSL(2, R ) ) contain two different points which project onto the same point of

q(LP W 2 , R ) ) . To that end let

e :’I

-+

P S L ( 2 , R ) represent a point of

Let 8 : I' -+ S L ( 2 , R ) be a representation that projects to 8. If 8’ : r + S L ( 2 , R ) is another representation projecting also to 8 then there exists a function ( on I' taking the values f l such that 8'(y) = ((y)8(y) holds for all y E r. Let us show that there is no continuous path, in 7$(I', SL(2, R)), joining two different points of 7,"(I SL(2, ’, R ) ) , which both project to the

CHAPTER 4. TEICHMULLERSPACES

170

same point in q ( F , PSL(2, R)). Assume that one these points corresponds to 8 and the other to 8’. To that end, consider the function t satisfying 8’ = 6 8. Since 8 # 8’ there exists an y E r such that ((7) = -1. Considering the function

-

t , : 73r, SL(2, R))

+

R, PI I-+ x

and repeating the argument of Lemma 2.1 we conclude that [8] and [O’] do not belong to the same component of T ( r ,SL2(R)).

4.12

The algebraic structure

The Teichmiiller space of a compact and oriented surface C can be given the local structure of an affine real algebraic variety. We will use here traces of elements of SL2(R) to parametrize the Teichmiiller space and to embed it into an affine space RM in such a way that it becomes a component of an affine variety. The construction that we review here is due to Heinz Helling

(WI

Let l? be again the fundamental group of a compact and oriented surface C. Let 0 1 , . . .,ambe any set of generators for r satisfying certain defining relations. Let I,,, denote the ordered set of all ordered j-tupels

of natural numbers with 1 5 y I( = K ( m )be the set

1, IRezl < This group serves also to show that the half-plane H in Lemma 3.10.2 is the largest possible, since a closer analysis shows that a fundamental domain for the elliptic modular group is the interior of

+

i}.

W = {w E UI - 1/2 < Rew 5 1/2, IwI

1 1 and IwI > 1 for Rew 5 0).

A detailed proof for this fact can be found in the monograph of Siegel [87, Theorem 3, section 91 or in that of Norman Alling [7, Chapter 91. We have observed above that the elliptic modular function furnishes a bijection j : M' C. The complex plane can, on the other hand, be compactified to the Riemann sphere C by adding the point at the infinity. That is also the usual compactification for M'. It is necessary to try to understand this compactification also in concrete geometrical terms. What kind of a Riemann surface corresponds to the point at the infinity? Here the situation is different from the general case. The reason really lies on the fact that the natural metric of a torus has curvature 0. Therefore each class of holomorphically homeomorphic torii has representatives whose area and/or diameter are either arbitrarily large or arbitrarily small. In the case of Rieiiiann surfaces of genus > 1 the situation is completely different. By the Gaua-Bonnet theorem, the area of a Ftiemann surface of genus g, g > 1, is always 4n(g - l ) , i.e., the area depends only on the genus and not on the particular complex structure. In order to understand properly the degeneration of torii it is, therefore, necessary to take particular representatives of each class of holomorphically isomorphic torii. One possibility is, for instance, to consider only those torii which have diameter < 2, i.e., which are of the form C / L , ,1.1 < 1.

-

5.3. MODULI SPACES O F G E N U S 1 SURFACES

183

This can be done, since for each T' we can always find an element A of the elliptic modular group such that IA(T')I < 1. Considering these torii one can interpret the point at the infinity of M' geometrically as a circle. The situation here is the following. As a sequence C / L , approaches the infinity of the moduli space M ' , then we may suppose that the parameters T, are chosen in such a way that T,, + 0, i.e., that the radius of injectivity of C / L , + 0 as n -+ 00. This means that the limiting surface is a circle, and, as we let a point in M' approach the infinity, then the corresponding Riemann surfaces collaps everywhere. The limiting object is not anymore a surface but a manifold of real dimension 1. The case of Riemann surfaces of genus > 1 is completely different as we will see in the proceeding sections. In the case of these hyperbolic Riemann surfaces collapsing can happen only in such a way that certain simple closed curves get replaced by points. The limiting structure is still a surface but it has finitely many simple singularities. Let us next consider non-classical surfaces of genus 1. Such a (topological) surface C has, by the definition, the torus T' as its complex douT' such ble, and there exists an orientation reversing involution u : T' that C = T ' / ( u ) . Such an involution induces a self-mapping u* of the Teichmiiller space T' of the torus. Assume that u1 and 6 2 are different orientation reversing involutions of the torus. They induce self-mappings a; and uz of the Teichmiiller space T ' , which usually are also different. Since 6 1 and 6 2 are both orientation reversing, u1 o u2 is orientation preserving. Hence the induced mapping (u1 o u2)* = a; o u; belongs to the mapping class group r'. Since M' = T’/I’’ we conclude that both orientation reversing involutions u1 and u2 induce the same self-mapping of the moduli space M ' . This induced mapping is also an involution. In order to find out what is the induced mapping at the level of the moduli space, consider the mapping .--)

u* : u

+

u, a*(.)

= -7.

A straightforward computation shows that this mapping u* : T' + T' is induced by an orientation reversing involution u : T' + T' for which T * / ( u ) is an annulus. By Theorem 4.5.1 (on page 149), the Teichmiiller space of an annulus can then be identified with the fixed-point set of the involution u* :T' + T', i.e., with the imaginary axis. Let 7-

:

c

--$

c,.(z)

= z.

On basis of the construction and the commutative diagram (5.3) we have: j o u* = T o j .

184

CHAPTER 5. MODULI SPACES

Figure 5.1: The unshacled part of the complex plane is a fundamental domain of the elliptic modular group. Those points of the fundamental domain for which j : T' -+ C is real are indicated by a thick line. Let 7r : T' -+ M' denote the projection. We conclude now that be the above considerations, ~(2’2.) C R. Here Ti. is the fixed-point set of the involution o* : T' + T'. In order to see what kind of torii may have real moduli, let

( M 1 ) =~ { ~[ X I E M’ 1 X has a n antiholomorphic involution}. The set ( M ' ) N c consists of ordinary isomorphism classes of Riemann surfaces which have a symmetry. It is best t o do the computations a t the level of the the Teichmiiller space. In figure 5.1 those points of a fundamental domain of the elliptic modular group, for which the fuiiction j takes real values, are indicated by a thick line. The elliptic modular function j takes real values also at many other points which do not belong to the fundamental domain of Figure 5.1. Detailed computations related t o computing real values of the elliptic modular function can be found in the monograph of Norman Alling. [7, Chapter 91 We will skip the details here. Direct computations show that { z I Im z = +} and { z 1 1.1 = 1, Im z > 0} are both models for the Teichmuller space of the Mobius band while the

5.4. S T A B L E R I E M A N N SURFACES

185

imaginary axis is a model for Teichmuller spaces of the annulus and the Klein bottle. This implies that

( M ' ) N c = R. The above considerations are only technical and do not have any hidden difficulties. They can be interpreted in terms of algebraic geometry. Classical compact genus 1 Riemann surfaces are simply complex algebraic curves, i.e., they can be ernbedded into a projective space in which they are defined by a finite number of polynomial equations satisfying certain regularity conditions. The moduli space M' is, therefore, the moduli space of genus 1 complex algebraic curves. Complex algebraic curves, which have an antiholomorphic involution, are, on the other hand, isomorphic t o curves defined by real polynomials, i.e., they are real algebraic curves. The set ( M ' ) N c consists of complex isomorphism classes of real algebraic curves. By the above observations we have now: Theorem 5.3.3 The set of real points of the moduli space M ' , M'(R), consists of complex isomorphism classes of real algebraic curves. Theorem 5.3.3 is actually well known in algebraic geometry and easy to prove. For j # 1728, the j-invariant of the curve y 2 = 4 2 - a x - a , a = 27j/(j-l728),equals this given j . For j = 1728, take the curve y2 = 4x3--5.

5.4

Stable Riemann surfaces

A main goal of this monograph is to understand, from a geometric point of view, the compactification given by Muinford and Bailey (cf. [65])for the moduli space of smooth compact Riemann surfaces. To that end we have to extend our considerations to Rieiiiann surfaces that are allowed to have singular points. In the proceeding definitions we follow the presentation of Bers ([la]).

Definition 5.4.1 A surface with nodes C is a Huusdorflspace whose every point has a neighborhood hoineomorphic either to the open disk i n the complex plane or to N = ( ( z , ~ f) C 2 J z w= 0, J z J< 1, J w 0 then 10, - O,?I < 6.

Here we have used the notation f*(P)to indicate the pants decomposition of Y whose pairs of pants are inverse images of pairs of pants of the decomposition P of X , and l j and 6, are the Fenchel-Nielsen coordinates of X with respect to P . The sets U , , S ( [ x ] )form a basis for the topology of

mg.

Theorem 5.6.1 The set of isomorphism classes of smooth genus g Riemann surfaces, M g , is dense in the moduli space of stable compact genus g Riemann surfaces, MY, which is a connected Hausdorfl space.

xg.

Proof. By the construction of the topology, Mg is dense in This is clear, since each u,,S([x]), E > 0, neighborhood of a stable Riemann surface contains smooth Riemann surfaces. Proof for the fact that this topology is a Hausdorff topology is rather obvious and is left to the reader. The connectedness of MYfollows from that of Mg (Lemma 5.2.1). The following result follows directly from the definition and from Lemma 5.5.2.


194

Lemma 5.6.2 Consider the Teichmdler space T g of compact genus g Riemann surfaces, g > l, equipped with the usual topology. The projection ~g 2”is a continuous mapping. --$

5.7

Compactness theorem

In this section we show that the moduli space MY, g > 1, is compact. In view of Theorem 5.6.1 it is enough to show the following:

Theorem 5.7.1 The closure of Mg in M g is compact. ) an infinite sequence of points of Mg c M .It Proof. Let ( [ ( C , X ) ] be suffices t o show that there exists a subsequence ([(C, X,,)]) that converges in Zg. By Theorem 3.19.1 on page 131 we can always find a decomposition P, of ( C , X n ) into pairs of pants by simple closed geodesic curves of length < 219. This is the key point in our argument. There are only finitely many topologically different decompositions of the surface C into pairs of pants. Therefore we may - by passing to a subsequence - assume that there is a fixed decomposition P of X into pairs of pants and orientation preserving homeomorphisms f n : C -+ C such that fn(P,) = P for each n. Let Y, be that complex structure of C for which the mapping

is holomorphic.

Let 17, S,: j = 1 , 2 , . . .,39 - 3, be the Fenchel-Nielsen coordinates of Y, with respect to the pants decomposition P. Let a1,...,a3~-3be the decomposing curves of the pants decomposition P. Then l? = la,(Y,) is the length of the simple closed geodesic curve freely homorotopic to the closed curve aj on the Ftiemann surface Y,. By the choice of the pants decompositions Pn of the Riemann surfaces (C, X n ) and by the definition of the complex structure Y, we have now, for M = 219: 1; = la,(Y,) < M

for each j and n. We have also 0 l , is a connected Hausdorfl space.

P m f . By Theorem 5.9.1, h4k is dense in

Mk. Hence it suffices to show

that the closure of h4; in Mk is connected. We will achieve this in two steps: 0

0

We show first that the union of the closures of components V ( g ,n, 1) is connected by showing that the closure of each V ( g ,n,l),n > 0, intersects the closure of V ( g ,0,l). Next we show that the closures of V ( g ,n, 0) and V ( g ,n - 1 , l ) always intersect for all possible values of n.

Consider now a coxnponent V ( g ,n, l ) , n > 0. To show that the closures of V ( g ,n, 1) and V ( g ,0 , l ) intersect we construct a point [(X,u)]that lies in the closures of both of them. To that end, let XObe the Riemann sphere punctured at the g + 1 points 0 , 1, 2 , . . ., g . Let Xi be the complex conjugate of XO,i.e., if z is a local coordinate of X o , then Z is that of Xi. Let X be the stable Riemann surface obtained by identifying the punctures 0, 1 , . ., g of XO with those of X,. The identity mapping induces an antiholomorphic mapping XO -+ Xi which, in turn, induces a symmetry o : X + X. We conclude that (X, CT) is a stable genus g Riemann surface. The stable Riemann surface X has two parts, they correspond to the punctured spheres XO and Xi. For notational convenience, call these parts X1 and X2. The symmetry u maps X Ionto Xz. Next we show that the point [(X, u)]lies in the closures of V ( g ,0 , l ) and V ( g ,n,1). To that, let 6 > 0 and S > 0 be arbitrary. We constructs a point of V ( g ,0 , l ) and another one of V ( g ,n, 1) which both lie in the U,,s([(X,u ) ] ) neighborhood of the point [(X, u ) ] . To that end, take first g - 1 pairs of pants PI, P2, . . .,Pg-l such that all boundary geodesics have length € 1 2 . identify a boundary component of Pj with a boundary coxnponent of Pj+l for all indices j = 1 , . . , g - 2 . Do the identifications in such a manner that the corresponding base points always agree. In this way one obtains a Riemann surface Y’ with ( g - 1) 2 = g 1 boundary components. Let Y'' be the complex conjugate of Y’. As above, we form the Riemann surface Y by identifying the boundary points of Y' with the corresponding

.

.

+

+


200

points of Y". The identity mapping of Y' induces then an antiholomorphic Y. The mapping Y' -+ Y" which, in turn, induces a symmetry u : Y Riemann surface Y is, of course, smooth. The point [(Yu)]lies in the component V ( g ,g 1,O). Next we deform the complex structure of Y in such a way that we get the desired points of V(g,O,1) and of V ( g , n ,1). This is a delicate part of the argument and is based on the considerations of Section 3.7 (starting on page 93). Let 011, a2,. . ., ag+lbe the closed geodesic curves of Y corresponding to the boundary geodesics of the Riemann surface Y'. Assume that the curves aj are oriented in such a way that they are positively oriented3 as boundary curves of Y'. Let k = g 1- n, and let Yk be the Riemann surface obtained from the Riemann surface Y be performing left Dehn twists along the curves ~ 1 ,...,crk. For the definition of the left Dehn twist see page 95. Recall now the construction of the mapping fa given in Definition 3.7.1 on page 94. Considering carefully the construction one concludes that

+

+

the homotopy class of the mapping fa, o fa, 0. an antiholoinorphic involution T : Yk Yh.

-

- 0 fak

ou contains

--f

It is immediate4, by the definition of fa,, that the symmetric Riemann surface (Y,T ) is of type ( 9 , n, l),i.e., that [ ( Y T, ) ] E V ( g ,n, 1). It follows, also directly from the definitions, that [(Y, T)] E UL,6([(X, u)]).This argument shows that the point [(X,u)]lies in the closure of V ( g ,n, 1). Replacing, in the above construction, k by g 1 one shows that [(X,u ) ] lies also in the closure of V ( g ,0 , l ) . This completes the proof of the first part. Observe that in the above we actually showed that the point [(X, u)]lies in the closures of the components V ( g , n ,l), V(g,O,1 ) and V ( g , g l , O ) , i.e., that also the closures of V(y, g t 1,O) and V ( g ,0 , l ) intersect. In order to show that the closure of V ( g ,0 , l ) intersects the closures of all the components V ( y , n , O ) 0, < n 5 g 1, n = g 1 (mod 2), we repeat the above argument by replacing the punctured sphere Y' by a Riemann surface of genus ( g 1 - n ) / 2 from which n disks (with disjoint closures) have been removed. (Here we repeat the construction of Section 3.7 starting on page 93.) Details are, word by word, same as in the above considerations. This completes the proof.

+

+

+

+

+

-.

The main point in the above proof is that the involutions T x fal o f a , o u and u are not homotopic to each other. The composition T O

o

- 0fa,

3Any orientation will do. For technical convenience we have to fix some orientation. 'Details here are exactly the same as in the considerations following Definition 3.7.1 on page 94.

5.10. COMPACTNESS OF THE REAL MODULI SPACE

201

-

is, of course, homotopic to the product of Dehn twists fal o f a 2 o .. o f a , . As a self mapping of the smooth surface C this product of Dehn twists is not homotopic to the identity and is of infinite degree. If one deforms the surface C in such a way that the closed curves al,...,crk get pinched t o nodes, then the mapping fal o fa2 o -.. o fa, induces a self-mapping of this deformed surface which is homotopic to the identity. It only turns the nodes around.

5.10

Compactness of the real moduli space

We use Theorem 5.9.1 and study the moduli space Mk as the union of the closures of the parts: V ( g ,n, k) = {[(X, u)]E M”,X

smooth, n ( u ) = n, k(u) = k}.

We will first show that the closure of each V ( g ,n , k) is compact in M k . The proof of the compactness is an extension of the arguments presented in Section 5.7 for the classical case and relies on Theorem 3.19.1 (on page 131). Observe especially, that the following result follows immediately from the definitions.

Lemma 5.10.1 The projection

is continuous.

We will have to deal with several different symmetries of the surface C. To make this distinction clear we write sometimes (C, X,u) to denote the symmetric Riemann surface X = ( C , X ) together with the symmetry u : ( C , X ) + ( C , X ) which is then an antiholomorphic involution.

Theorem 5.10.2 The closure of V ( g ,n, k) in M

i

is compact.

Proof. Let u : C -+ C be an orientation reversing involution, k ( a ) = k and n ( u ) = n . Let ([(C,X,,a)]),be an infinite sequence of points of V ( g , n , k ) in

xk. It suffices to show that there exists a subsequence ([(C, X,,

7

u>1)

that converges in %&. We shall, at various stages of the proof pass from a sequence t o its subsequence. To keep the notation as simple as possible we use the same


202

notation for a sequence and its suitable subsequence when there is no danger of confusion. Use first Theorem 3.19.1 (page 131) to find, for each index n, a decomposition P, of C into pairs of pants in such a way that each decomposing curve a3 of each pants decomposition Pn has length < 219 on ( C , X , ) . By Theorem 3.19.1 we can furthermore choose these pants decompositions Pn in such a way that, for each n, .(Pn) = Pn. There are only finitely many topologically different decompositions of the surface C into pairs of pants. Therefore we may - by passing to a subsequence - assume that there is a fixed decomposition P of C into pairs of pants and orientation preserving homeomorphisms f n : C + C such that f,(P,) = P for each n. Let Y, be that complex structure of C for which the mapping fn : (C, X,) -, (C, Y,) is holomorphic. Let T, = f, o c o f;'. Each mapping T, : (C, Y,,) + (C, Y,) is then an antiholomorphic involution. Furthermore, T,(P) = P for each R . Up to Dehn twists around the decomposing curves of the pants decomposition P there are only finitely many different involutions 7, that map P onto itself. Therefore we may assume - by passing again to a subsequence and choosing the mappings f n in a suitable way - that all the involutions r, agree. Let T = T,, for all values of n. After these choices we have a. decomposition P of C into pairs of pants and representatives (C,Y,,.) for the points [ ( C , X , , c ) ] in Mk such that the following holds:

1. T ( P )= P . 2. Each

T

: (C, Y,)

+

(C, Y,) is an antiholomorphic involution.

3. Each decomposing curve crj, j = 1 , 2 , . . .,3g - 3, of the pants decomposition P is of length < 219 on the hyperbolic Riemann surface (C, Yn). Let l?, e?, j = 1,2,. ..,3g - 3, be the Fenchel-Nielsen coordinates of Y, with respect to the pants decomposition P. Then by the above mentioned property 3, we have e? < 219 for each j and n. Also we have 0 5 0: < 27r. Therefore - by passing again to a subsequence - we may assume that all the sequences l?;,e!, lg, . . . and O f , B j , Og,. . . converge. Let l , = limndm ly and Oj = liin,-,m 03, j = 1 , 2 , . . .,3g - 3. We deform next the surface C in the following fashion. If l j = 0 then we replace the decomposing curve aj of the pants decomposition P by a node. Do that for each j with lj = 0. That construction yields a stable surface C* of genus 9. The identity mapping C -+ C induces a strong deformation f : C + C*.

5.10. COMPACTNESS OF T H E REAL MODULI SPACE

203

By the above mentioned properties 1 and 2 of the representative (C,Yn, 7) we deduce that r induces an orientation reversing involution T : C ’ +C ’ . Also the decomposition P of C into pairs of pants gives a similar decomposition P of C* into pairs of pants. On C* define a complex structure Y' by the Fenchel-Nielsen coordinates e j and 0, with respect to the decomposition P of C' into pairs of pants. That complex structure Y * is uniquely defined up to Dehn twists around those decomposing curves aj of P that are not nodes and up to deformations by mappings homotopic to the identity mapping in each pair in each pair of pants belonging to P. The diagram

commutes. Here f : (C,Y,) + ( C * , Y * )is a strong deformation and each mapping T : (C,Y,) + (C,Y,) is an antiholomorphic involution mapping P onto it self. Consider next the decomposition P on C into pairs of pants as an miented decomposition. That gives an orientation also for the decomposition P of C* into pairs of pants. The mapping r maps each pair of pants belonging to P onto some other (or possibly the same) pair of pants in P. Therefore, as oriented decoinpositions of C into pairs of pants r ( P ) and P are different. Since each mapping T : (C, Y,) (C, Y,) is antiholomorphic, the FenchelNielsen coordinates of Y, with respect to the oriented pants decomposition T ( P )are obtained by a permutation of the coordinates e? and the coordinates 07 and by possibly replacing some of the coordinates 0: by the coordinates 27~- 0:. Observe that the coordinates of Y, with respect to r ( P ) are always obtained from the coordinates of Y, with respect to P by the same continuous transformation which does not depend on n. We conclude, by continuity, that also the coordinates of Y' with respect to the oriented pants decomposition r ( P ) are obtained by the same transformation from the coordinates of Y* with respect to the pants decomposition P. Then we conclude that by deforming the complex structure Y' by additional Dehn twists around some of the decomposing curves aj that are not nodes of C*, and by deforming Y * in each pair of pants of P by a mapping homotopic to the identity mapping, we may actually suppose that the mapping r : ( u * ,Y') + (C*, Y * )is an antiholomorphic involution. Details here are easy but tedious. Therefore (C", Y ' ) defines a point in Mk. --f


2 04

From the construction it follows then immediately that

[(C,Xn)I in

--$

“C*,Y*)l as n

-+

0

Mk.This proves the theorem. By Theorem 5.9.1,Mk is the closure of M i . The moduli space M g is, R

on the other hand, the union of finitely many components V ( g ,n,k). Since, by Theorem 5.10.2,each one of them has compact closure in M&,also the closure of M& in n/r& is compact. That remark proves the main result of this Section:

Theorem 5.10.3 The real moduli space of symmetric genus g Riemann surfaces, ZL, is compact for g > 1.0 Observe that in the above we have considered only the case of hyperbolic Riemann surfaces, i.e., we have assumed everywhere that g > 1. The results hold also in the case of genus 1 or 0 Riemann surfaces but have to be shown by the methods of Section 5.3. The case of symmetric genus 0 Riemann surfaces is trivial, since, in that case, the moduli space reduces to a set of two points.

5.11

Review on results concerning the analytic structure of moduli spaces of compact Riemann surfaces

The focus of this monograph has been on that part of the theory of Teichmuller spaces that can be derived studying multipliers of Mobius transformations. That leads to parametrizations of Teichmuller spaces by geodesic length functions. These are real analytic parameters, while the Teichmiiller space of genus g compact Riemann surfaces is actually a 39 - 3 dimensional complex manifold. Teichmuller spaces of non-classical surfaces are only r e d analytic (and not complex) manifolds, but the complex structure of the clas-. sical Teichmuller spaces plays an important r6le in this non-classical theory. For that reason we provide, in this Section, a review of this complex theory. Everything here will be presented without proofs, but exact references will be given. During recent years three excellent monographs, which treat the complex analytic theory of Teichmuller spaces, have appeared. They were written by Fredrick P. Gardiner ([32]),Olli Lehto ([60]) and by Subhasis Nag ([SS]). Especially the above cited monograph of Gardiner and that of Nag give a good introduction the modern parts of this complex theory.

5.11. ANALYTIC STRUCTURE OF MODULI SPACES

205

On page 148 we have described the complex structure of the Teichmuller space of compact smooth genus g Riemann surface, g > 1. Main result concerning the complex structure is Theorem 4.4.2 (on page 148), which states that the Teichmuller space Tg of compact genus g , g > 1, Riemann surfaces is a complex manifold of complex dimension 39 - 3. By the constructions related to the complex structure it is easy to see that elements of the modular group are holomorphic automorphisms of the Teichmiiller space of compact genus g Riemann surfaces. Royden has shown (in [71] and in [72]), furthermore, that the modular group is, for g > 2, the full group of holomorphic automorphisms of Tg. Using this result one can draw some conclusions concerning the structure of the moduli space. To that end we need the following theorem of Henri Cartan [24, Th6orBme 11 :

Theorem 5.11.1 Let X be a complex manifold and G a properly discontinuous group of holomorphic automorphisms of X . The quotient X / G is a normal complex space. The normality is a condition concerning the singularities of the moduli space. This result simply means that the singularities are not too bad. By Theorem 4.10.5 (on page 163), the modular group I'g acts properly discontinuously on the Teichmuller space Tg. Since elements of the modular group are holomorphic automorphisms of Tg, Theorem 5.11.1 implies:

Theorem 5.11.2 The moduli space M g , g > 1, of smooth compact genus g Riemann surfaces is u normal complex space. By means of the Geometric Invariant Theory algebraic geometers, David Mumford, Walter L. Baily, David Gieseker, Finn Knudsen and others, have shown a stronger result ([50][511 ,[52]$351, [66] ,[67], [68], [38]):

Theorem 5.11.3 The moduli space, M g , of smooth compact genus g Riemann surfaces, g > l, is a quasiprojective algebraic variety. Its compactification Mg is u projective algebmic variety. This is one of the most important results in the theory of Riemann surfaces and algebraic curves. After the success of the Geometric Invariant Theory in compactifying the moduli space and studying its structure a natural problem was to prove the same results using Teichmuller theory. That this is possible was announced as early as in 1973 ([ll]). The details turned out to be difficult. Concerning the compactification it was, for many years, possible to prove, using exclusively Teichmuller theory, only that the moduli space of stable

206


Riemann surfaces of a given genus is a compact Hausdorff space (Theorem 5.11.2 above). Ideas to this proof are due to Lipman Bers and appeared already in [ll]. For another account of these facts see also the notes of Joe Harris ([37]). To prove, using arguments of analytic geometry, that the moduli space of stable Riemann surfaces is a projective variety, took much longer. First geometric proof for this fact is due to Scott Wolpert ([loll). Recently Frank Herrlich ([41]) has been able to complete the original arguments of Bers ([I 11) completing this line of investigations. In view of the rich complex analytic theory of Teichmiiller spaces and moduli spaces of classical Riemann surfaces, it is natural to ask what can be said about the possible analytic structure of the Teichmiiller spaces and the moduli spaces of non-classical Riemann surfaces. Let C be a non-classical topological surface and C" its complex double. Then C" has an orientation reversing involution u : Cc -+ Cc such that C = X c / ( u ) .Recall that by Theorem 4.5.1 (on page 149) we may identify the Teichmuller space T ( C )of the non-classical compact surface C with the fixed-point set T(C'),. of the induced involution u* : T(C")+ T ( C C ) .This involution is antiholomorphic. Locally the involution u* can be modelled as the complex conjugation in C3g-3. Since the fixed-point set of such an antiholomorphic involution is always a real analytic manifold we have:

Theorem 5.11.4 The Teichmiiller space of a non-classical compact surface C of genus g , g > 1 , is a real analytic manifold of real dimension 39 - 3 Next question is what can be said about the moduli space M i of nonclassical Riemann surfaces of genus g . We consider here only the general caseg > 1 . In [79, Theorems 2.1 and 9.31 (see also [75]) Mika Seppda and Robert Silhol have shown the following result:

Theorem 5.11.5 Components of the moduli space M&, of non-classical Riemann surfaces of genus g , g > 1, are semi-algebraic varieties and imducible real analytic spaces. Let x : Mg + M gbe the mapping that forgets the real structure. We observed above that the moduli space MYis a complex projective variety and a normal complex space. Another natural problem is to study the structure of T ( Zin~Zg. ) To that end, recall that the moduli space MYof genus g compact stable Riemann surfaces can be viewed as the moduli space of stable complex algebraic curves. Likewise, M i is the moduli space of stable real algebraic curves. Let T :MY+ MYbe the mapping that takes the isomorphism class

5.11. ANALYTIC STRUCTURE OF MODULI SPACES

207

of a complex algebraic curve onto that of its complex conjugate. Using techniques described in [67] it is possible to embedd M ginto a complex projective space PN(C) in such a way that the diagram

Mg,

+

lfd

Mk

+

I.

MY

MY

L*

-

PN(C) lcompl. conj. PyC)

(5.10)

is commutative. This means, in particular, that

where

(M“)T denotes the

fixed-point set of the involution r : MY-+ M , MYand we have interpreted M g as a

Mg(R) is the set of real points of subset of PN(C).

- Regarding diagram (5.10) Clifford Earle showed in [26] that Mk # Mg(R). Another proof and a direct construction for this fact has been given by Goro Shimura in [84]. Earle’s proof uses Teichmiiller theory while Shimura gives an explicit family of polynomials that define complex algebraic curves, with real moduli, that are not real curves. This observation has been a motivation in the study of the properties of the moduli space

Mk.

In [78] Mika Seppala has shown:

Theorem 5.11.6 Assume that g > 3 . The image n ( M k ) of Mg in MYis a connected semialgebraic variety and the quasi-regular real part of MY.

Recall that the quasi-regular real part, as defined by Aldo Andreotti and Per Holm in [9], consists of those points ofMY(R)where the local dimension of x g ( R )is maximal. This result has recently been extended to the case g = 3 by Robert Silhol (see [SS]). In that pepr Silhol has also shown that the result cannot be extended ot the case g = 2. Silhol has also extensively studied moduli problems of real abelian varieties and obtained important results for them. Observe that Theorem 5.10.2 implies that n(ML) is a compact subset of MY. We can characterize T(%&) as the subset of MY consisting of isomorphism classes of those Riemann surfaces X on which Zz = Z/2Z acts via an antiholomorphic involution. This formulation leads to the following generalization. Let G be a finite group. Set M $ = {[XI I G c IsomX}. (5.11) Here IsomX denotes the isometry group of X. Theorem 4.1 of my other lecture in this volume means that it is not possible to use the construction of Sections 1 - 3 to study this set % : in the general case.

208


In the general case MG is not compact. It is an interesting problem to characterize the closure of ML in Zg. We conclude this Section by the following remarks. The modern generalizations of these arguments to moduli problems of real algebraic geometry were started by Clifford Earle ([26]) and by Goro Shimura ([84]). Earlier these moduli problems were extensively studied by Felix Klein (cf. e.g. [46]). These investigations were continued independently by S. Natanzon ([70]), Mika Seppala (see e.g. [77] and [78]), Robert Silhol (see e.g. [88])and jointly by Seppala and Silhol ([79]).

Appendix A

Hyperbolic metric and Mobius groups A.l

Length and area elements

The Mobius transformation g(2) =

az+b bz+ii

7

with 1aI2 - lb12 = 1 maps the unit disk D = { z E C I 1 1 1< 1) conformally onto itself. For 21,z2 E D let wj = g(zj), j = 1,2. From (A.l) we get

and

1 -m1w2 =

1-7122

(671

+

.)(5~2

+ E)

Hence

Letting z1 approach 2 2 (A.2) becomes

-Id4

1 - 1212

-

Id4 1 - lW12’

This shows that the Riemannian metric whose element of length is

209

210 APPENDIX A . HYPERBOLIC METRIC A N D MOBIUS GROUPS is invariant under conformal self-mappings of the unit disk. In this metric every rectifiable arc y has the length

and every measurable set E has the area

Jl,

(ly:;)2-

The metric defined by the line element (A.3) is called the hyperbolic metric of D . In this metric the shortest arc from 0 to any other point is along the radius. Hence geodesics are euclidean circles orthogonal to dD = { z E C I IzI = 1). Such geodesics are also called h-lines or hyperbolic lines. D together with the hyperbolic metric is called the hyperbolic plane. Let .(z) = 2/(1- 1 ~ 1 ~ ) Then . the Gaussian curvature of the hyperbolic metric is

The constant 2 was added in (A.3) in order t o get a metric of constant curvature -1. Generally speaking any Riemannian metric of curvature -1 is called hyperbolic. The hyperbolic geometry can also be carried over to the upper halfplane U = { z E C I Im z > O}. The element of length that corresponds to (A.3) is

The upper half-plane together with this metric is another model for the hyperbolic plane. Sometimes it is more convenient t o work in U instead of D. Therefore we frequently switch back and forth between these two hyperbolic planes. The hyperbolic distance from 0 to T > 0 is

2dr J, -=log-

1-l-r 1-r'

For the hyperbolic distance d D ( z1, z2) of arbitrary points the formula l+t b ( z 1 , .2) = 1% 1 - 7 where t = Iz1 - z21/11 - 21221. A similar formula with

holds for the hyperbolic metric of the upper half-plane.

z1,z2

E

D we have

(A4

A.2. ISOMETRIES OF T H E HYPERBOLIC METRIC

A.2

21 1

Isometries of the hyperbolic metric

By (A.2) all Mobius-transformations mapping the unit disk onto itself are isometries of the hyperbolic metric. Conversely, let f : D -+ D be an isometry of the hyperbolic metric. We want t o show that f is either a Mobiustransformation of the complex conjugate of a Mobius-transformation. To that end consider, in the stead of f , the mapping

Then F is an isometry as well and it maps the interval [0,1] onto itself keeping the origin fixed. It suffices to show that F is either a Mobiustransformation or the complex conjugate of a Mobius-transformation. Let C, denote the euclidean circle { z E C I 1.1 = r } for 0 5 T 5 1. Observe that C,. is a circle in the hyperbolic metric as well. Since F is an isometry of the hyperbolic metric and since F ( 0 ) = 0, F ( C , ) = C, for each T.

Let reiq be an arbitrary point of C,. Then F maps the arc of C,. with end-points T and z = reiq onto some arc with end-points T and reiqf(4. F being an isometry of the hyperbolic metric these two arcs must have the same hyperbolic length. We conclude that either ' ~ f ( ~ ) = corp c p f ( * ) = -cp for each z = re'q. Since F is continuous we conclude then that either vj(Z) = cp or cpj(z) = -9 for all z = re'q E D . This proves the following result:

Theorem A.2.1 The group of isometric self-mappings of the hyperbolic unit disk D is the group of conformal or anticonformal self-mappings of D .

A.3

Geometry of the hyperbolic metric

A ray from a poizat z in the hyperbolic plane is an infinite arc on a h-line

with an end-point z. Angles in the hyperbolic plane can be defined just like angles in the Euclidean geometry; they are bounded by two hyperbolic rays starting from one point. Angles can be measured just like angles in the Euclidean geometry. It follows that hyperbolic angles are euclidean angles. The following properties of the hyperbolic metric follow easily from the above: a

Hyperbolic circles are Euclidean circles but their centers do not usually coincide.

a

There is a unique 11-line through any two distinct points of the hyperbolic plane.

212 APPENDIX A . HYPERBOLIC METRIC AND MOBIUS GROUPS

Figure A.1: Points whose distance from 1 is at most 0

0

0

T.

Two distinct h-lines intersect in at most one point of the hyperbolic plane. Given any two h--lines It and s ( 4 ) = 12.

12,

there is an isometry g such that

Given any h-line 1 and any point z , there is a unique h-line through z and orthogonal to 1.

The distance of a point z froin a h-line 1 is measured along the unique h-line through z perpendicular to 1. The imaginary axes 1 is a h-line in the upper half-plane. Computing by (A.6) we conclude that the set of points of U which lie a t a given distance T from 1 consists of two euclidean rays starting froin the origin and forming equal angles with 1. Let cp be that angle. One computes further that T + 00 as cp + 7~12. Let d"(z,l) denote the distance of the point z from the h-line 1. We conclude that for an arbitrary h-line 1 in U the set {z

E

u I dv(z,l) < .I

is the crescent in figure (A.1). Let D, be the disk of hyperbolic radius T with center at the origin of the hyperbolic unit disk D. Then D, is also an euclidean,disk. Its euclidean radius R equals R = arctanh(r/2) by (A.6). Straightforward integration then gives the following result.

Theorem A.3.1 The hyperbolic area of a hyperbolic disk of mdius 2lr(coshr - 1). The hyperbolic length of a hyperbolic circle of mdius 27~sinhT .

T T

is is

Observe that the area of the hyperbolic disk D , can also be expressed as 4asinh2 ( ~ / 2 ) .

A.4. MATRIX GROUPS

A.4

213

Matrix groups

Denote by M ( n , R) the real vector space of real n x n matrices all

a12

..-

an1

an2

* * *

A= ann

The linear mapping R” -+ Rn defined by A is bijective if and only if det A 0. In this case A is nowsingular and the inverse matrix A-’ satisfies

#

where Sjj is the Kronecker symbol. Denote by GL(n, R) (“General Linear Group”) the group of real non-singular n x n-matrices. The complex vector space of complex n x n-matrices A = (a;j) is denoted by M ( n , C).Also in this case A is non-singular if and only if det A # 0. Let GL(n, C) denote the group of complex non-singular n x n-matrices. In a natural way, G L ( n , R ) is a subgroup of GL(n, C). If A, B E M ( t a , C ) ,then det(AB) = det A det B = det(BA). If A E GL(n, C), then det A det A-’ = 1 and det A-’ = (det A)-’. cially, det(ABA-I) = det(AA-’B) = det B.

Espe-

It follows that matrices A E GL(n, C) with det A = 1 constitute a group. This subgroup of GL(n, C) is denoted by SL(n, C) (“Special Linear Group”). Similarly, SL(n,R) = {A E G L ( n , R ) I d e t A = l} is a subgroup of GL(n, R). Moreover, SL(n, R) C SL(n, C). Example. Let A = E GL(2, C). Then we have det A = a d - bc #

(

0. If A-’ =

(;

)

$),then

aa+by ap+b6 c a + dy c p + d S

214 APPENDIX A. HYPERBOLIC METRIC AND MOBIUS GROUPS and hence

(

)

Let I = E SL(2,C) and -I = ( - I ) A = - A for all A E M(2, C ) . Denote

( -,,’

!l

).

Then A ( - I ) =

N = { I , -I}. Since I 2 = I, ( - I ) 2 = I and I ( - I ) = (-1)1 = - I , N is a subgroup of SL(2,C). Moreover, since AIA-’ = I and A(-I)A-’ = - I for all A E SL(2, C), N is a normal subgroup of SL(2, C ) and of SL(2, R). The quotient groups SL(2, C ) / N = PSL(2, C), SL(2, R)/N = PSL(2, R)

(“Projective Special Linear Group”)

consist of equivalence classes ( A ,- A ) , A E SL(2, C). The vector space M ( n , C ) can be identified with the euclidean space c n x n . Hence M ( n , C) has a natural topology. The E-neighbourhood of the matrix A = ( a i j ) consists of the matrices B = ( b ; j ) for which

The same topology is obtained if E-neighbourhoods are defined by the metric

If Ak = ( a i j ( k ) )and A = ( u i j ) , then

Hence the mapping det : M ( n , C ) + C , A

H

detA, is continuous. For

A = ( a i j ) E M ( n , C ) , let

Then also the mapping tr : M ( n , C ) -, C is continuous. We have seen that GL(n, R), SL(n, R), GL(n, C ) and SL(n, C ) are both groups and topological spaces. Moreover, A,’

+ A-’

and An& -+ A B

A.4. MATRIX GROUPS

215

whenever An + A and B n + B . Generally, let G be an arbitrary group. Suppose that G is also a topological space. If the mappings

x

(X,Y)

H

z-’

(G-tG)

Z Y (G x G

+

G)

are continuous, then G is a topological group. Suppose that every point x E G has a countable base of neighbourhoods. Then the mappings H z-l and (x,y) H z y are continuous if z ; ’ + z-’ and znyn + z y whenever x , + x and yn + y. Hence all subgroups of GL(n,C) and SL(n,C) are topological groups. For any y E G , the space G x {y} has a natural topology whose open sets are of the form A x {y}, A c G open. Then the mapping z H (x, y) is a homeomorphism G G x {y}, and the mapping (x,y) H z y is a continuous surjection G x {y} -+ G. The composition --f

5

ZY

is a continuous surjection G -+ G whose inverse x Hence we have

H

xy-’ is continuous.

Theorem A.4.1 Let G be a topological group. Then for every y E S, the mappings z I+ z y and x H yx are homeomorphisms G + G .

Definition A.4.1 A topological group G is discrete if the topology of G is discrete, i.e., if all subsets of G are open. Theorem A.4.2 Let G be a topological group. Suppose there exists g E G such that { g } is open. Then G is discrete.

Proof. Let y E G. Since the mapping x set { g ( g - ' y ) } = {y} is open. o

H

x(g-’y)

is homeomorphic, the

Let H c G be a normal subgroup. We state without a proof that the quotient group G / H becomes a topological group if it is equipped with the topology co-induced by the projection map G + G / H . In the following, we shall consider subgroups of M ( 2 , C). Let

P : SL(2, C)-+ PSL(2, C) by the projection for which P ( A ) = P ( - A ) . Then P is locally injective. If PSL(2,C) is equipped with the topology co-induced by P and if A E PSL(2,C) and P-'(A) = { A l , A z } , then A has a neighbourhood U such

216 APPENDIX A . HYPERBOLIC M E T R I C AND MOBIUS GROUPS

that P-'(V) consists of disjoint neighbourhoods of A1 and Az. Moreover, PSL(2, C) and SL(2, C) and all their subgroups are topological groups. By the above definition, a topological group G is discrete if G contains no accumulation points, i.e., if the conditions

A,,A€G

A,+A;

always imply that there exists an no such that A , = A for all n 2 no. We show next that a discrete group G C SL(2,C) has in fact no accumulation points in SL(2, C).

Theorem A.4.3 Let G c SL(2,C) be a group. Then G is discrete if and only if the conditions

A,

+

A and A , E G, A E M ( 2 , C )

always imply that there exists an no such that A , = A for all n 2 no. Proof. Suppose that G is discrete. Let {A', A 2 , . ..} c G and A E M(2, C ) such that A , + A . Since

1 = clet A,

+

det A ,

we have det A = 1, i.e., A E SL(2, C). Since SL(2, C) is a topological group,

A;'

i

A-' and T, = A,'A,+1

+ A-'A

=I.

Since T,, E G , there exists an no such that n 2 no

+-T, = I.

Then A,, = An,+' = Ano+2 = . . .. Since A , + A , we have A, = A for all n 2 no. Conversely, suppose that the conditions of the theorem hold. Let { A l , A 2 ,..., A } c G such that A , + A . Then there exist an no such that A , = A for all n >-no. Hence G is discrete. Let G c PSL(2, C) be a group and let A, E G. Suppose that we can choose A,, E P-'(A,&)such that A, + A E M(2, C ) . Since A, E SL(2, C ) , we have A E SL(2, C ) by the preceding proof. Moreover,

A;'

+

A-' and T, = A,'A,+l

+

I.

Suppose that the matrices A , are distinct. Then the sequence T, contains infinitely many distinct elements since otherwise T, + I would imply that T, = I and A , = A,+1 for n 1 no. Since P ( T , ) E G and P(T,) P(I) E G, the group G would have P ( 1 ) as a accumulation point. Then G would not be discrete. Conversely, if no sequence A , -+ A of the above type can be found, the group G is discrete. Hence Theorem A.4.3 has the following corollaries: --.)

A.4. MATRIX GROUPS

217

Corollary A.4.4 A group G C PSL(2,C) is discrete if and only if the conditions A , + A, P ( A , ) E G and A E M(2, C) always imply that there exists a n no such that A , = A for all n 2 no. Corollary A.4.5 Suppose that G C PSL(2, C) is not discrete. Then there exists a n infinite sequence A , E SL(2, C ) , n = 1, 2, ..., of distinct elements such that P ( A , ) E G and A , I. --f

Consider discreteness of a group G c SL(2, C). By Corollary A.4.2, it suffices t o find a point A E G such that the set { A } is open. For instance, it suffices to show that { I } is open, i.e., inf{ IIX - Ill

I X

E G,

X # I } > 0.

In other words, it suffices to show that the conditions A , + I , A , E G, imply that A , = I for all n 2 no. Theorem A.4.6 A group G

is finite for all t

c SL(2, C) is discrete i f and only if the set

> 0.

Proof. If the set in question is finite for all t > 0, then G has no accumulation points and G is discrete. Suppose conversely that the set { A E G IlAll 5 t } is infinite for some t > 0. Let A l , A2, ..., E G be distinct elements such that llAnll 5 t. If

I

( %; 2 ), and, -

= 1, then lanl I t , lbnl 5 t , lcnl 5 t and ldnl 5 t. Hence the sequence a, has a convergent subsequence ani, the sequence bni has a convergent subsequence and so on. Therefore, we may assume that

A, =

an C,

(

),

+

a,

bn

y, d ,

-+

-+

P,

6.

f then A , + A . Since the matrices A , are distinct, G is not discrete by Theorem A.4.3.

If A =

(:

Example. The modular group consists of all matrices A = : ) € SL(2,R) for which a , b , c , d are integers. By Theorem A.4.6 the modular group is discrete. The Picard group consists of all matrices A = E S L ( ~ , C for ) which a, b, c and d are complex integers, i.e., of the form m ni where m and n are integers. Also the Picard group is discrete.

(

:)

+

218 APPENDIX A . HYPERBOLlC METRIC A N D MOBlUS

A.5

GROUPS

Representation of groups

Let X be a non-empty set and let F ( X ) be the set of all bijections f : X --f X. Then F ( X ) is a group whose group operation is the composition of transformations. The transformation group F ( X ) is said t o act in X . Let G be a subgroup of F ( X ) . A homomorphism cp : G + GL(2, C ) is a representation of G . If y is injective, the representation is faithful. In this case we have an isomorphism y :G

--f

cp(G) C GL(2,C).

We shall also consider representations y : G + SL(2,C) and cp : G + PSL(2, C). Let X = C be the extended complex plane and let G1 be the group of all translations S, : z H z w, w E C. Then

+

If we interpret S, as a linear fractional transformation of the form (az + b)/(cz

+ d ) , then

Sw(z)= z

+ w = lo-. zz ++ 1w'

Hence we obtain a mapping cp : G1 + SL(2, C ) for which

Clearly, y(S;') = (y(S,))-' and y(S,, 0 S,,) = y(S,,)y(S,,). Hence cp is a representation of G1 in SL(2, C). Moreover, y is faithful since

i.e., kercp = {id}. Secondly, let G2 be the group of all stretchings Vk : z Since v k ( z ) = kZ =

&+O

o . z + 1/&'

we define now y : G2 -+ SL(2,R) by setting

H

hz, h

> 0.

A.5. REPRESENTATION O F GROUPS

219

Again, cp(V;') = (cp(Vk))-' and v(V&0 Vk2)= c p ( V ~ , ) c ~ ( vIt , ~f)o. ~ o w s that cp is a faithful representation of G2 in SL(2, R). Thirdly, let GJbe the group of all rotations 1O,

since AUA-’ = U by Theorem A.7.3. Now

@.

if and only if p 2 = Since bc < 0, we have -c/& > 0. Hence p > 0 is well-defined. We still have to show that p does not depend on the transformation V E G , V ( 0 )# 0 . Suppose that G contains the transformations

and

Let V2 E G such that Vz(0) # 0. Then by (A.12)

Corollary A.7.7 If G C M is a finite group then hGh-l c

hEM.

K

for some

Proof. A finite group G C M is purely elliptic since the powers gn, n = 0,5 1 , f2,. . .are all distinct transformations whenever g # id is non-elliptic.

0

238 APPENDIX A . HYPERBOLIC METRIC AND MOBIUS

A.8

GROUPS

Discrete groups of Mobius transformations

+

+

+

SL(2,C) is a topological group with the norm IlAll = lbI2 lcI2 ld12)'/2, A = ad - bc = 1. A subgroup I' c SL(2, C) is discrete, if all subsets of r are open in I', i.e., if for every A E I' there exists E > 0 such that r n { B E s ~ ( 2 , c I) I I A- B I 0. D Let G be a cyclic group. We may suppose that G is generated by a standard transformation mk. Since

Ilrnill = (lkl" t lkl-n)’/2 and

Ilm7ll = (2 + n 2 11 / 2

if k

#

1

9

it follows from Theorem A.8.1 that G is discrete if g is parabolic or loxodromic. If r n k is elliptic, i.e., k # lkl = 1, G is discrete if and only if g is of finite order. There exist also non-cyclic Abelian discrete groups G c M :

G is generated by translations z

t+

z +ol and z

++

z+w2, Im

2 # 0,

G is conjugate to the quadratic group, G is generated by an elliptic h f M of finite order and a loxodromic g E M such that Fg = Fh.

A.8. DISCRETE GROUPS OF MOBIUS TRANSFORMATIONS

239

Theorem A.8.2 Let g , h E M . If g is loxodromic and Fg f l Fh contains exactly one point, then the group G generated by g and h is not discrete.

Proof. We may suppose that F'flFh = (m}. Replacing g by g-' if necessary we have

Then g - n ( h ( g n ( z ) ) ) = az

+ a-"b and

1.(

Since 11g-" o h o g'*)l Theorem A.8.1.

+-

the group G is not discrete by

Theorem A.8.3 Let G C A4 be a group. If there exists un infinite sequence ( g n ) C G such thut gn(w) + w

for w = 1, 0, 00,

then G is not discrete.

Proof. Let the representations

be chosen such that Red,, 2 0. Since

we have

240 APPENDIX A . HYPERBOLIC METRIC A N D MdBIUS GROUPS

Since Re d , 2 0, also dn

-+

1. Then

and it follows that a, -+ 1. On the other hand,

and

bn = dngn(0)

+ 0.

Since the sequence ( g , ) contains infinitely many distinct elements, the group

G is not discrete by Theorem A.8.1. Let G c M be a group. A disk D is G-invariant if g(D) = D for all g E G . If D is G-invariant, the group G is said to act in D. Denote by U the upper half-plane { z I Ini z > 0). Lemma A.8.4 Let g(4 =

at

+b

5

7

ad - bc = 1 .

Then g ( U ) = U if and only if a , b, c and d are real.

Proof.If a , b, c , d are real, g maps the extended real axis onto itself. Then g ( U ) is either U or the lower half-plane. Since Img(i) = 1 / ( c 2 d 2 ) , we have g(U) = U . Conversely suppose that g ( U ) = U . Denote

+

1 -c~-d g ( t ) - az b *

h ( z ) = --

+

Then h ( U ) = U and ( - c ) b - ( - d ) a = a d - bc = 1 . If c = 0, then a # 0. Hence, replacing g by la if necessary, we may suppose that c # 0. Since g ( U ) = U , g is not strictly loxodromic. Then a d is real by Theorem A.6.12. Since g ( m ) = a / c and g-'(oo) = - d / c are red, also ( a + d ) / c is real. Hence c , a and d are real. Finally, it follows from ad-bc = 1 that b is real.

+

Theorem A.8.5 If a non-Abelian purely hyperbolic group G acts in then G is discrete.

U,

A.8. DISCRETE GROUPS OF MOBIUS TRANSFORMATIONS

241

Proof. We consider G as a subgroup of PSL(2, C)and use matrix notation for the transformations of G. If A E G, A = ad - bc = 1, then a, b, c, and d are real by Lemma A.8.4. We may normalize by conjugation such that G contains a transformation A with 0 and 00 as fixed points. Then

( 1),

by Theorem A.6.11. Suppose that G is not discrete. Then by Corollary A.4.4 we can find a sequence Vn E G of distinct elements such that V, -+ I . We show that there exists an integer no such that

for all n

D,

> no. To that

end, let

= AC,A-'C,'

with Vn -

(zi

f;",

), a,,&

- b,,c, = 1. Then (cf. Theorem A.7.2)

Since SL(2, C)is a topological group and V, -, I , we have Cn + I . Hence trC, --t 2 and b,cn 4 0. Then andn = 1 b,cn -+ 1 and and, > 0 for all sufficiently large values of R. Since G is purely hyperbolic, we have tr2C, 2 4 and tr2D, 2 4. Since b,c,(A - X-')2 + 0, there exists an nb such that

+

bncn 5 0 for n > nb.

Similarly anbnc,d, 2 0 for sufficiently large values of n. Since andn large values of n, we can find an ng such that

b , ~ , 2 0 for n > n;. Hence

> 0 for

242 APPENDIX A. HYPERBOLIC METRIC A N D MOBIUS GROUPS Moreover, trCn = 2 for n > no. Since G contains no parabolic elements, we have Cn = I for n > no. Since G contains no elliptic elements, we have by Theorem A.7.3, FA = Fv,, and bn = cn = 0 for n > no. Since G is non-Abelian, G contains an element B for which FB nFA = 8 (Theorem A.7.4). Hence

.=(7

a

6),

Crs-p7=1,

Ia+SI>2,

pzo,

7#0.

then X, + I. Then it follows similarly as above that

for sufficiently large values of n. Since p i # 1 # p i 2 , we have ap = 76 = 0. Suppose that a = 0. From a6 - ,By = 1 it follows that y # 0. Hence 6 = 0 which contradicts the condition la Sl > 2. If a # 0, then p = 0 which is also impossible. Hence there exists no sequence V, -+ I and G is discrete.

+

Theorem A.8.5 can be complemented as follows:

Theorem A.8.6 If a group G C M is purely hyperbolic, there exists a G-invariant disk D .

Proof.Let g, h

E G\{id}. By Theorem A.7.2, either Fg = Fh or FgrlFh = 8. If Fg = Fh for all g and h, any disk D whose boundary contains Fg is G-invari ant. Suppose that there exist g, h E G \ {id} such that Fg n Fh = 8. We may suppose that Fg = {O,oo}. Let f E G and

Since f and g o f are hyperbolic, the traces

and t2

= tr(g o f ) = a u + S/u

A.8. DISCRETE GROUPS OF MOBlUS TRANSFORMATIONS

243

are real by Theorem A.6.12. Then it follows that a and 6 are real. Let

Then a and d are real and ( a of h:

x = Y

Since x

#

00

=

+ d)2 > 4.

Consider the fixed points x and y

a-dtda

-

d

2c 2c

d

w

# y , we have c # 0, and

is real. It follows that the fixed points of g and h lie on the same line through the origin. If we conjugate by a rotation z H ei'z, the matrix of g is not changed but the fixed points x and y of h can be mapped e.g. on the r e d ' axis. Hence we may suppose that g and h map U onto itself. Then by Lemma A.8.4, a , b, c and d are real. I f f E G, f = ,* f a6 - /37 = 1, then (I! and S are real and

(

),

+

+

Since f o h E G, also a a pc and yb Sd are real. Since bc # 0, it follows that also p and 7 are real. then f ( U ) = U by Lemma A.8.4. o Combining Theorems A.8.5 and A.8.6 we obtain a quite useful result on purely hyperbolic groups.

Theorem A.8.7 A non-Abeliarz purely hyperbolic group is discrete.


Appendix B

Traces of matrices B.l Trace functions In order t o complete certain arguments of Section 4.12 we need to consider traces of matrices. Here we review the results of Heinz Helling (1391).

Definition B . l . l Let 0, satisfying for any a,P E

r

be a group. A function t : l? + R,not identically

r, is called a trace function.

Let B : ’I + SL2(R) be an injective homomorphisms. A computation shows that t = trB is a trace function on I?. Here trB is the usual trace of a matrix. It is a straightforward verification to show that a trace function has the following properties:

1. Let

E

be the identity element of the group l?. Then t ( ~=) 2.

2. t(a-1) = t ( 0 ) .

3. t(@) = @a). 4. Permuting the arguinents of the function

q a , P, 7) = N a P 7 ) - t ( 4 t ( P 7 )- t(PP(7 Q)-t(r)t(@ + t ( Q ) t ( P ) t ( T ) its value gets multiplied by the sign of the permutation. 5. For a,@E l?, n E 2,t(a"p) is a polynomial in t ( a ) ,t(p), and t ( @ ) with rational coefficients. 245

APPENDIX B. TRACES OF MATRICES

246 6 . Let w be a word in

a1,.

..,a, E r and t a trace function.

Then t ( w )

is a polynomial of the values

7. Let a, P, 7, 6 E .’I The value of a trace function at aP76 satisfies

In particular, t(apr6)is a polynomial with rational coefficients of the values of the trace function t at products of at most three elements of { a , P , 7761. 8. Let a, ,8 E r. For a trace function t , let kt(a,,B)= a b c - a 2 - b 2 - c 2 + 4 where a = t ( a ) ,b = t(/3), c = t(ap). Let K(a,P;

u, v) = a"pa-U,B-u,u, 2) = f l ,

be a commutator of a and D. Then

Let F be a topological group. We use the notation Aut(F) for the group of continuous automorphisms of F.

Theorem B.l.l ([39,Proposition 11) Let I' be a group, t : I’ 4 R a trace function such that kt : r x r -+ R is not identically 0. We suppose, furthermore, that k t ( a , P )5 0 ijlt(a)l 5 2 . There ezists a representation 8 : r -+ SL2(R) such that t = tr8. If and 82 are both such representations, then there ezists an g E Aut(SL2(R)) such that 82 = g o 81.

Proof. Let a, /3 E r be elements for which k t ( a , @ ) # 0. Let A, B E SL2(R) be matrices for which t ( a )= t r A,

t ( p ) = t r B , t ( @ ) = trAB.

The condition k t ( a , P ) # 0 guarantees that a matrix by the numbers tr C, tr AC, tr BC, t r ABC.

C is uniquely defined

B.1. TRACE FUNCTIONS For every 7 E

247

r we form the equations

The condition k t ( a , p ) # 0 implies that the above system of equations has a unique solution B(y) E SLz(R). It is a straightforward verification to check that ’I 3 SLz(R), y H B(y), is a homomorphism.o Let t : I' -, R be a function. The topology induced by t on G is the weakest topology of G for which G is a topological group and t : G + R continuous.

Corollary B.1.2 Assume that r is not commutative, t is a trace function on r satisfying k t ( a , p ) 5 0 whenever It(a)l 5 2. If the topology induced by t on r is Hausdorff, then there exists a faithful representation 8 : r + SLz(R) with t = t r 8. If the induced topology is discrete and the virtual cohomological dimension of r is 2, then SLz(R>/B(I') is compact. One has to find a and ,8 E r such that k t ( a , p ) # 0. Assume that a and p do not commute. Since the topology induced by t on I' is Hausdorff, there exists an element y E r such that t ( a p a - ' P - ' y ) # t(7). If this holds already f o r 7 = E , then k t ( a , P ) = 2-t(apa-'p-l) # 0. Ift(apa-'P-') = 2, then a computation shows that kt(apa-'P-',7) # 0. Therefore we can apply the preceding theorem to find a representation 8 : G + SL2(R) such that t ( y ) = tr8(y) for all y E r. This homomorphism is injective because of the assumption concerning the topology induced by t on r. The last assertion about the compactness of SLz(R)/e(I') is an immediate application of the results in [83, Corollary to Proposition 181.


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Geometry of Riemann surfaces and Teichmüller spaces

Geometry of Riemann Surfaces and Teichmüller Spaces

Geometry of Riemann surfaces and Teichmüller spaces