Geometry and Topology of Surfaces 9783985470006

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Z U R I C H L E CT U R E S I N A DVA N C E D M AT H E M AT I C S

Sebastian Baader

Geometry and ­Topology of ­Surfaces

Zurich Lectures in Advanced Mathematics Edited by Habib Ammari, Alexander Gorodnik (Managing Editor), Urs Lang (Managing Editor), Michael Struwe Mathematics in Zurich has a long and distinguished tradition, in which the writing of lecture notes volumes and research monographs plays a prominent part. The Zurich Lectures in Advanced Mathematics series aims to make some of these publications better known to a wider audience. The series has three main ­constituents: lecture notes on advanced topics given by internationally renowned experts, in particular lecture notes of “­Nachdiplomvorlesungen”, organized jointly by the Department of Mathematics and the Institute for Research in Mathematics (FIM) at ETH, graduate text books designed for the joint graduate program in Mathematics of the ETH and the University of Zürich, as well as contributions from researchers in residence. Moderately priced, concise and lively in style, the volumes of this series will appeal to researchers and students alike, who seek an informed introduction to important areas of current research. Previously published in this series: Yakov B. Pesin, Lectures on partial hyperbolicity and stable ergodicity Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry Sergei B. Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions Pavel Etingof, Calogero–Moser systems and representation theory Guus Balkema and Paul Embrechts, High Risk Scenarios and Extremes – A geometric approach Demetrios Christodoulou, Mathematical Problems of General Relativity I Camillo De Lellis, Rectifiable Sets, Densities and Tangent Measures Paul Seidel, Fukaya Categories and Picard–Lefschetz Theory Alexander H. W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles Michael Farber, Invitation to Topological Robotics Alexander Barvinok, Integer Points in Polyhedra Christian Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis Shmuel Onn, Nonlinear Discrete Optimization – An Algorithmic Theory Kenji Nakanishi and Wilhelm Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations Erwan Faou, Geometric Numerical Integration and Schrödinger Equations Alain-Sol Sznitman, Topics in Occupation Times and Gaussian Free Fields François Labourie, Lectures on Representations of Surface Groups Isabelle Gallagher, Laure Saint-Raymond and Benjamin Texier, From Newton to Boltzmann: Hard Spheres and Short-range Potentials Robert J. Marsh, Lecture Notes on Cluster Algebras Emmanuel Hebey, Compactness and Stability for Nonlinear Elliptic Equations Sylvia Serfaty, Coulomb Gases and Ginzburg–Landau Vortices Alessio Figalli, The Monge–Ampère Equation and Its Applications Walter Schachermayer, Asymptotic Theory of Transaction Costs Anne Thomas, Geometric and Topological Aspects of Coxeter Groups and Buildings Todd Fisher and Boris Hasseblatt, Hyperbolic Flows

Published with the support of the Huber-Kudlich-Stiftung, Zürich

Sebastian Baader

Geometry and ­Topology of ­Surfaces

Author: Sebastian Baader Mathematisches Institut Universität Bern Sidlerstrasse 5 3012 Bern, Switzerland E-mail: [email protected]

2020 Mathematics Subject Classification: 57K20 Key words: mapping class group, Dehn twist, pseudo-Anosov diffeomorphism, hyperbolic surface, Basmajian identity, measured foliation, Teichmüller theory, Thurston classification

ISBN 978-3-98547-000-6 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. Published by EMS Press, an imprint of the European Mathematical Society – EMS – Publishing House GmbH Institut für Mathematik Technische Universität Berlin Straße des 17. Juni 136 10623 Berlin, Germany https://ems.press © 2021 European Mathematical Society Typeset using the author’s tex files: WisSat Publishing + Consulting GmbH, Fürstenwalde, Germany Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

For Alessandra Iozzi

Acknowledgments

First and foremost, I would like to thank the Department of Mathematics of ETH Zurich and the director of the Institute for Mathematical Research, Tristan Rivière, for inviting me to give a graduate course at ETH on a topic of my choice. I had a fantastic audience that kept me motivated throughout these lectures, very demanding at times, spotting errors, correcting and simplifying proofs. Special thanks go to Alessandra Iozzi, Emmanuel Kowalski and Merlin Incerti-Medici from the front row, Marc Burger, Peter Feller and Alessandro Sisto from the back row, and Matthew Cordes, Yannick Krifka, Jean-Claude Picaud from the middle row. Jean-Claude read the entire manuscript, helped fixing it in several places, served me wine and cheese, and got me excited about stretch factors of pseudo-Anosov maps. Altogether, his support with this project was enormous. All the figures in these notes, forty in number, were produced by Raphael Appenzeller. I thank him for the enormous amount of time and effort he put into this. In fact, even the notes are not entirely new: in 2011, I gave a similar course at EPFL. I thank Livio Liechti for sharing his excellent notes with me, as well as for proving interesting results that found their way into this manuscript. The chapter on the Basmajian identity may come as a surprise, and would not exist without Hugo Parlier, who explained it to me last summer in Les Diablerets. Thank you for that, Hugo. These notes are dedicated to Alessandra Iozzi, in honor of her 60th birthday, and in appreciation of her support and inspiration. Thank you so much, Alessandra!

Contents Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Étude on the flat case: S 1  S 1 . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2

Plane hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3

Simple closed geodesics and systoles . . . . . . . . . . . . . . . . . . . . . . .

13

4

Fenchel–Nielsen coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

5

Dehn twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

6

Normal generators for mapping class groups . . . . . . . . . . . . . . . . . . .

31

7

Measured foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

8

The .9g

9/-theorem for measured foliations . . . . . . . . . . . . . . . . . .

45

9

Compactification of Teichmüller space . . . . . . . . . . . . . . . . . . . . . .

49

10 Classification of mapping classes . . . . . . . . . . . . . . . . . . . . . . . . .

55

11 Perron–Frobenius theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

12 Pseudo-Anosov maps with small stretch factor . . . . . . . . . . . . . . . . . .

65

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

ix

Introduction

The main purpose of this series of lectures is to get acquainted with hyperbolic structures on surfaces and the classification of mapping classes. Soon enough, it will transpire that it is all about understanding curves on surfaces. The basic object of study is a surface †, always smooth, orientable, compact and connected. We define S.†/ to be the set of noncontractible (smoothly) embedded circles in †, up to isotopy. This set is countable, since the fundamental group of † is. We refer to elements of S.†/ as simple closed curves. Every diffeomorphism of † acts on S.†/. This action is invariant under continuous deformations of a diffeomorphism. Therefore, the mapping class group, defined as the set of isotopy classes of orientation-preserving diffeomorphisms, MCG.†/ D DiffC .†/=Diff0 .†/; acts on the set S.†/, as well. As we will see, the isotopy type of a diffeomorphism is determined by its action on simple closed curves. Fact 1. For a closed surface † of genus at least three, MCG.†/ < Perm.S.†//: Surfaces of negative Euler characteristic carry hyperbolic structures. We will study the space T .†/ of hyperbolic structures on a surface †, up to isometries isotopic to the identity map. A hyperbolic metric attributes a unique length to each isotopy class of closed curves, in turn defines an element of RS.†/ >0 : Fact 2. The natural map lW T .†/ ! RS.†/ >0 is a proper embedding. with the natural projection to the real The composition of the map lW T .†/ ! RS.†/ >0 projective space W RS.†/ ! P .RS.†/ / >0 is still an embedding, but not a proper one. The limit points, whose union we denote by PMF .†/, correspond to a certain kind of degenerate length functions called (projective) measured foliations on the surface †. The homeomorphism types of the spaces T .†/ and its compactification in P .RS.†/ / are T .†/ ' R6g

6

and T .†/ [ PMF .†/ ' D 6g where g  2 denotes the genus of † and D important statement.

6g 6

6

;

is a closed ball. This leads us to the third

Fact 3. The mapping class group MCG.†/ acts on T .†/ [ PMF .†/ by homeomorphisms. In particular, by Brouwer’s fixed point theorem, each (class of) diffeomorphism f W † ! † has a fixed point in T .†/ [ PMF .†/.

2

Introduction

There are two fundamentally different possibilities: (1) f has a fixed point in T .†/, in which case it is periodic, up to isotopy. (2) f has a fixed point in PMF .†/. Then f is either reducible or of pseudo-Anosov type, two features we will shortly explain. It is the latter type which is the most interesting, and also generic, for that matter. Pseudo-Anosov maps f W † ! † come with a real stretch factor .f / > 1. We will put a special emphasis on studying pseudo-Anosov maps with small stretch factors. On our way, we will come across numerous beautiful results, old and new. Here is a somewhat random sample of theorems we will prove. Theorem (Dehn [14]). The mapping class groups MCG.†/ are generated by Dehn twists along simple closed curves. Theorem (Basmajian [4]). The length of the boundary curve of a hyperbolic surface † with one geodesic boundary component @† is determined by its ortho-spectrum, in the following way:   X 1 l.@†/ D 2 arcsinh : sinh.l. //

?@†

A similar identity involving only simple closed curves was later obtained by McShane [43]. This paved the way to Mirzakhani’s famous counting result for the number of simple closed geodesics on a fixed hyperbolic surface [46]. The next result by Lanier and Margalit implies that pseudo-Anosov maps with a small stretch factor normally generate the mapping class group. Theorem (Lanier and Margalit [35]). Let N < MCG.†/ be a non-trivial normal subgroup p and let f 2 N be a pseudo-Anosov map with stretch factor .f /. Then .f /  2. The structure of this manuscript is derived from the graduate course I gave at ETH in the autumn 2018. There are twelve chapters, each corresponding to a two hours lecture. Of the many literature sources that inspired these notes, three stand out: Braids, links, and mapping class groups by Birman [7], the Primer on mapping class groups by Farb and Margalit [19], and Travaux de Thurston sur les surfaces by Fathi, Laudenbach and Poénaru [20], of which there exists an English translation by Kim and Margalit [21]. I would also like to highlight the excellent books by Benedetti and Petronio [5], Casson and Bleiler [11], Ivanov [31], as well as Thurston’s original work [60, 61]. The reader only interested in hyperbolic surfaces and mapping classes can stop reading in the middle, that is at the end of Chapter 6, and possibly add Chapters 10 and 12. Cutting it further down, I would recommend the hyperbolic part, Chapters 1–4, or the Dehn twist part, Chapters 5–6. For the most minimalistic approach, just read Chapter 2.

1 Étude on the flat case: S 1  S 1

The torus S 1  S 1 has an abelian fundamental group, generated by the two oriented curves S 1  1 and 1  S 1 : 1 .S 1  S 1 ; .1; 1// ' Z2 : The action of an orientation-preserving diffeomorphism f W S 1  S 1 ! S 1  S 1 on the first homology group H1 .S 1  S 1 ; Z/ is given by a matrix f 2 SL.2; Z/. Every matrix A 2 SL.2; Z/ arises in this way, since the linear map AW R2 ! R2 descends to a diffeomorphism on the torus fA W R2 =Z2 ! R2 =Z2 , whose induced action in homology tautologically coincides with the action of A on Z2 . Proposition 1. We have MCG.S 1  S 1 / ' SL.2; Z/: Proof. Let f W S 1 S 1 ! S 1 S 1 be an orientation-preserving diffeomorphism. By a version of Fact 1, which we are going to prove later on, the isotopy type of f is determined by the action of f on the set of oriented simple closed curves on S 1  S 1 , which can be seen as a subset of 1 .S 1  S 1 ; .1; 1// ' H1 .S 1  S 1 ; Z/, and certainly contains the two generating curves S 1  1 and 1  S 1 . Therefore, the isotopy type of f is determined by the homological action f 2 SL.2; Z/.  Remark 1. The set of oriented simple closed curves on S 1  S 1 coincides with the orbit of the element .1; 0/ 2 Z2 under the action of SL.2; Z/, i.e. the set of pairs of coprime integers .p; q/ 2 Z2 . Indeed, let c  S 1 S 1 be an oriented simple closed curve. The complement S 1 S 1 nc is diffeomorphic to an annulus, since c is not contractible. In particular, there exists an orientation-preserving diffeomorphism between the annulus S 1 S 1 nS 1 1 and S 1 S 1 nc. After a suitable isotopy near the boundary, this can be extended to a diffeomorphism f W S 1  S 1 ! S 1  S 1 with f .S 1  1/ D c. Then the element of 1 .S 1  S 1 ; .1; 1// represented by the curve c is simply the first column of the matrix f 2 SL.2; Z/, i.e. a pair of coprime integers. For surfaces of higher genus, most closed geodesics have self-intersection points. However, we will only make use of the simple ones. The coarse classification of mapping classes into periodic, reducible and pseudoAnosov ones mentioned in the introduction admits a particularly easy interpretation in the case of SL.2; Z/, in terms of the trace. Let A .t / D t 2

tr.A/t C 1

4

1 Étude on the flat case: S 1  S 1

be the characteristic polynomial of a matrix A 2 SL.2; Z/. There are three possibilities for the eigenvalues of A: (1) tr.A/ 2 ¹0; 1; 1º. Then A is periodic of order 4, 6, or 3, respectively. (2) tr.A/ 2 ¹˙2º. Then A is a unipotent matrix, up to sign. If, in addition, A ¤ ˙Id, then A has precisely one rational invariant line. The corresponding mapping class fixes has an invariant simple closed curve and is called reducible. (3) jtr.A/j > 2. Then A has two reciprocal real eigenvalues ; 1 . In this case, the real plane admits two invariant foliations by parallel lines of irrational slope, which are stretched by  and 1 , respectively. These foliations carry over to invariant foliations on the quotient torus. In dynamics, such a map is called Anosov or hyperbolic. The definition of pseudo-Anosov maps for surfaces of genus g  2 is slightly more involved; one has to allow foliations with certain types of singularities. They also come with a real stretch factor  > 1, which turns out to be the spectral radius of a positive matrix A 2 SL.n; Z/, where n is bounded above by a linear expression in the genus g. At this point, we would like to mention an elementary open question about the smallest possible spectral radius of matrices, depending on the size n. Recall that the spectral radius .A/ 2 R of a matrix A 2 SL.n; Z/ is the largest modulus among its complex eigenvalues. We define n D inf¹.A/ j A 2 SL.n; Z/ with .A/ > 1º: The following question is due to Schinzel and Zassenhaus [55]; a proof was recently announced in [15]. Question 1. Does there exist a constant c > 0 so that for all n 2 N, log.n /  We note that the order of magnitude An 2 SL.2n; Z/: 0 0 0 B0 0 B B1 0 B B An D B0 1 B :: :: B: : B @0 0 0 0

c ‹ n

1 n

is realized by the following family of matrices 1


0 0 2 1


0 0 1 1 C C


0 0 0 0 C C


0 0 0 0 C C: : : : : C :: : :: :: :: :: C C


1 0 0 0 A


0 1 0 0

Indeed, the n-th power of An is 0

2 1 0 B1 1 0 B B .An /n D B0 0 2 B0 0 1 @ :: :: : :

0 0 1 1

1





C C C C; C A :: :

5

Geometry and Topology of Surfaces

so

s .An / D

n

p 3C 5 2

and

p   1 3C 5 log..An // D log : n 2 We will return to stretch factors of pseudo-Anosov maps in more detail towards the end of these lectures. The above question is preceded by the well-known Lehmer problem on the Mahler measure of polynomials [36]. For convenience, we define the Mahler measure m.A/ of a matrix A 2 SL.n; Z/ to be the product of absolute values of all eigenvalues of A that lie outside the unit circle S 1  C. Question 2. Is there a constant d > 0 so that for all A 2 SL.n; Z/, m.A/  1 C d ‹ This is indeed stronger than Question 1, since m.A/  .A/n , thus log..A// 

1 log.m.A//; n

in turn

1 log.1 C d /: n Lehmer’s question comes in many guises, for example in terms of growth of certain groups of matrices [8], and in terms of stretch factors of orientation-reversing mapping classes [39]. We conclude this chapter by a well-known fact about PSL.2; Z/, the quotient of SL.2; Z/ by its center ˙Id. log.n / 

Proposition 2. We have PSL.2; Z/ ' Z=2Z ? Z=3Z: It is this free product structure that makes the group SL.2; Z/ stand out in the family of special linear groups SL.n; Z/, in many ways. The following short proof is inspired by Alperin [1]; a more insightful one can be found in Serre’s course in arithmetic [58]. Proof. The two matrices 

0 AD 1

 1 ; 0

 BD

1 1

 0 ; 1

AB

 1 0

satisfy A2 D B 3 D



1 0

 0 ; 1

 AB D

1 1

1

D AB 2 D

 1 0

 1 ; 1

hence generate PSL.2; Z/ (see for example [3] for the fact that elementary matrices generate SL.n; Z/). Every finite word in A and B involving both letters is conjugate to a word of the form AB ˙1 AB ˙1


AB ˙1

6

1 Étude on the flat case: S 1  S 1

or AB ˙1 AB ˙1


AB ˙1 A: The former represents a matrix with at least three strictly positive entries; the latter has at least three non-zero entries. Therefore, the identity matrix cannot be represented by a non-trivial word in A; B; B 2 . We conclude that PSL.2; Z/ is indeed isomorphic to the free product Z=2Z ? Z=3Z. 

2 Plane hyperbolic geometry

The Uniformization Theorem states that every simply-connected Riemann surface is conformally equivalent to one of the following homogeneous models: the Riemann sphere, the complex plane, or the open unit disc [13]. While the former two have an obvious complete constant curvature metric underlying the conformal structure, this is less obvious for the third one. We will introduce the hyperbolic plane from that point of view, constructing a metric on the upper half-plane H D ¹z 2 C j Im.z/ > 0º, which is conforz i mally equivalent to the open unit disc D 2 via the Cayley transform f .z/ D zCi . Recall that the conformal (or holomorphic) automorphisms of the upper half-plane are precisely the Möbius transformations 'A W H ! H of the following type [53]:   az C b a b 'A .z/ D with A D 2 PSL.2; R/: c d cz C d These are generated by horizontal translations, dilations in zero with a real stretch factor  > 0, and the inversion z 7! z1 . Up to scaling, there is precisely one Riemannian metric on H invariant under these transformations: gH D y12 g, where y D Im.z/ and g is the restriction of the Euclidean metric to the upper half-plane. For symmetry reasons, the positive imaginary axis is a geodesic with respect to gH . In fact, the following calculation shows that the curve .t / D i e t is a unit speed parametrization of iR>0 with .0/ D i : 1 hie t ; i e t iR2 D 1: e 2t Here h
;
iH D y12 h
;
iR2 , where h
;
iR2 denotes the standard scalar product. By taking into account that Möbius transformations 'A with A 2 PSL.2; R/ act simply transitively on the unit tangent bundle of H and preserve angles, that geodesics are determined by an element of the unit tangent bundle; moreover, that orientation-preserving isometries of a geodesically complete surface are determined by the image of an element of the unit tangent bundle, we obtain the following classification of geodesics and orientationpreserving isometries of H . h P .t /; P .t /iH D

Proposition 3. The following statements hold: (1) Geodesics in H are half-lines and half-circles perpendicular to the real axis. (2) IsoC .H / ' PSL.2; R/. Making use of the geodesic .t / D i e t , we compute the hyperbolic distance between two points of the form i a; i b 2 i R>0 with a  b:   b : dH .i a; i b/ D log.b/ log.a/ D log a This is a special case of the following formula.

8

2 Plane hyperbolic geometry

Proposition 4. For all z; w 2 H , cosh.dH .z; w// D 1 C

jz wj2 : 2 Im.z/ Im.w/

Proof. The formula is true for z D i a; w D i b with 0 < a  b:      b 1 b a .b a/2 cosh log D C D1C : a 2 a b 2ab We conclude by observing that both sides of the equation are invariant under horizontal translations, dilations in zero, and the inversion z 7! z1 , which allow to map every pair of points z; w 2 H on the positive imaginary axis.  As an application, we will prove an elegant trigonometric formula for quasi-ideal Lambert quadrilaterals, i.e. quadrilaterals with three right angles and one vertex at infinity, as shown in Figure 2.1. The original interest in Lambert quadrilaterals stems from the parallel postulate, which is equivalent to the statement that all quadrilaterals with three right angles, i.e. Lambert quadrilaterals, are right-angled.

a b

Figure 2.1. Quasi-ideal Lambert quadrilateral.

Proposition 5. Let a; b > 0 be the lengths of the two compact sides of a quasi-ideal Lambert quadrilateral. Then sinh.a/ sinh.b/ D 1: Proof. Let Q  H be a quasi-ideal Lambert quadrilateral and let z 2 H be the vertex opposite to the vertex at infinity. The geodesic line connecting these two vertices dissects the right angle at z into two angles ˛; ˇ < 2 , adjacent to the sides of length b and a, respectively. The angles ˛ and ˇ are called the angles of parallelism of the sides b and a. We claim that sin.ˇ/ cosh.a/ D 1:

Geometry and Topology of Surfaces

9

i a z

1 0 Figure 2.2. Angle of parallelism.

This is most easily seen by letting the side a be the geodesic segment connecting the two points i and z D cos.ˇ/ C i sin.ˇ/, as seen in Figure 2.2. According to Proposition 4, cosh.a/ D cosh.dH .i; z// ji zj2 2 Im.z/ .1 sin.ˇ//2 C cos.ˇ/2 1 D1C D : 2 sin.ˇ/ sin.ˇ/

D1C

Likewise, cosh.b/ D

1 . sin.˛/

Now ˛ C ˇ D

 2

implies sin.ˇ/ D cos.˛/. We compute

.sinh.a/ sinh.b//2 D .cosh.a/2 1/.cosh.b/2 1/    1 1 1 1 D sin.ˇ/2 sin.˛/2    1 sin.˛/2 1 cos.˛/2 D D 1: cos.˛/2 sin.˛/2



Quasi-ideal Lambert quadrilaterals are the key ingredient in the proof of the Basmajian identity. Postponing a formal discussion on hyperbolic surfaces to the next chapter, let us now take for granted that the universal covering space of a closed hyperbolic surface is isometric to the hyperbolic plane H . More generally, the universal covering of a hyperbolic surface † with finitely many geodesic boundary components is isometric to a connected domain in H delimited by a countable union of disjoint geodesics. An easy way of seeing this is by doubling †, i.e. by gluing two copies of † together along their boundary components. This produces a closed surface, whose covering is H . The gluing curves lift to countably many disjoint geodesics.

10

2 Plane hyperbolic geometry

Let † be a hyperbolic surface with precisely one geodesic boundary component @†. Referring again to Chapter 3, we will see that every continuous relative arc ˛W Œ0; 1 ! † is relatively homotopic to a unique length-minimizing arc W Œ0; 1 ! †, which is necessarily geodesic and perpendicular to the boundary @†, i.e. an orthogeodesic. The set of orthogeodesics is therefore countable; the corresponding set of lengths ¹l. /º  R>0 is called the orthospectrum of †. Theorem 1 (Basmajian [4]). We have l.@†/ D

X

 2 arcsinh

?@†

 1 : sinh.l. //

We would like to stress that this identity is independent of the hyperbolic metric on † and, perhaps more remarkably, independent of the genus of †. Q  H , which is a connected domain with countProof. Consider the universal covering † Q be an arc of ably many complete geodesic boundary components. Let aW Œ0; 1 ! @† length l.@†/, so that the orthogonal geodesics ı0 ; ı1 at a.0/; a.1/ converge to @H . This means that the projections of these curves on † do not return to the boundary @†. The Q whose other boundary components are three curves a; ı0 ; ı1 delimit a subdomain of †, countably many half-circles ¹si ºi 2N , see Figure 2.3.

a

1

ai

i

0

Figure 2.3. Half-circles and lifts of orthogeodesics.

For each of these half-circles si , there is a unique minimal geodesic arc i connecting si and a, perpendicular at both ends. All of these are lifts of orthogeodesics on †, and there is precisely one lift for each orthogeodesic. To each orthogeodesic i corresponds a subarc ai  a, so that the orthogonal geodesics at ai .0/; ai .1/ converge to the boundary

Geometry and Topology of Surfaces

11

points of si on @H . By the formula on quasi-ideal Lambert quadrilaterals, we have   l.ai / sinh sinh.l. i // D 1: 2 We conclude by summing up the lengths of all the subarcs ai : l.@†/ D l.a/ X l.ai / D i 2N

 2 arcsinh

 1 sinh.l. i // i 2N   X 1 : D 2 arcsinh sinh.l. // D

X

?@†

A little care is needed with the second equality, since the union of subarcs ai does S not cover the arc a, in general. A limit set argument shows that the complement a n i ai is a Cantor set of measure zero, whose points are starting points of orthogonal geodesics that stay inside † forever. 

3 Simple closed geodesics and systoles

Let † be a standard surface, orientable, compact and connected, possibly with boundary. We recall that a hyperbolic structure on † is an atlas with charts in H (if @† D ;) or HR D ¹z 2 H j Re.z/  0º (if @† ¤ ;), and transition maps (restrictions) of isometries of H , i.e. Möbius transformations 'A with A 2 PSL.2; R/. According to this definition, hyperbolic surfaces have geodesic boundary components. Remark 2. Closed hyperbolic surfaces are Riemann surfaces with constant curvature 1. This follows from Minding’s theorem, stating that Riemann surfaces of constant curvature are locally isometric to a standard model, in this case the hyperbolic plane H ([45], see also [16]). Proposition 6. A standard surface † admits a hyperbolic structure with geodesic boundary, if and only if .†/ < 0. Proof. Let † be a hyperbolic surface. Then the Gauss–Bonnet formula Z 2.†/ D K dA †

and constant curvature K D 1 imply 2.†/ D area.†/ < 0. For the reverse, let † be a standard surface with .†/ D n < 0, where n 2 N. Then † admits a finite collection of embedded circles C  †, so that † n C is a union of n pairs of pants P , i.e. spheres with three disjoint discs removed, as shown in Figure 3.1. This is easily proved by induction on n D j.†/j, observing .P / D 1. We will shortly see that pairs of pants admit a hyperbolic structure with three geodesic boundary components of arbitrarily prescribed lengths a; b; c > 0. A hyperbolic structure on † can then be obtained by gluing together n hyperbolic pairs of pants with matching boundary lengths, for example all equal to one.  Q is Proposition 7. Let † be a closed hyperbolic surface. Then the universal covering † isometric to H . In particular, the fundamental group 1 .†/ is isomorphic to a subgroup of (in fact, a lattice in) PSL.2; R/. As explained in the previous chapter, Proposition 7 implies that the universal covering of a surface with geodesic boundary components is isometric to a connected domain in H delimited by a countable union of geodesic lines. Q is locally isometric to H , complete, and simply Sketch of proof. The universal covering † connected. These three features are enough to construct an isometry to H . For that, choose Q and a linear isometry LW Ti .H / ! Tp .†/ Q between the tangent spaces at a point p 2 † Q Q via the exponential map. i 2 H and p 2 †; extend L to a local isometry 'W H ! †, Q Q is Observe that ' is surjective, since † is complete. Moreover, ' is injective, since †

14

3 Simple closed geodesics and systoles

Figure 3.1. Pair of pants P .

simply connected. The inverse map, called a development map of †, is constructed in detail in [5, Proposition B.1.3.].  Now let  H be a hyperbolic geodesic and fix a positive number  > 0. We define the -neighborhood of as N . / D ¹p 2 H j dH .p; /  º  H: The following statement is the key ingredient in proving that simple closed geodesics are unique in their homotopy class. Lemma 1. The only complete geodesic line contained in N . / is . Proof. We may assume that is the half-line i R>0  H . Let p 2 @N . / be a point at distance  from . Then p 2 @N . / for all  > 0, since all dilations of the form z 7! z are isometries of H . As a consequence, N . / is a cone delimited by two symmetric Euclidean rays emanating from the origin, one of which contains the point p. Therefore, no complete geodesic line other than is contained in N . /.  Proposition 8. Every non-contractible closed curve ˛  † on a hyperbolic surface is homotopic to a unique closed geodesic curve . Moreover, if ˛ is a simple closed curve, then so is . An analogous statement holds for relative arcs, i.e. for images of continuous maps ˛W Œ0; 1 ! † with endpoints on @†; we simply need to replace homotopies by relative homotopies and geodesics by orthogeodesics. Proof. Let ˛W S 1 ! † be a smooth closed curve of length l.˛/, and let ı > 0 be smaller than the injectivity radius of the surface †. Then every pair of points in † of distance at most ı is connected by a unique geodesic arc of length at most ı; we can thus replace

Geometry and Topology of Surfaces

15

Q2

Q1

Figure 3.2. Intersecting neighborhoods of geodesics.

the curve ˛ by a homotopic curve which is piecewise geodesic with N geodesic arcs of length at most ı, where N is chosen to be larger than l.˛/ . Now consider the set of ı curves I.˛; N / consisting of all closed piecewise geodesic curves homotopic to ˛, with N geodesic arcs of length at most ı. There exists a curve 2 I.˛; N / of minimal length, since the set I.˛; N / is compact. Length minimality excludes kinks, hence is a closed geodesic curve. As for the uniqueness, suppose that 1 ; 2  † are closed geodesics, homotopic via Q D H . Then the map h lifts to a continuous map hW Œ0; 1  S 1 ! †. Fix a lift Q1  † Q a homotopy hW Œ0; 1  R ! H between Q1 and a lift Q2 of 2 . Moreover, there exists a constant  > 0 so that the image of hQ is contained in N . Q1 /. Now Lemma 1 implies

Q1 D Q2 , in turn 1 D 2 . Finally, suppose that  † is a closed geodesic with a self-intersection point p 2 †, where at least two segments of meet in an angle strictly between zero and . Then there exist two lifts Q ; Q 0  H which intersect in a point pQ 2 H , with the same angle. Now let ˛  † be a curve homotopic to . As before, there exist two lifts ˛Q  N . Q / and ˛Q 0  N . Q 0 /. By the Mean-Value Theorem, these curves must intersect in a point qQ 2 H (see Figure 3.2), which projects to a self-intersection point q 2 † of the curve ˛.  The set of lengths of closed geodesics on a closed hyperbolic surface † admits a minimum sys.†/ > 0. Curves realizing this minimum, so-called systoles of †, are necessarily simple closed geodesics. As we will see, the length of systoles cannot exceed the logarithm of the genus of a surface. Lemma 2. There exists a constant C > 0 so that for all closed hyperbolic surfaces †g of genus g  2, sys.†g /  2 log.g/ C C: Families of hyperbolic surfaces with increasing genus g and systoles of order of growth log.g/ (more precisely, 43 log.g/) do exist, as shown by Buser and Sarnak [9]. The proof of Lemma 2 is an easy consequence of the fact that the area of a disc in the hyperbolic plane has exponential growth in the radius. Proof. Recall from Chapter 2 that the hyperbolic distance of pairs of points z; w 2 H satisfies jz wj2 cosh.dH .z; w// D 1 C : 2 Im.z/ Im.w/

16

3 Simple closed geodesics and systoles

This transforms to the following formula for the distance dD 2 .z; w/ of pairs of points z i z; w 2 D 2 in the open unit disc model, via the Cayley transform f .z/ D zCi : cosh.dD 2 .z; w// D 1 C

2jz wj2 : jzj2 /.1 jwj2 /

.1

In particular, for all r 2 .0; 1/, cosh.dD 2 .0; r// D 1 C

2r 2 : 1 r2

Using the Poincaré metric .1 4r 2 /2 ds 2 on the unit disc D 2 , we compute the hyperbolic area of the disc D0 .r/  D 2 centered at zero with Euclidean radius r < 1: Z r r dr area.D0 .r// D 8 .1 r 2 /2 0 r2 D 4 1 r2 D 2.cosh.dD 2 .0; r// 1/  .exp.dD 2 .0; r//

1/:

Now let † be a closed hyperbolic surface of genus g  2. By the Gauss–Bonnet formula, area.†/ D

2.†/ D 4.g

1/;

so the area of an embedded disc in † of hyperbolic radius x > 0 cannot exceed 4.g Since this is roughly  exp.x/, for large x > 0, we obtain

1/.

C 2

x  log.g/ C

for some universal constant C > 0. This provides the desired upper bound 2 log.g/ C C for the diameter of an embedded disc in †, in turn for the systole sys.†/.  The next theorem lies at the heart of Mumford’s compactness criterion [47]: the subset of moduli space Mg ./  T .†g /=MCG.†g / consisting of surfaces † of genus g with sys.†/  , for a fixed  > 0, is compact. Theorem 2 (Collar Theorem [33]). Let be a simple closed geodesic on a closed hyperbolic surface † of genus g  2. Then the tubular neighborhood N . / D ¹p 2 † j d.p; /  º  †g is an embedded annulus, for all 

1

 < arcsinh sinh

l. / 2


:

17

Geometry and Topology of Surfaces

ˇ

˛ z

x

s y

Figure 3.3. Decomposing P into two right-angled hexagons.

Proof. Choose two simple closed geodesics ˛; ˇ  †, which, together with , delimit an embedded pair of pants P  †. Let x; y; z  P be three geodesic arcs of minimal length connecting the pairs of curves .ˇ; /, .˛; /, .˛; ˇ/, respectively. These arcs decompose P into a pair of right-angled hexagons S (see Figure 3.3 and the next chapter for a detailed argument, which will also reveal that these two hexagons are isometric). Let s  S be yet another shortest geodesic arc connecting the two sides labelled ˛ and x, decomposing S into a pair of right-angled pentagons (see again Figure 3.3). Referring back to Proposition 5 about quasi-ideal Lambert quadrilaterals, we obtain   l. / sinh sinh.l.y// > 1: 2 / A symmetric argument yields sinh. l. / sinh.l.x// > 1. We conclude that the tubular 2 neighborhood N . / is embedded, for all   1  < arcsinh : 
/ sinh l. 2

Theorem 3 ([48, Corollary 1.4]). The maximal number of systoles on a closed hyperbolic g2 surface of genus g  2 cannot exceed log.g/ , up to a universal multiplicative constant. The precise order of growth for the maximal number of systoles on closed hyperbolic 4 surfaces of genus g is conjectured to be g 3 . It comes with a dual conjecture, stating that the maximal length of a systole is of the order 43 log.g/. Both conjectures are due to Schmutz Schaller [57], see also [48]. The proof of Theorem 3 is based on the fact that two systoles cannot intersect in more than one point. Surprisingly, the maximal number of simple closed curves on a surface of genus g with pairwise at most one intersection point is not known. It is believed to be of order g 2 , but the best known upper bound by Greene

18

3 Simple closed geodesics and systoles

is g 2 log.g/, see [25]. We will use a simplified version of Parlier’s argument and derive an upper bound of order g 4 for the maximal number of systoles. Proof of Theorem 3, weak version. Let † be a closed hyperbolic surface of genus g  2 with sys.†/ D l  2 log.g/ C C and let   1 1 r.l/ D arcsinh :
2 sinh 2l We define B.†/ to be the least number of hyperbolic discs of radius r.l/ needed to cover †. If r.l/  2l , then pairs of systoles cannot intersect each other, by the Collar Theorem. In this case, the number of systoles cannot exceed the maximal number of pairwise non-isotopic, disjoint simple closed curves, which has linear growth 3g 3. In the other case, r.l/ < 2l , all the discs of radius r.l/ are embedded, and the number B.†/ is bounded above by the maximal number of disjointly embedded hyperbolic discs of radius r.l/ in †. The latter admits an obvious upper bound by the following quotient of areas: 2 B.†/ 

2.g  exp

r.l/ 2

1/

: 1

We note that this quotient cannot exceed g 2 , up to a constant factor. Indeed, the extreme case l D 2 log.g/ yields   1 r.l/ 1 r.l/  and exp 1 : g 2 2g Now every systole has to pass through at least two discs of radius r.l/, since these are embedded discs. Moreover, if two systoles pass through the same disc, they must intersect in that disc, by the Collar Theorem and our choice of the radius r.l/. Altogether, this 2
yields an upper bound of order g2 , that is g 4 , for the number of systoles. 

4 Fenchel–Nielsen coordinates

The goal of this chapter is to understand the space of hyperbolic structures on a closed surface † of negative Euler characteristic. We define T .†/ to be the set of hyperbolic structures on †, up to isometries homotopic to the identity map of †. Identifying homotopic hyperbolic structures allows us to view the set T .†/ as the space of discrete and faithful representations from 1 .†/ to PSL.2; R/, up to conjugation. We endow T .†/ with the natural topology coming from either of these two interpretations: small deformations of a Riemannian metric of curvature 1 on † correspond to small deformations of the corresponding lattice in PSL.2; R/. The space T .†/ together with its topological structure is called Teichmüller space. As we will see, for a closed surface † of genus g  2, T .†/ is homeomorphic to R6g 6 . Moreover, we will see that, up to homotopy, a hyperbolic structure is uniquely determined by the lengths of 9g 9 well-chosen simple closed curves on †. The choice of these curves is not canonical; rather, it depends on a decomposition of † into pairs of pants. Recall from the previous chapter that a closed surface of genus g  2 can be decomposed into a union of j.†/j D 2g 2 pairs of pants by a collection of simple closed curves. Since every pair of pants has three boundary components, this collection consist of 3g 3 simple closed curves. Whenever a hyperbolic structure on † is fixed, these curves can be represented by simple closed geodesics. The crucial point is that these geodesics are still pairwise disjoint. The proof of that is completely analogous to the third part of the proof of Proposition 8 in Chapter 3. It is therefore important to understand the Teichmüller space T .P / of a pair of pants P with three geodesic boundary components ˛; ˇ; . Proposition 9. The map LW T .P / ! R3>0 sending a homotopy class Œ of a hyperbolic structure  on P to the three lengths .l .˛/; l .ˇ/; l . // is a homeomorphism. Remark 3. The map L is continuous, since the lengths of the three boundary curves depend continuously on the Riemannian metric . For a fixed  and closed geodesic curve ˛, the lift ˛Q  H comes with a primitive deck transformation '˛ 2 PSL.2; R/ that translates along the axis ˛Q by the length l .˛/. The translation length l.A/ of a Möbius transformation 'A with tr.A/ > 2 satisfies   l.A/ 1 cosh D tr.A/; 2 2 a fact that is easily verified for the matrix 

a AD 0

0 a

 1 ;

20

4 Fenchel–Nielsen coordinates

where a > 0 and l.A/ D log.a2 / D 2 log.a/. Therefore,   l .˛/ 1 cosh D tr.'˛ /: 2 2 This formula holds for all closed geodesics ˛  † on hyperbolic surfaces. Proof of Proposition 9. Let  be a hyperbolic metric on P and let x; y; z  P be geodesic arcs of shortest length joining the pairs of boundary curves .ˇ; /, .˛; /, .˛; ˇ/, respectively, see Figure 3.3 in Chapter 3. Once again, these curves do not intersect each other, and are orthogonal to the boundary curves at both ends. Therefore, we obtain a decomposition of P into two right-angled hexagons S1 ; S2 , a fact that we already used in the previous chapter. We claim that a right-angled hexagon is uniquely determined by the lengths of three fixed non-adjacent sides, say a; b; c > 0. The limit hexagon with c D 0 can be seen as a union of two quasi-ideal Lambert quadrilaterals that share one of their infinite sides, see Figure 4.1, for t D 0. Thanks to Proposition 5 in Chapter 2, the limit hexagon is determined by the side lengths a; b > 0. Moving the two Lambert quadrilaterals apart by a distance t > 0 gives rise to a right-angled hexagon with side lengths a, b, and c D f .t /, where f W R>0 ! R>0 is a strictly increasing continuous function, whose inverse is thus also continuous.

f .t /

a

b t

Figure 4.1. Family of right-angled hexagons.

We conclude that the two right-angled hexagons making up P are isometric, since they share the sides x; y; z. Their other side lengths are exactly half of the lengths of the corresponding boundary geodesics of P . This provides the desired homeomorphism between T .P / and R3>0 .  Before we proceed to the Teichmüller space of a closed surface †, let us state a classical fact about its group of orientation-preserving isometries IsoC .†/.

21

Geometry and Topology of Surfaces

Theorem 4 (Hurwitz [30]). We have jIsoC .†/j  84.g

1/:

Surfaces realizing the upper bound exist and are called Hurwitz surfaces. They tend to have large systole growth. The conjectural maximal systole growth ( 34 log.g/, as discussed in the previous chapter) is attained by Hurwitz surfaces associated with certain arithmetic lattices in SL.2; R/, see [32]. Sketch of proof. We first prove the finiteness of IsoC .†/. Recall that an orientationpreserving isometry of a connected, compact hyperbolic surface † is determined by the image of one element, say v 2 Tp .†/, of the unit tangent bundle T 1 .†/. The set of images of v under all isometries of † is a closed subset of the compact set T 1 .†/, so IsoC .†/ is compact. Moreover, the identity map is isolated among all isometries of †: let 1 ; 2  † be two simple closed geodesics that intersect in one point p 2 †. Choose  > 0 so that the neighborhoods N . i / are embedded annuli for i D 1; 2. Then every isometry ' 2 IsoC .†/ which is -close to the identity leaves both geodesics i invariant, hence fixes the point p. By choosing  even smaller, we make sure that the point symmetry in p is not -close to the identity, leaving ' D Id† as the only possibility. As a consequence, IsoC .†/ is compact and discrete, hence finite. In order to further bound the number of elements of IsoC .†/, we consider the quotient X D †=IsoC .†/, which is a hyperbolic orbifold with finitely many cone singularities p1 ; p2 ; : : : ; ps of orders n1 ; n2 ; : : : ; ns  2. The rational Euler characteristic of the orbifold X is  s  X 1 ; .X / D 2 2g.X / 1 nk kD1

where g.X / is the genus of X, viewed as a topological surface. The smallest possible 1 absolute value of .X / turns out to be 42 ; it is realized by a sphere with three singularities of orders 2; 3; 7. The orbifold Euler characteristic is multiplicative under quotients, so j IsoC .†/j D

j.†/j 2g 2   84.g 1 j.X /j 42

1/:

For more details, we refer the reader to [19, Chapter 7].



Coming back to the Teichmüller space T .†/ of a closed surface †, let † S be pairwise disjoint simple closed curves, so that † n k ck is a union of 2g 2 pairs of pants. We assume that all pairs of pants have three different boundary curves. As before, we consider the continuous surjective map c1 ; c2 ; : : : ; c3g

3

LW T .†/ ! R3g >0

3

that maps the class of a hyperbolic metric Œ 2 T .†/ to the lengths l . 1 /; : : :, l . 3g 3 / of the unique simple closed geodesics k (with respect to the metric ), homotopic to the 3 curves ck . Fix x D .x1 ; : : : ; x3g 3 / 2 R3g >0 . We will describe the preimage L

1

.x/ 2 T .†/:

22

4 Fenchel–Nielsen coordinates

ck

Figure 4.2. The twisted metric ˛ .

Choose a metric 0 2 L 1 .x/ obtained by gluing pairs of pants with the appropriate boundary lengths, via arbitrary isometries between all pairs of boundary curves. We define a twisted metric ˛ 2 L 1 .x/, for all ˛ D .˛1 ; : : : ; ˛3g 3 / 2 R3g 3 , by gluing the pants with a shift of ˛k along the curve ck , with respect to the gluing of 0 , as sketched in Figure 4.2. The twisting direction is specified by a fixed auxiliary orientation of †. In order to distinguish the metrics ˛ , we choose another set of simple closed curves 0 c10 ; : : : ; c3g 3, the curve ck0 intersects ck in two points, 3  †, so that for all k  3g and intersects no other curve ci , as shown in Figure 4.3. Here we use the fact that all the curves ck are adjacent to two different pairs of pants. The next lemma is well-explained in [20, Exposé 7]. Lemma 3. For all ˛ 2 R3g strictly convex.

3

and for all k  3g

3, the function t 7! l˛Ct ek .ck0 / is

Here l˛Ct ek .ck0 / denotes the length of the unique geodesic k0 , with respect to the metric ˛Ctek , homotopic to ck0 . Now observe that l˛C2ek .ck0 / D l˛ .Tck .ck0 //; where Tck W † ! † denotes a positive Dehn twist along the curve ck (see the next chapter for a detailed discussion on Dehn twists). We define a third set of curves ck00 D Tck .ck0 /  †: Together, the lengths of the curves ck0 and ck00 determine the twist parameter ˛k around the curve ck . What is more, we obtain an embedding of T .†/ into R9g 9 , thanks to the following easy fact. Lemma 4. Let f W R ! R be a strictly convex and proper function. Then the function F W R ! R2 , t 7! .f .t /; f .t C 1//, is a proper embedding.

Geometry and Topology of Surfaces

23

ck00 ck0

Figure 4.3. The curves ck0 and ck00 .

Q T .†/ ! R9g 9 sending a homotopy class Œ of a hyperbolic Theorem 5. The map LW >0 structure  on † to the set of 9g 9 lengths .l .ck /; l .ck0 /; l .ck00 //1k3g

3;

is an embedding. We have now identified the preimage L 1 .x/ as a homeomorphic copy of R3g 3 , 3 1 for all x 2 R3g >0 . Lemma 3 allows for a canonical choice of metric 0 2 L .x/, by 0 minimizing the lengths l0 .ck /, for all k  3g 3. Therefore, we obtain the desired 3 homeomorphism between T .†/ and R3g  R3g 3 ' R6g 6 . >0 Theorem 6. We have T .†/ ' R6g

6

:

The coordinates .l .ck /, ˛k /, 1  k  3g 3, are known as Fenchel–Nielsen coordinates [22]. It is an interesting problem to determine the least number of (not necessarily simple) closed curves whose lengths determine all hyperbolic metrics, up to homotopy. The answer is 6g 5, see [56] and [26].

5 Dehn twists

In this chapter we derive several relations involving Dehn twists and determine the center of the mapping class groups for surfaces of genus at least three. Let A D S 1  Œ0; 1 be an annulus. The map T W A ! A defined by sending .e i ; t / to i. C2 t/ .e ; t / is called positive Dehn twist along the curve S 1  ¹ 21 º. Now let a  † be a simple closed curve on an oriented surface †. Choose a smooth embedding 'W A ! † that maps S 1  ¹ 12 º to the curve a, so that the differential .d'/p at any point p 2 A maps the vectors e , e t to an oriented basis of T'.p/ †. Define the positive Dehn twist Ta W † ! † along the curve a as ´ 'T ' 1 .p/; p 2 '.A/; p 7! p; p … '.A/: The homotopy type of the map Ta W † ! † does only depend on the homotopy type of the curve a, and not on the choice of embedding 'W A ! †, since all these are isotopic. This is a consequence of Epstein’s theorem (homotopic simple closed curves are isotopic [17]) and the Jordan–Schoenflies Theorem [10]. Therefore, Dehn twists along curves are welldefined elements of the mapping class group MCG.†/. Example 1. On the torus † D S 1  S 1 , the positive Dehn twists along the curve S 1  ¹1º lifts to the linear map associated with the unipotent matrix   1 1 AD : 0 1 Now let .p; q/ be a pair of coprime integers and let p;q  S 1  S 1 be the simple closed geodesic of slope pq . Then there exist c; d 2 Z, so that the matrix   p c BD q d is in SL.2; Z/. The positive Dehn twist T p;q lifts to the matrix   1 pq p2 BAB 1 D : q2 1 C pq In particular, all positive Dehn twist along simple closed curves on the torus are conjugate and have infinite order. In order to prove that Dehn twists have infinite order on any surface, we need to consider the intersection number of simple closed curves a; b  †, defined as i.a; b/ D min¹#˛ \ ˇ j ˛  a; ˇ  bº; where  stands for homotopy between curves, and the minimum is taken over all simple closed curves homotopic to a and b.

26

5 Dehn twists

Proposition 10 (Bigon criterion). Let a; b 2 S.†/ be represented by simple closed curves ˛; ˇ  † with transverse intersection. The following statements are equivalent: (1) i.a; b/ D #˛ \ ˇ. (2) For all subarcs ı  ˛,   ˇ with two common endpoints on ˛ \ ˇ, the union ı [  does not bound an embedded disc in †. Q of ˛; ˇ, we have #˛Q \ ˇQ  1. Here † Q denotes the universal (3) For all lifts ˛; Q ˇQ  † 2 covering of †, which is either R or H , depending on the genus of † (g D 1 or g  2, respectively). Corollary 1. Let ˛ ¤ ˇ  † be two simple closed geodesics. Then i.Œ˛; Œˇ/ D #˛ \ ˇ: Example 2. Let .p; q/ and .p 0 ; q 0 / be two pairs of coprime numbers and let

p;q ; p0 ;q 0  S 1  S 1 be the corresponding simple closed geodesics. Then ˇ  ˇ p i. p;q ; p0 ;q 0 / D ˇˇdet q

ˇ p 0 ˇˇ : q0 ˇ

This is most easily proved by verifying the case .p; q/ D .1; 0/ and observing that both sides of the equation are invariant under isometries, i.e. under multiplication by elements of SL.2; Z/. Proof of Proposition 10. (1) ) (2) A bigon, i.e. an embedded disc bounding ı [  allows to reduce the number of intersection points ˛ \ ˇ by two. Q of ˛; ˇ. Then there exists (2) ) (3) Assume # ˛Q \ ˇQ  2, for some lifts ˛; Q ˇQ  † Q Q a bigon between ˛Q and ˇ, i.e. an embedded disc D  † with @D D ıQ \ Q for some subarcs Q D ;. ıQ  ˛, Q Q  ˇQ with two common endpoints. We may assume that DV \ .˛Q [ ˇ/ Indeed, transversality implies that ˛ \ ˇ is finite, so there exists an innermost bigon Q Now the projection of D in † is a bigon between the Q between the curves ˛Q and ˇ. D† curves ˛ and ˇ. (3) ) (1) Assume #˛ \ ˇ > i.a; b/. Then there exists an isotopy hW Œ0; 1  S 1 ! † with h.¹0º  S 1 / D ˛ and #h.¹1º  S 1 / \ ˇ < #˛ \ ˇ: Q that intersect in one point, and a lift of the isoMoreover, there exist two lifts ˛; Q ˇQ  † Q Q Q in an r-neighborhood of ˛Q that topy hW Œ0; 1  R ! † between ˛Q and a curve ˛  Nr .˛/ Q zero. This is impossible, since the curves ˛; has fewer intersection points with ˇ: Q ˇQ are Q contained in the neighborhoods of two intersecting geodesics in †.  Using the bigon criterion, one can show the following formula for the intersection number of curves under iterated application of a Dehn twist [19, Proposition 3.2].

Geometry and Topology of Surfaces

27

Lemma 5. For all k 2 Z, we have i.Tak .b/; b/ D jkji.a; b/2 . As an application, Dehn twists along non-contractible simple closed curves a  † have infinite order. Indeed, if a is non-separating, then there exists a simple closed curve b  † with i.a; b/ D 1. If a is separating, then there exists a simple closed curve b  † with i.a; b/ D 2, see Figure 5.1. Here we use the bigon criterion to show that the curves a and b intersect minimally.

a

b

Figure 5.1. Curve b with i.a; b/ D 2.

Here is another set of interesting consequences of the bigon criterion. As before, we let  stand for homotopy between curves, Œf; g D fgf 1 g 1 stands for the commutator of maps, and equality between maps is understood in the mapping class group MCG.†/. Proposition 11. For all a; b 2 S.†/ and for all f 2 MCG.†/: (1) Ta D Tb , a  b, (2) Œf; Ta  D 1 , f .a/  a, (3) ŒTa ; Tb  D 1 , i.a; b/ D 0, (4) for all a 6 b, we have Ta Tb Ta D Tb Ta Tb if and only if i.a; b/ D 1. The last equality is called braid relation; see Figure 5.2 for an illustration of its origin. Dehn twists along curves with intersection number at least two generate free subgroups of MCG.†/. A beautiful argument of this fact, based on the Ping Pong Lemma, is presented in [19, Proposition 3.14]. Proof of Proposition 11. (1) As mentioned before, the homotopy type of a Dehn twist Ta only depends on the homotopy type of the simple closed curve a  †. For the reverse implication, suppose a; b 2 S.†/ are non-homotopic curves. Then there exists a third curve c 2 S.†/ with i.a; c/ ¤ i.b; c/. Indeed, if i.a; b/ ¤ 0, we can simply take c D b. Otherwise, one can construct c disjoint from b, so that i.a; c/ D 1 or 2, depending on the connectivity properties of the three surfaces † n a, † n b, † n .a [ b/, see Figure 5.3. Now Lemma 5 tells the actions of Ta and Tb on S.†/ apart: i.Ta .c/; c/ D i.a; c/2 ¤ i.b; c/2 D i.Tb .c/; c/: (2) If the curves f .a/ and a are isotopic, then f is isotopic to a diffeomorphism whose support is disjoint from an annular neighborhood of a, hence f commutes with Ta . For

28

5 Dehn twists

Figure 5.2. Braid relation.

b a

c c a

c

a

b

b

Figure 5.3. Curve c with i.a; c/ ¤ i.b; c/.

the converse, observe that f Ta f

1

Ta

1

D Tf .a/ Ta 1 ;

so Œf; Ta  D 1 implies that the two Dehn twists Tf .a/ ; Ta are isotopic. We conclude f .a/  a, by (1). (3) If a and b are disjoint, then the corresponding Dehn twists Ta , Tb commute. Conversely, if ŒTa ; Tb  D 1, then Tb .a/  a, by (2). Now Lemma 5 implies i.a; b/2 D i.Tb .a/; a/ D i.a; a/ D 0: (4) The braid relation Ta Tb Ta D Tb Ta Tb is equivalent to .Ta Tb /Ta .Ta Tb /

1

D Tb :

The left-hand side of the latter is a Dehn twist: TTa Tb .a/ . Therefore, the braid relation is equivalent to Ta Tb .a/  b, again by (1). Now suppose that the curves a, b intersect in one point. Then Tb .a/  Ta 1 .b/, as shown in Figure 5.4. Conversely, if Ta Tb .a/  b, then i.a; Ta Tb .a// D i.a; b/:

Geometry and Topology of Surfaces

29

Tb .a/

b a

Figure 5.4. Tb .a/  Ta 1 .b/.

The left-hand side is invariant under the Dehn twist Ta , so i.a; Tb .a// D i.a; b/. Now Lemma 5 implies i.a; b/2 D i.a; b/. We have to rule out i.a; b/ D 0, which is easy, since that together with the braid relation implies Ta D Tb , in turn a  b.  The second item of Proposition 11 is all we need to determine the center of the mapping class groups. Indeed, an element f 2 MCG.†/, which commutes with all Dehn twists, has to preserve all simple closed curves, up to isotopy. For g  3, one can choose a system of 2g simple closed curves on † that intersect in the pattern of a tree with one branch, whose complement is an open disc, as shown in Figure 5.5. All pairwise intersection points of these curves are fixed points of f . After a suitable isotopy, f fixes the 2g curves pointwise. At last, we apply the Alexander trick (every homeomorphism of the closed unit disc fixing the boundary is homotopic to the identity map) and conclude that f is homotopic to the identity map.

Figure 5.5. Tree-like pattern of curves.

Theorem 7. The center of the mapping class group of an orientable, closed, connected surface of genus at least three is trivial. The assumption on the genus is necessary, since the center of the mapping class group of the genus two surface contains one non-trivial mapping class, the hyperelliptic involution.

6 Normal generators for mapping class groups

In the present chapter we prove Dehn’s theorem and a recent generalization thereof, due to Lanier and Margalit [35]. For a set of diffeomorphisms to generate the mapping class group MCG.†/, it is necessary that it acts transitively on the set of non-separating simple closed curves on †. This is true for the set of Dehn twists along non-separating curves. Lemma 6. Let a; b  † be two non-separating, simple closed curves. Then there exists a finite sequence of non-separating simple closed curves c1 ; : : : ; cn in †, so that Tcn ı


ı Tc1 .a/  b: Remark 4. Separating simple closed curves split into finitely many orbits of the mapping class group, according to the topological type of their complement. The proof of Lemma 6 is by induction on the intersection number i.a; b/. Proof. The case i.a; b/ D 1 turns out to be the easiest, since then Ta Tb .a/  b, as seen in the previous chapter. In all other cases, that is if i.a; b/ ¤ 1, we will construct a finite sequence of curves in †, starting at a and terminating at b, whose consecutive members have intersection number one. Then we obtain a finite sequence of Dehn twists mapping a to b, as desired. If i.a; b/ D 0, then there exists a single non-separating, simple closed curve c  † with i.a; c/ D i.b; c/ D 1. There are two cases to consider, depending on whether † n .a [ b/ is connected or not, as shown in Figure 6.1.

b

c

a

c a

b

Figure 6.1. Curve c with i.a; c/ D i.b; c/ D 1.

32

6 Normal generators for mapping class groups

b

b

c1 c

a

c2

a

Figure 6.2. Reducing the intersection number i.a; b/.

Now let us assume that i.a; b/ D n  2. Consider two intersection points between a and b, which are consecutive along the curve a. Choose an auxiliary orientation of b. We distinguish two cases, see Figure 6.2: (a) The two arcs of b cross a in the same direction. Then we find a simple closed curve c with i.a; c/  n 1 and i.b; c/ D 1. The latter implies that c is non-separating. (b) The two arcs of b cross a in opposite directions. Then we find two simple closed curves c1 ; c2 with i.a; ck /  n 2 and i.b; ck / D 0. At least one of the two curves c1 ; c2 must be non-separating, since otherwise b would be separating. This concludes the induction on i.a; b/.  Remark 5. The set of simple closed curves on a closed surface † gives rise to an infinite graph called curve graph, whose vertex set is in bijection with the set S.†/, and whose edges correspond to pairs of simple closed curves whose intersection number is zero. One interesting consequence of the proof of Lemma 6 is that the curve graph associated with a closed surface of genus at least two is path-connected [28]. Indeed, the proof of Lemma 6 shows that every pair of non-separating simple closed curves is related by a chain of curves whose consecutive members have intersection number one. Now for all pairs of simple closed curves a; b  † with i.a; b/ D 1, there exists a simple closed curve c  † with i.a; c/ D i.b; c/ D 0, provided the genus of † is at least two. This is because a small tubular neighborhood of a [ b in † is homeomorphic to a torus with one boundary component. As a consequence, every pair of non-separating simple closed curves is related by a chain of curves whose consecutive members have intersection number zero. At last, for every separating simple closed curve x  †, there exists a non-separating simple closed curve y  † with i.x; y/ D 0. A much more refined argument reveals that the curve graph is hyperbolic [42]. Theorem 8 (Dehn [14]). The mapping class group MCG.†/ of a closed orientable surface † is generated by Dehn twists along non-separating simple closed curves. In fact, a stronger statement is true: one only needs a finite number of Dehn twists. An explicit set of generating Dehn twists along 3g 1 simple closed curves was exhib-

33

Geometry and Topology of Surfaces

ited by Lickorish [37]. This is easily verified in the special case g D 1, since the group MCG.S 1  S 1 / ' SL.2; Z/ is generated by the two matrices     1 1 1 0 and ; 0 1 1 1 which correspond to the Dehn twists along the two curves S 1  ¹1º and ¹1º  S 1 , respectively. Proof of Theorem 8. Let f 2 DiffC .†/. Consider a chain of curves a1 ; a2 ; : : : ; a2g  † with i.ak ; akC1 / D 1, and all other intersection numbers equal to zero, as shown in Figure 6.3 for g D 3.

a2

a3

a4

a5

a6

a1

Figure 6.3. Chain of curves.

By Lemma 6, there exists a finite product of Dehn twists along non-separating curves 1 D Tcn ı


ı Tc1 with 1 .a1 /  f .a1 /. Moreover, we may choose the sequence of Dehn twists so as for the map f1 D 1 1 ı f to preserve the orientation of a1 . After a suitable isotopy, the map f1 fixes a1 pointwise. We then proceed by adjusting the curve a2 , making sure to use a sequence of Dehn twists along curves that do not intersect a1 . An iteration of this process provides a sequence of products of Dehn twists 2 ; 3 ; : : : ; 2g along non-separating curves, so that the composition h D 2g1 ı


ı 1 1 f fixes the union of curves a1 [


[ a2g  † pointwise. The complement of this union of curves is homeomorphic to an open disc. Using the Alexander trick, as in the previous chapter, we conclude that h is homotopic to the identity map. In turn, f is homotopic to the finite product of Dehn twists 1 ı


ı 2g .  We recall that all Dehn twists along non-separating curves are conjugate, since the mapping class group acts transitively on the set of non-separating curves, and since Tf .a/ D f Ta f

1

for all f 2 MCG.†/ and all a 2 S.†/. In particular, the normal closure of a single Dehn twist along a non-separating curve a  † is the entire mapping class group MCG.†/. This fact admits a beautiful generalization.

34

6 Normal generators for mapping class groups

Theorem 9 ([35, Lemma 2.2]). Let † be a closed surface of genus at least three, let f 2 MCG.†/ and suppose there exists a simple closed curve c  † with i.f .c/; c/ D 1. Then f normally generates the mapping class group MCG.†/. In fact, this is a weak variation on Lanier and Margalit’s main theoremp stated in the introduction: pseudo-Anosov maps f W † ! † with stretch factor .f / < 2 normally generate the mapping class group MCG.†/. Observe that the above theorem applies to Dehn twists Ta along non-separating curves a  †. In that case, every curve c  † with i.a; c/ D 1 will do, by Lemma 5: i.Ta .c/; c/ D i.a; c/2 D 1: The following example, taken from Lanier and Margalit’s paper (see [35, Section 4]), shows the existence of pseudo-Anosov maps that normally generate the mapping class groups.

b1

a2

b2

a3

a1

b3 a4

Figure 6.4. Another chain of curves.

Example 3. Consider a chain of curves a1 ; b1 ; a2 ; b2 ; : : : ; ag ; bg ; agC1 on a surface † of genus g  3, as shown in Figure 6.4 for g D 3. As we will see later on, the map f D Tbg1 Tbg1 1


Tb11 Ta1 Ta2


TagC1 is of pseudo-Anosov type. Moreover, i.f .a1 /; a1 / D i.Tb11 .a1 /; a1 / D 1; implying that f normally generates MCG.†/. Proof of Theorem 9. Suppose f 2 MCG.†/ and a simple closed curve c  † satisfy i.f .c/; c/ D 1. Consider the commutator ŒTc ; f  D Tc f Tc 1 f

1

1 D Tc Tf .c/ ;

which is contained in the normal closure hf iN of f in MCG.†/. We claim that for all pairs of simple closed curves a; b  † with i.a; b/ D 1: ŒTa ; Tb  2 hf iN :

35

Geometry and Topology of Surfaces

For that, we observe that there exists h 2 MCG.†/ with h.c/ D a; h.f .c// D b, again since a small tubular neighborhood of two simple closed curves intersecting in one point is homeomorphic to a torus with one boundary component in †, and the mapping class group acts transitively on those. The latter is a consequence of the classification of closed orientable surfaces. Therefore, Ta Tb

1

1 D hTc Tf .c/ h

1

2 hf iN :

Now the purely algebraic fact Œx; y D y.y

1

xyx

1

/y

1

2 hxy

1

iN

implies ŒTa ; Tb  2 hf iN , for all pairs of curves with intersection number one. Moreover, this trivially extends to pairs of curves with intersection number zero. In order to conclude this proof, we need to invoke two more facts. Firstly, the mapping class group is generated by Dehn twists along 3g 1 non-separating curves whose pairwise intersection number is zero or one [37]. Secondly, the mapping class groups of closed surfaces are perfect, as soon as the genus is at least three [52]. Therefore, MCG.†/ D ŒMCG.†/; MCG.†/ D hf iN : For the last equality, we use another purely algebraic fact: if x1 ; : : : ; xn 2 G generate a group G, then the set of pairwise commutators Œxi ; xj  generates the commutator subgroup ŒG; G. This follows from the commutator identity Œx; yz D Œx; yyŒx; zy

1

:



7 Measured foliations

Let † be a closed surface of genus two or more. Then DiffC .†/ acts on the Teichmüller space T .†/ by push-forward. If ' 2 DiffC .†/ has a fixed point  2 T .†/, i.e. if '  D , then ' is periodic, as seen in Chapter 4. A classical theorem by Wiman states that the order of ' is at most 4g C 2 (see [62] and [24] for a graph-theoretical proof of this fact). Now suppose ' has infinite order, pick any hyperbolic metric , and consider the sequence of metrics .'n /n2N . In general, we cannot expect this sequence to have an accumulation point, let alone a limit point, in T .†/. Recall that ' also acts on the space of lengths of n simple closed curves RS.†/ >0 . As we will see, the sequence .' /n2N has a limit point in S.†/ the projective space P .R /. For example, if a; b  † are simple closed curves with i.a; b/ D 1, then l.Tan /  .a/ lim D 0: n!1 l.T n /  .b/ a  The limit points admit a natural interpretation as projective measured foliations, objects we will describe next. A measured foliation f is a foliation with isolated singularities of simple type which is endowed with a measure on curves transverse to the leaves. Formally, a measured foliation f on a surface † is defined by a smooth atlas with regular and singular charts, with the following two requirements: (1) all transition maps between regular charts are restrictions of horizontal maps of the following type: .x; y/ 7! .h.x; y/; ˙y C c/; where hW R2 ! R is smooth, and c 2 R is a constant, (2) the singular charts are locally foliated by the level sets of the smooth “map” z 7! Im.z k=2 /; where k 2 N, k  3, is called the index of the singularity. Examples of a regular chart, as well as two singular charts, together with the leaves of their foliation, are depicted in Figure 7.1. The index of a singular point is equal to the number of singular leaves, i.e. leaves running into the singular point. The measure of a transverse curve is defined by integrating the form jdyj over that curve, chart by chart. The definition of a measured foliation extends to surfaces with boundary, with one additional requirement: all boundary components are unions of leaves or transverse to the leaves of the foliation, as shown in Figure 7.2. For a measured foliation f on a surface †, we denote the finite set of singularities of f by Sing.†; f /, and the index of a singularity p 2 Sing.†; f / by indf .p/.

38

7 Measured foliations

Figure 7.1. Foliated charts.

Figure 7.2. Foliated boundary charts.

Proposition 12. We have 2.†/ D

X

.indf .p/

2/:

p2Sing.†;f /

Proof. First suppose that the leaves of f admit a transverse orientation. In particular, all singularities have even index. Then we may construct a vector field X on †, which is transverse to the foliation and whose set of singularities Sing.†; X / coincides with that of f , with 1 .indf .p/ 2/ indX .p/ D 2 for all p 2 Sing.†; X /, see Figure 7.3. The desired formula follows from the theorem of Poincaré–Hopf, which also holds in the relative case, i.e. for vector fields tangent or transverse to each boundary component of † (see [44]): X .†/ D indX .p/: p2Sing.†;X /

Q For foliations f without transverse orientation, we consider a suitable double covering † Q of †, with branching points at singularities of odd index. The Euler characteristic .†/ is determined by the theorem of Hurwitz (see [27, Chapter IV]). We conclude by applying Q which admits a transverse orientation. the above argument to the lift of f to †, 

Geometry and Topology of Surfaces

39

Figure 7.3. Vector field around even singularity.

Remark 6. Foliations whose leaves admit a transverse orientation can be described by closed 1-forms with the appropriate types of singularities. In general, equivalence classes of measured foliations (up to Whitehead moves defined below) are described by so-called quadratic differentials [29]. We note that the index formula of Proposition 12 implies that surfaces admitting a measured foliation have non-positive Euler characteristic. In particular, the closed unit disc does not admit a measured foliation. The annulus S 1  Œ0; 1 and the torus S 1  S 1 admit measured foliations without singularities, all whose leaves are circles. A pair of pants P admits a measured foliation with two singularities of index three, whose leaves are transverse to all the three boundary components, as sketched in Figure 7.4. The transverse measure is not specified in the picture; we will discuss this in more detail shortly.

Figure 7.4. Foliated pair of pants.

40

7 Measured foliations

By pasting together foliated pairs of pants in a suitable way, we end up with measured foliations on all closed surfaces with negative Euler characteristic. It is important to mention that not all topological foliations on a surface admit an invariant transverse measure. In particular, spiraling leaves with a limit circle and Reeb type components, as shown in Figure 7.5, are ruled out, since they would create an infinite accumulation of measure near the boundary.

Figure 7.5. Forbidden foliations.

For a surface † with .†/  0, we define MF .†/ to be the set of measured foliations on †, modulo measure-preserving isotopy and the Whitehead operation, which merges two singularities, as shown in Figure 7.6. In the context of measured foliations, the relation f ' f 0 will always include the Whitehead operation, as well as its inverse.

Figure 7.6. Whitehead operation.

As stated above, curves  † transverse to the leaves of a foliation f have a length, defined by integrating the transverse measure jdyj associated with f over : Z . / D jdyj:

This extends to piecewise smooth curves consisting of finitely many smooth arcs transverse to the foliation f . We define the length of a simple closed curve c 2 S.†/ as lf .c/ D inf . /;

c

Geometry and Topology of Surfaces

41

where the infimum is taken over all simple closed curves isotopic to c, consisting of finitely many smooth arcs transverse to the foliation f . The length of a curve is invariant under the Whitehead operation. Therefore, we obtain a map I W MF .†/ ! RS.†/ 0 : As we will see, the point zero is not contained in the image of I , so we can compose I with the projection W RS.†/ ! P .RS.†/ / and define the space of projective measured 0 foliations on † as PMF .†/ D  ı I.MF .†//: Our goal is to identify the space of projective measured foliations as the boundary of Teichmüller space, in the following sense: PMF .†/ D @. ı l.T .†///: For this, we need an analogue of geodesics in the context of measured foliations. A simple closed curve  † is called quasi-transverse to a measured foliation f on † if it consists of finitely many smooth arcs, each of which is a transverse arc or contained in a leaf of f , so that the succession of arcs is monotone with respect to the transverse measure associated with f . This means the following: the final segment of a transverse arc and the starting segment of the next transverse arc of need to be contained in two different local components of the complement of the leave (or graph of singular leaves) adjacent to these two arcs, as shown in Figure 7.7. The length function lf naturally extends to quasi-transverse simple closed curves on †.

Figure 7.7. Quasi-transverse arc.

Proposition 13. Let c 2 S.†/ be represented by a simple closed curve  †, quasitransverse to a measured foliation f . Then lf .c/ D . /. Proof. Let 0 be a curve isotopic to . We need to prove lf . 0 /  lf . /. For this, we may assume that and 0 intersect transversally. First suppose that and 0 are disjoint. Then there exists an embedded annulus A  † with @A D [ 0 . Let ˛ be a leaf of the foliation f entering A through . There are four possibilities for ˛ to consider: (1) ˛ exits A through 0 , (2) ˛ runs into a singularity of f in A, (3) ˛ exits A through , (4) ˛ stays in A forever.

42

7 Measured foliations

˛

D

D Ž

D

Figure 7.8. Doubling a disc.

The last possibility would entail an accumulation of mass, as sketched in Figure 7.5, which is impossible. The third possibility would imply that the part of ˛ inside A cobounds an embedded disc D  A, together with a segment of the curve . By doubling this disc, we would obtain a disc with a measured foliation, whose boundary is a union of leaves, as shown in Figure 7.8. This is again impossible. The second case only happens for finitely many arcs ˛. Therefore, all but finitely many leaves entering A through must exit through 0 . This implies lf . 0 /  lf . /: Now suppose \ 0 ¤ ;. Then there exists a bigon between and 0 , i.e. an embedded disc D  † whose boundary consists of two subarcs ı  , ı 0  0 . A variation of the above argument implies lf .ı 0 /  lf .ı/I we conclude by induction on the number of intersection points between and 0 .



A quasi-transverse curve without horizontal arcs (i.e. without arcs contained in leaves) is called a transverse curve. As we will see next, simple closed curves of strictly positive length are isotopic to transverse curves, provided we allow to change the foliation by Whitehead moves. Proposition 14. Let  † be a simple closed curve and f a measured foliation on †. (1) If lf .Œ / > 0, there exist a simple closed curve 0 , isotopic to , and a measured foliation f 0 , Whitehead equivalent to f , so that 0 is transverse to f 0 . (2) If lf .Œ / D 0 and is non-separating, there exist a simple closed curve 0 , isotopic to , and a measured foliation f 0 , Whitehead equivalent to f , so that 0 is contained in a finite union of (possibly singular) leaves of f 0 . The idea of proof is fairly simple: one gets rid of unnecessary flat arcs and summits by the two moves shown in Figure 7.9. However, one needs to be extremely careful, since the curve might get self-intersections after the second move, as shown in Figure 7.10. This is where the Whitehead operation comes to rescue. In the case of length zero, the curve might even completely collapse, as shown in Figure 7.11. This can only be avoided if is non-separating. A detailed proof is presented in [20, Exposé 5].

Geometry and Topology of Surfaces

Figure 7.9. Simplifying moves.

Figure 7.10. Avoiding a self-intersection.

Figure 7.11. Collapsing curve of length zero.

43

8 The .9g

9/-theorem for measured foliations

In this chapter we prove that the natural map I W MF .†/ ! RS.†/ is an injection. Here 0 again, all the material is based on [20]. We start by classifying measured foliations on a pair of pants P with three distinguished boundary curves a1 ; a2 ; a3 . Let f 2 MF .P / and define mk D lf .ak /  0, for k 2 ¹1; 2; 3º. The extreme case mk D 0 happens precisely if the boundary component ak consists of a union of leaves of f . One might hope to identify MF .P / with R30 , but this is wrong, for the following reason: if a boundary component of P is a union of leaves, then one can just as well glue an annulus foliated by circles of arbitrary thickness to that boundary component, without changing the lengths of the three boundary curves. For this reason, we define MF 0 .P / D MF .P /=boundary annuli: Proposition 15. The set MF 0 .P / is in bijection with R30 , via the map f 7! .lf .a1 /; lf .a2 /; lf .a3 //: Proof. We need to show that every triple .m1 ; m2 ; m3 / 2 R30 is realized by a unique measured foliation on P , modulo isotopy, the Whitehead operation, and boundary annuli. By the index formula, every measured foliation f 2 MF 0 .P / has either two singularities of index three, or one singularity of index four. Moreover, all leaves are circles or compact intervals with endpoints on the boundary @P or in the set of singularities Sing.†; f /. We may assume m1  m2  m3 . There are five cases to distinguish: (1) m1 D m2 D m3 D 0, (2) m1 > m2 D m3 D 0, (3) m1 D m2 > m3 D 0, (4) m1 > m2 > m3 D 0, (5) m3 > 0. The last case subdivides into three further cases, according to whether (a) m1 > m2 C m3 , (b) m1 D m2 C m3 , (c) m1 < m2 C m3 . Each of these cases, seven in total, is realized by a unique foliation, as shown in Figure 8.1, in the same order as in the above description. Here the outer boundary component is a1 ; the inner ones are a2 and a3 , on the left and right, respectively. 

46

8 The .9g

(1)

9/-theorem for measured foliations

(2)

(5a)

(3)

(5b)

(4)

(5c)

Figure 8.1. Seven measured foliations on P .

The starting point for the .9g 9/-theorem was a decomposition of † into embedded pairs of pants along 3g 3 simple closed curves c1 ; c2 ; : : : ; c3g 3 , which we now suppose to be non-separating. Using the same set of dual curves ck0 (1  k  3g 3) with i.cj ; ck0 / D ıj k , and ck00 D Tck .ck0 /, as in Chapter 4, we obtain a map I W MF .†/ ! R9g

9

;

mapping a measured foliation f to the set of lengths .lf .Œck /; lf .Œck0 /; lf .Œck00 //. Theorem 10. The map I is injective; its image is homeomorphic to R6g

6

n ¹0º.

Proof. Let f be a measured foliation on †, and fix one of the curves ck  †, where 1  k  3g 3. We write P1 ; P2 for the two embedded pairs of pants adjacent to ck . The measured foliations on P1 ; P2 are determined by the lengths of their boundary curves, up to annuli parallel to boundary curves of length zero. As in the setting of hyperbolic structures, we need additional parameters that record how two neighboring pairs of pants are glued together. We distinguish two cases. Case (1): lf .Œck / > 0. Thanks to Proposition 14, we may assume that the curve ck is transverse to the foliation f . The restriction of f to the union P1 [ P2 is determined by a twist parameter tk 2 R, very much like in the classical case. More precisely, there is a unique gluing of P1 and P2 that minimizes the length of the dual curve ck0 . We choose the parameter tk D 0 to correspond to this gluing. Then the length lf .Œck / is a linear function of jtj. The sign of the parameter t is again determined by the length of the third curve, lf .Œck00 /. Case (2): lf .Œck / D 0. Here we may assume that the curve ck is contained in a finite union of leaves of the foliation f . Twisting along the curve ck has no influence on the measure of f . However, in this case, we need a parameter mk > 0 that records the thickness of the largest embedded annulus between P1 and P2 , foliated by circles, one of which is the curve ck , as sketched in Figure 8.2. The parameter mk is determined by the length

47

Geometry and Topology of Surfaces

Figure 8.2. Largest annulus between two pants.

of the curve ck0 , which coincides with the length of ck00 . Indeed, let a1  P1 ; a2  P2 be two shortest non-contractible relative arcs with both endpoints on ck . Then lf .Œck0 / D lf .Œck00 / D .a1 / C .a2 / C 2mk : While it is true that a measured foliation f 2 MF .†/ is determined by the set of parameters lf .Œck /, tk , mk , 1  k  3g 3, it is not true that all sets of values are realized: if all lf .Œck / are zero, then at least one parameter mk has to be strictly positive, since otherwise f would have singular leaves only, which is impossible. All the other sets of values are realized. A careful analysis of the expression of the parameters lf .Œck /; lf .Œck0 /; lf .Œck00 / in terms of lf .Œck /, tk , mk , reveals that the image of the map I W MF .†/ ! R9g

9

is homeomorphic to R6g 6 n ¹0º, where the point zero is really the point 0 2 R9g [20, Exposé 6] for details).

9

(see 

We are finally in a position to define a topology on the set of measured foliations: the one induced by the length parameters lf .Œck /; lf .Œck0 /; lf .Œck00 /. Equivalently, the topology on PF .†/ is the one induced by the topology of RS.†/ via the injective map I W PF .†/ ! RS.†/ : Corollary 2. The space of projective measured foliations PMF .†/ D  ı I.MF .†//  P .RS.†/ / is homeomorphic to the sphere S6g

7

.

9 Compactification of Teichmüller space

The goal of this chapter is to identify the limit points of Teichmüller space in the projective space of length functions P .RS.†/ /. We start by two important facts on length functions induced by measured foliations and hyperbolic metrics. Proposition 16. The following statements hold: (1) The images I.MF .†// and l.T .†// in RS.†/ 0 are disjoint. (2) The restriction of the projection W RS.†/ ! P .RS.†/ / to l.T .†// is injective. >0 Proof. For the first item, we recall that every closed hyperbolic surface has a strictly positive systole, whereas measured foliations admit simple closed curves of arbitrarily small length. Indeed, if a measured foliation f admits a closed leaf ˇ  †, we have a simple closed curve of length zero. Otherwise, there exist a non-compact leaf ˛  † and a smooth chart containing infinitely many horizontal segments of ˛. Given  > 0, we can find two segments that are less than  apart with respect to the transverse measure. Now an appropriate subarc of ˛ together with a transverse arc joining these two segments (see Figure 9.1) gives rise to a simple closed quasi-transverse curve  † with lf .Œ / < .

Figure 9.1. Short simple closed curve.

For the second item, consider a pair of simple closed curves a; b  † intersecting in one point p which we view as the base point of the fundamental group of †. Let now A; B 2 PSL.2; R/ be corresponding deck transformations, with respect to a hyperbolic metric  on †. The length of a simple closed curve x  † is determined by the trace of the corresponding deck transformation X 2 PSL.2; R/, as discussed in Chapter 4:   1 l .x/ D tr.X /: cosh 2 2 In addition to a; b, we consider the simple closed curves c; d , defined by the deck transformations AB and A 1 B, respectively. Now the formula tr.A/tr.B/ D tr.AB/ C tr.A

1

B/;

50

9 Compactification of Teichmüller space

valid for all A; B 2 SL.2; R/, implies         l .b/ l .c/ l .d / l .a/ cosh D cosh C cosh ; 2 cosh 2 2 2 2 the left-hand side of which is equal to     l .a/ C l .b/ l .a/ l .b/ cosh C cosh : 2 2 Suppose there exists a hyperbolic metric 0 , stretching the length of all simple closed curves by the same factor  > 1; then the above formula would hold with all l .x/ replaced by l .x/. A careful application of the strict convexity of the function x 7! cosh. cosh

1

.x//;

explained in the next paragraph, reveals that the two sets of numbers ¹l .a/ C l .b/; l .a/

l .b/º and

¹l .c/; l .d /º

would have to coincide. However, for a simple geometric reason, both lengths l .c/; l .d / are strictly smaller than l .a/ C l .b/. Therefore, the length functions associated with two hyperbolic metrics on † cannot be proportional. The convexity argument mentioned above is fairly general: suppose f W R>0 ! R>0 is strictly convex and invertible. Then for all  > 1, the function x 7! f .f

1

.x//

is convex, and even strictly convex in the case f .x/ D cosh.x/. Suppose there exist  > 1 and a; b; c; d > 0 with f .a/ C f .b/ D f .c/ C f .d / and f .a/ C f .b/ D f .c/ C f .d /: By setting x D f .a/; y D f .c/; z D f .a/ C f .b/, the second equation becomes x// D f .f

1

Now the strict convexity of the function x 7! f .f in turn ¹a; bº D ¹c; d º.

1

f .f

1

.x// C f .f

1

.z

.y// C f .f

1

.z

y//:

.x// implies x D y or x D z

y, 

The images of the spaces T .†/ and MF .†/ in RS.†/ being disjoint, we need to construct a projection qW T .†/ ! MF .†/ in order to determine the limit points of sequences in  ı L.T .†//  P .RS.†/ /. Let now c1 ; : : : ; c3g 3 be the non-separating curves of an embedded pants decomposition, geodesic with respect to a fixed hyperbolic metric  on †. We construct q./ 2 MF .†/ with lq./ .Œck / D l .Œck / for all k  3g 3.

51

Geometry and Topology of Surfaces

Let P  † be a pair of pants with geodesic boundary curves a1 ; a2 ; a3 2 ¹c1 ; : : : ; c3g 3 º and set li D l .ai /, for i 2 ¹1; 2; 3º. We may assume l1  l2 ; l3 . The construction of q./ splits into two cases. Case (1): l1  l2 C l3 . For all pairs i < j  3, let gij  P be the unique shortest geodesic arc joining the two boundary curves ai ; aj . Fill P with arcs that are parallel to one of the geodesics gij . In each of the two component of P n ¹g12 ; g13 ; g23 º, there are three extremal arcs that cobound an embedded triangle   P with vertices on @P , as shown in Figure 9.2, on the left. The location of these vertices can be determined by solving the set of linear equations: l1 D 2x C 2y l2 D 2x C 2z l3 D 2y C 2z The restriction of the measured foliation q./ to P is defined by collapsing the two triangles to tripods; the transverse measure between leaves is defined via their hyperbolic distance, measured along the boundary curves. Case (2): l1  l2 C l3 . Apply the same procedure as in Case (1), with g23 replaced by g11  P , the unique shortest geodesic arc with both endpoints on a1 , separating P into two connected components, as shown on the right of Figure 9.2. Again, there are two embedded triangles, one in each connected component of P n g11 , which we replace by tripods.

Figure 9.2. Collapsing two triangles.

The case l1 D l2 C l3 is covered by Cases(1) and (2) and gives rise to a single singularity of index 4 in P . Proposition 17. The map qW T .†/ ! MF .†/ is a homeomorphism onto its image q.T .†// D ¹f 2 MF .†/ j lf .Œck / > 0 for all k  3g

3º:

Proof. We need to reconstruct a hyperbolic metric  2 T .†/ from its image q./. Let f 2 MF .†/ with lf .Œck / > 0 for all k  3g 3.

52

9 Compactification of Teichmüller space

As before, let a1 ; a2 ; a3 be the boundary curves of P and set li D lf .ai / for i 2 ¹1; 2; 3º. Here again, we may assume l1  l2 ; l3 . If l1 ¤ l2 C l3 , then the union of singular leaves of the restriction of f to P separates P into three connected components. Each of these components contains a unique leaf which is at equal distance from the two singularities of f . Up to isotopy, there is a unique hyperbolic metric  2 T .P / with li D l .ai /, so that these three distinct leaves become shortest geodesic arcs between the respective boundary curves. In the special case l1 D l2 C l3 , one of the three distinct leaves is replaced by the singular arc with both endpoints on the curve a1 .  Our goal is to show that the two length functions l and lq./ 2 RS.†/ are not too far apart. For a quantitative statement, we fix  > 0 and consider the set U./ D ¹ 2 T .†/ j l .Œck /   for all k  3g

3º:

Lemma 7 (Comparison Lemma). Let ˛ 2 S.†/. There exists a constant C.˛; / > 0 so that for all  2 U./, lq./ .˛/  l .˛/  lq./ .˛/ C C.˛; /: Corollary 3. For all sequences ¹n º  U./ whose image in RS.†/ tends to infinity, the projected sequence ¹ ı L.n /º  P .RS.†/ / converges if and only if the following sequence converges: ¹ ı I.q.n //º  P .RS.†/ /: Moreover, in the case of convergence, the limits of these two sequences coincide. Proof of Lemma 7. The first inequality, lq./ .˛/  l .˛/, is fairly obvious, by construction, since the transverse measure of the foliation q./ is defined in terms of the hyperbolic distance of leaves. For the second inequality, let ˛  † be a non-contractible simple closed curve. The difference between l .˛/ and lq./ .˛/ comes from the fact that ˛ can traverse pairs of pants along leaves of the foliation q./, without picking up length. Recall from Chapter 3 that hyperbolic pairs of pants with all boundary curves longer than  > 0 have geodesic arcs of uniformly bounded length connecting each pair of boundary components. More generally, there exists an upper bound C./ for the length of all leaves joining two boundary components of pants with all boundary curves longer than  > 0. Now let b 2 N be the number of times the curve ˛ traverses a pair of pants. Then we obtain l .˛/  lq./ .˛/ C bC./:



We have all ingredients needed to identify the limit points of  ıL.T .†//  P .RS.†/ / as PMF .†/ D  ı I.MF .†//. There is one subtlety that requires a careful consideration: not every sequence ¹n º  T .†/ whose image in P .RS.†/ / converges, is contained in a set U./, for some fixed  > 0. Indeed, one could have, for some k  3g 3, lim ln .ck / D 0:

n!1

Geometry and Topology of Surfaces

53

However, in that case, one can choose an alternative embedded pants decomposition along a suitable set of curves cQ1 ; : : : ; cQ3g 3 , so that there exists an  > 0 with ln .cQk /   for all k  3g 3 and for all n 2 N. We refer the reader to [20, Exposé 8] for a detailed argument. In the same exposé, one finds a proof that the union  ı L.T .†// [ PMF .†/ is homeomorphic to the closed unit ball D 6g 6 . We refer to this union as the compactified Teichmüller space T .†/.

10 Classification of mapping classes

The mapping class group MCG.†/ acts on the space of simple closed curves S.†/ by permutation. This action descends to a continuous action of MCG.†/ on the compactified Teichmüller space T .†/  P .RS.†/ /. In particular, each individual diffeomorphism 'W † ! † induces a continuous map ' W T .†/ ! T .†/. Now we just saw that T .†/ is homeomorphic to the closed ball D 6g 6 , so ' has a fixed point, by Brouwer’s fixed point theorem. Depending on whether this fixed point is in the interior or on the boundary of T .†/, the mapping class Œ' fixes a hyperbolic metric , or the projective class of a measured foliation f 2 MF .†/. In the first case, the map 'W † ! † is isotopic to an isometry, hence to a periodic map. In the latter case, there exists a real number  > 0 with ' .f / D f . Up to replacing ' by its inverse ' 1 , we may suppose   1. We will see that  D 1 implies that ' is periodic or reducible, i.e. there exists an embedded closed 1-submanifold C  †, consisting of a finite union of disjoint non-contractible simple closed curves, with '.C / D C . If  > 1, then there exists a transverse invariant measured foliation f with 1 ' .f / D : f The leaves of f are transverse to the ones of f , except at the singular points, which f shares with f , as shown in Figure 10.1. A diffeomorphism 'W † ! † with two invariant measured foliations of that type is called a pseudo-Anosov diffeomorphism; the number  > 1 is called the stretch factor of '.

Figure 10.1. Transverse measured foliations.

Theorem 11 ([60, Theorem 4]). Let 'W † ! † be an orientation-preserving diffeomorphism. Then either a finite power of ' fixes a curve in S.†/, or ' is of pseudo-Anosov type. In the former case, ' is periodic or reducible. Diffeomorphisms of pseudo-Anosov type cannot be periodic or reducible. Their action on the curve graph defined in Chapter 6 can be thought of as a translation along an axis;

56

10 Classification of mapping classes

their stretch factor can be recovered from that action as an asymptotic stretch factor for curves with respect to an auxiliary Riemannian metric (see [19, Section 14.6]). In contrast, there exist diffeomorphisms that are both periodic and reducible, as can be guessed from Figure 10.2. We postpone the proof of Theorem 11 to the next chapter, and continue with a homological criterion that allows us to construct diffeomorphisms of pseudoAnosov type. Proposition 18. Suppose 'W † ! † is an orientation-preserving diffeomorphism and let A 2 SL.2g; Z/ be the matrix of its induced action ' W H1 .†; Z/ ! H1 .†; Z/ with respect to any basis. If the characteristic polynomial A .t / is irreducible, has no roots of finite order, and is not a polynomial in t n , for some n  2, then ' is of pseudo-Anosov type. Proof. We first observe that a periodic mapping class 'W † ! † of order p has a periodic action in homology, whose order divides p, so all the roots of A .t / are p-th roots of unity. Next, suppose that 'W † ! † is reducible. Then there exists a finite union C  † of simple closed curves with '.C / D C . Let n 2 N be the smallest positive number so that ' n preserves all the connected components of C . If one of these components is a non-separating curve c, then the homology class Œc 2 H1 .†; Z/ is a non-trivial fixed vector for 'n , thus A .t / has a root of order n. If all the components of C are separating, then there exists a connected component Y0  † n C that has precisely one boundary component. The complement of the orbit of Y0 in †, X D † n .Y0 [ '.Y0 / [


[ ' n

1

.Y0 //;

is invariant under ', and so is its homology under the action of ' . The irreducibility of A .t / implies that the relative homology group H1 .X; @X / is zero. Therefore, X is a sphere with n boundary components, as sketched in Figure 10.2, for n D 3. Moreover, the matrix A has the following n  n-block type form, with respect to a suitable basis of H1 .†; Z/ compatible with the action of ' on the union Y0 [ '.Y0 / [


[ ' n 1 .Y0 /: 0 1 0 0


0 B BId 0


0 0 C B C B :: :: C :: :: B : : : :C A D B0 C: B :: : : : :: 0 0 C @: A : 0


0 Id 0 As a consequence, the characteristic polynomial A .t / is a polynomial in t n , so n D 1, implying that X is a disc and the curve c is contractible, which contradicts the assumption on C .  Explicit examples of pseudo-Anosov diffeomorphisms, detected by the above homological criterion, are presented in [19, Chapter 14]. The following result, due to Penner, provides a geometric construction of pseudo-Anosov diffeomorphisms. Proposition 19 (Penner’s construction [49]). Let ¹a1 ; a2 ; : : : ; am º and ¹b1 ; b2 ; : : : ; bn º be two sets of pairwise disjoint simple closed curves on a closed surface †, with minimal

Geometry and Topology of Surfaces

57

Figure 10.2. An invariant sphere X with boundary C .

pairwise intersection number, so that the complement of the union of all curves ai and bj is a finite union of embedded discs in †. Then any finite product of positive Dehn twists Tai and negative Dehn twists Tbj 1 is a pseudo-Anosov diffeomorphism, provided that the product includes Dehn twists around all the curves ai and bj , in any order. The easiest example of a pseudo-Anosov diffeomorphism arising from Penner’s construction is the product of a positive Dehn twist around the meridian 1;0  S 1  S 1 and a negative Dehn twist around the longitude 0;1  S 1  S 1 . The corresponding matrix   2 1 AD 2 SL.2; Z/ 1 1 p

has eigenvalues  D 3C2 5 and 1 . A recent result by Liechti states that this is the smallest possible stretch factor for pseudo-Anosov maps arising from Penner’s construction [38]. Another interesting application of Penner’s construction is the family of diffeomorphisms presented in Chapter 6 (see Example 3). We recall that these diffeomorphisms normally generate the mapping class groups [35]. We will address the problem of determining the lowest possible stretch factor among all pseudo-Anosov diffeomorphism of a closed surface in the last chapter. The following statement provides a first step towards bounding the stretch factor of pseudo-Anosov diffeomorphisms from below. Proposition 20. The stretch factor of a pseudo-Anosov diffeomorphism of a closed surface of genus g  2 is an algebraic integer of degree at most 6g 6. Proof. Let 'W † ! † be a pseudo-Anosov diffeomorphism with stretch factor  > 0 and let f 2 MF .†/ a measured foliation with ' .f / D f . First suppose that f admits

58

10 Classification of mapping classes

a transverse orientation. Then f is defined by a closed 1-form ! on † satisfying ' .!/ D !; so  is an eigenvalue of the induced action ' W H 1 .†; R/ ! H 1 .†; R/. In particular,  is an algebraic integer of degree at most 2g. Here we use the fact that the 1-form ! cannot be exact, since differentials of functions on compact manifolds cannot be uniformly stretched under a diffeomorphism. Q of †, branched For the general case, we consider again the double branched covering † along all the singular points of f with odd index, as in the proof of the index formula (Proposition 12). Let us estimate the potential range of values for the Euler characterisQ The largest possible value is attained when there is no singular point of odd tic .†/. index. In this case, the covering would be a regular double covering, so Q D 2.†/ D 4 .†/

4g:

The smallest possible value is attained when all the singular points have index three. By the index formula, the number of singular points equals 2.†/ D 4g 4. All these points Q so lift to single points in †, Q D 8 8g: .†/ Q !† Q stretching the lift fQ of the foliation f by . The map 'W † ! † lifts to a map 'W Q † We conclude that  is an algebraic integer of degree at most 2 .8 8g/ D 8g 6, since fQ is defined by a closed 1-form. Moreover, one can show that the action Q Z/ ! H 1 .†; Q Z/ 'Q W H 1 .†; has an invariant subspace of rank 2g, induced by a lift of H 1 .†; Z/, which further reduces the possible degree of  to 6g 6.  The precise set of algebraic degrees of pseudo-Anosov diffeomorphisms of a closed surface of genus g was recently determined by Strenner [59]: it contains all the even natural numbers from 2 to 6g 6 and all the odd natural numbers from 3 to 3g 3 (see [40] for an argument excluding all the odd numbers larger than 3g 3). In the next chapter, we will see that the stretch factor  of a pseudo-Anosov diffeomorphism is in fact the dominant eigenvalue of a Perron–Frobenius matrix, i.e. of an integer matrix M with positive entries. In particular,  is the largest root (in absolute value) of the characteristic polynomial M .t /. We close this chapter by a classical result due to Kronecker [34], describing the polynomials whose largest root has absolute value one. Proposition 21. Let p.t / 2 ZŒt  be a monic polynomial with all roots on the unit circle S 1 . Then all the roots of p.t / are roots of unity. Proof. We follow Kronecker’s proof, declaring the roots of p.t / to be a1 ; a2 ; : : : ; an 2 S 1 , where n 2 N is the degree of p.t /. Equivalently, p.t / D .t

a1 /.t

a2 /


.t

an /:

We observe that the coefficients of p.t / are bounded in absolute value by a function of n, say by nŠ, to be on the safe side. As a consequence, the polynomials of fixed degree

Geometry and Topology of Surfaces

59

with the required properties are finite in number. Starting from p.t /, we define an infinite family of polynomials pk .t / of the same degree, indexed by k 2 N: pk .t / D .t

a1k /.t

a2k /


.t

ank /:

The coefficients of these polynomials are elementary symmetric functions in the roots a1 ; a2 ; : : : ; an , so pk .t / 2 ZŒt . By the above discussion, there exist h < k with ph .t / D pk .t /; from which we conclude ¹a1h ; a2h ; : : : ; anh º D ¹a1k ; a2k ; : : : ; ank º: If a1h D a1k , then a1 is a root of unity. Otherwise, after relabelling the roots, a1h D a2k , h a2h D a3k , and so on, until we reach a label m  n with am D a1k . Then there exists N > h h N with a1 D a1 , and again, a1 is a root of unity. 

11 Perron–Frobenius theory

Keeping in mind that each individual element of the mapping class group fixes a hyperbolic metric or a projective class of a measured foliation, we are not far from proving Thurston’s classification result. We have already seen that mapping classes fixing a hyperbolic metric are periodic. We are left to prove the following: (1) Mapping classes that preserve a measured foliation are reducible or periodic. (2) Mapping classes that stretch a measured foliation by a factor  > 1 are of pseudoAnosov type. Throughout this chapter, we will assume that 'W † ! † is an orientation-preserving diffeomorphism that preserves the projective class of a measured foliation f 2 MF .†/, i.e. there exists   1 with ' .f / D f : We will further assume that ' is not reducible, meaning that ' does not admit any essential invariant compact 1-submanifold C  †. Let €  † be the union of singular leaves of the invariant foliation f . The boundary of a small neighborhood of the union of compact singular leaves of € is a '-invariant 1-submanifold, since € is '-invariant, up to isotopy and Whitehead moves. All components of this 1-submanifold must be contractible, so € is a finite union of trees. After contracting all the compact edges of these trees, € turns into a finite union of star-shaped trees. Now the measured foliation ' .f / is isotopic to f , without invoking any Whitehead moves. Case (1):  D 1. The map ' permutes the singular points of f , as well as the finite set of singular leaves emanating from these singular points. There exists a number n 2 N so that ' n fixes all singular points and all singular leaves. We will show that ' n is isotopic to the identity. For this purpose, we will decompose † into an invariant union of horizontal rectangles. We define a horizontal rectangle with respect to the measured foliation f to be the image of a smooth map W Œ0; 12 ! † with (i)

is an immersion whose restriction to .0; 1/2 is an embedding,

(ii) for all t 2 Œ0; 1, the arc

.Œ0; 1  ¹tº/ is contained in a finite union of leaves of f ,

(iii) for all s 2 Œ0; 1, the arc

.¹sº  Œ0; 1/ is transverse to the leaves of f .

We denote by @T R D .¹0; 1º  Œ0; 1/ and @k R D .Œ0; 1  ¹0; 1º/ the union of the two transverse and parallel boundary components of a horizontal rectangle R, respectively. Two examples of horizontal rectangles are shown in Figure 11.1. Let p 2 † be a singular point of f and let  .p/ D 1 [


[ m be a finite union of embedded transverse arcs of equal length incident to p, one for each sector of the singular

62

11 Perron–Frobenius theory

Figure 11.1. Horizontal rectangles.

Figure 11.2. System of arcs .

point, as shown in Figure 11.2. Here m is the index of the singular point p. Choose a system of arcs  .p/ for each singular point p of f and let  be the finite union of these systems of arcs. Lemma 8. There exists a unique minimal finite collection of horizontal rectangles R1 ; : : : ; RN  † covering † with pairwise disjoint interior, so that N [

@T Rk D :

kD1

Proof. We specify a finite set of corner points V  † for the system of rectangles Rk , as follows: @  V , where @ denotes the union of endpoints of arcs of the system , including all singular points. The first intersection point of all leaves starting at a point of @ with an arc of the system  belongs to V , as shown in Figure 11.3. Consider the union of horizontal rectangles R1 ; : : : ; R N  † whose transverse boundary components are subarcs of  with endpoints in V . Let M be the union of these rectangles. We claim that † D M . Indeed, otherwise the boundary @M  † would be a non-empty closed cycle contained in a finite union of leaves of f .

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63

Figure 11.3. Corner points of horizontal rectangles.

This cycle would have to be parallel to a singular cycle of leaves of f , which is impossible, since the union of singular leaves € is a tree. Altogether, we constructed a finite union of horizontal rectangles with the desired properties. Moreover, the construction is unique, since all the points of V have to be corner points of horizontal rectangles.  Now we are in a position to argue that our diffeomorphism ' is periodic. We already chose n 2 N so that ' n fixes all singular points, as well as all singular leaves emanating from them. We may further assume that ' n restricts to the identity on , since  D 1. This implies that ' n maps each rectangle Rk of the above collection to itself, fixing its boundary. As a consequence, ' n is isotopic to the identity, by Alexander’s trick. In turn, ' is periodic. Case (2):  > 1. We need to construct a transverse invariant measured foliation f with 1 ' .f / D : f As in the first case, we choose a system of arcs  with one transverse arc for each sector of a singular point. The stretch factor being strictly larger than one, we can only assume '. /  . Let n 2 N be the smallest natural number so that ' n fixes all singular points of f and maps each sector of a singular point to itself. Every arc k   intersects a singular leaf Lk of f in its interior. We denote by ˛k the subarc of Lk joining the singular point of Lk to the arc k and define a finite set of singular arcs ! n [ [ j LD ' .Lk / : k 

j D1

The set L is not '-invariant, but satisfies L  '.L/. Shorten every arc k so that @k 2 L. Fill the surface with a finite union of rectangles R1 ; : : : ; RN  †, so that for all i  N , '.@T Ri / 

N [ j D1

@T Rj

64

11 Perron–Frobenius theory

and '

1

.@k Ri / 

N [

@k Rj :

j D1

We will determine the measure of the transverse foliation f by associating a width xi > 0 to each rectangle Ri , as follows. Define a matrix A 2 N N N whose coefficient aij is the number of connected components of '.Ri /ı \ Rjı : Since all leaves of the foliation f are dense in †, there exists K 2 N so that AK is strictly positive. By the theorem of Perron and Frobenius [51, 23], the matrix A has a unique positive eigendirection spanned by a vector v 2 RN >0 corresponding to a positive eigenvalue ˛ > 1 of maximal modulus. Let x1 ; : : : ; xN > 0 be the coordinates of that vector (or any scaling thereof), and declare the width of the rectangle Ri to be xi . This defines a transverse measure  for the foliation f , satisfying './ D

1 : ˛

The product of measures    defines an area form on † with ' .  / D

   ; ˛

which implies ˛ D , since ' is a diffeomorphism. We conclude that 'W † ! † is a pseudo-Anosov diffeomorphism. The reader is referred to [20, Exposé 9] for details for the construction of the transverse invariant foliation.

12 Pseudo-Anosov maps with small stretch factor

In the previous chapter, we saw that the stretch factor  of a pseudo-Anosov diffeomorphism 'W † ! † is a Perron–Frobenius eigenvalue, i.e. the largest eigenvalue of a matrix A 2 N N N , where N 2 N is the number of rectangles in a covering of † adapted to the action of '. From the construction of this covering, it is clear that the number N is bounded above by a linear expression in the genus g of †. In fact, it is bounded by a constant times the number of arcs of the system   †, which is proportional to the genus, thanks to the index formula, Proposition 12. Let v 2 RN >0 be an eigenvector corresponding to the maximal eigenvalue  > 0 of A. Knowing that the matrix A has a power with strictly positive coefficients, we conclude that at least one column of A has sum norm at least two. As a consequence, kAN ei k1  2

for all i  N ,

and kAN vk1  2kvk1 ; since v is strictly positive. This implies N  2; so log  is bounded below by a constant times g1 . Following Penner [50], we define the spectrum Spec.MCG.†// of the mapping class group of the surface † to be the set of logarithms of stretch factors of pseudo-Anosov diffeomorphisms of †. This set admits a reinterpretation as the logarithmic length spectrum of the moduli space Mg D T .†g /=MCG.†g /; endowed with the Teichmüller metric, as explained in [19, Section 14.2.2]. In particular, the minimal stretch factor among all pseudo-Anosov diffeomorphisms of a fixed surface can be thought of as the length of a systole of the corresponding moduli space. By the above argument, due to Penner, Spec.MCG.†// has a lower bound of order g1 . Moreover, Spec.MCG.†// is discrete, since stretch factors are algebraic numbers whose degree is bounded by the genus (Proposition 20, see also [2]). In [50], Penner constructs families of pseudo-Anosov diffeomorphisms with log  of order g1 . These examples admit a simple description on a surface with a rotational symmetry of order g  2. Let a; b; c  † be a chain of three simple closed curves, as shown in Figure 12.1. Define a diffeomorphism ' D RTc Ta 1 Tb , where RW † ! † denotes a rotation of order g. Writing P D Tc Ta 1 Tb ;

66

12 Pseudo-Anosov maps with small stretch factor

Figure 12.1. Pseuso-Anosov map with small stretch factor.

we compute successive powers of ': ' 2 D .RP /2 D R2 .R 3

3

g

g

3

' D .RP / D R .R ' D .RP / D R

1

PRP /;

2

PR2 R

.g 1/

PR

g 1

1

PRP /;

R

.g 2/

PRg

2




R

1

PRP;

and realize that ' g is a product of 3g Dehn twists, iterate conjugates of P D Tc Ta 1 Tb by the rotation R. A closer look at the twist curves reveals that ' g arises from Penner’s construction, Proposition 19. One can show that the stretch factor of ' g is smaller than a constant (in fact smaller than 11, see [50]), for all g, so log  grows like g1 . Here we recall that the stretch p factor of a diffeomorphism arising from Penner’s construction cannot be smaller than 3C2 5 (see [38]). The positive answer to Schinzel and Zassenhaus’ question, recently announced in [15], implies that the smallest spectral radius n > 1 among all matrices A 2 SL.n; Z/ has a logarithmic growth of order n1 . An interesting reformulation of this fact in terms of reciprocal and skew-reciprocal polynomials was recently formulated by Liechti [39]. The last result of this lecture states that all pseudo-Anosov diffeomorphisms with small stretch factor can be decomposed as a product of a periodic map and a reducible map. According to Margalit, this result is due to Leininger, see [41, Section 10]. Proposition 22. Let 'W † ! † be a pseudo-Anosov diffeomorphism with stretch factor  < 23 . Then there exist a periodic diffeomorphism P and a reducible diffeomorphism R of † with ' D PR. Penner’s construction illustrates this fact, with the roles of the two maps P and R interchanged, a notational inconsistency that will help keeping the reader alert on these last couple of pages.

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As mentioned in Chapter 7, the maximal order of a periodic diffeomorphism of a closed surface of genus g is 4g C 2. Moreover, the set of conjugacy classes of periodic mapping classes of a fixed surface is finite. A combinatorial model for periodic diffeomorphisms, based on A’Campo’s tête-à-tête graphs, was recently described by Graf [24] and de Bobadilla, Pe Pereira and Portilla Cuadrado [12]. Not all diffeomorphisms 'W † ! † admit a product decomposition into a periodic and a reducible diffeomorphism, due to the existence of a wealth of unbounded quasi-morphisms on the mapping class groups [6]. The proof of Proposition 22 makes use of the following statement, which is due to Farb, Leininger and Margalit [18]. Lemma 9 ([18, Proposition 2.7]). Let 'W † ! † be a pseudo-Anosov diffeomorphism with i.c; '.c//  n  3 for all c 2 S.†/. Then the stretch factor  of ' satisfies   n : log./  log 2 Proof of Lemma 9. We endow † with a flat metric q with cone points in all singular points of the invariant measured foliation f of ', so that the action of ' away from the singular points is locally modelled on the matrix    0 : 0  1 Let c  † be a shortest non-contractible simple closed curve with respect to the metric q. We may suppose that c is embedded, otherwise we could work with a sufficiently small perturbation of q which is a Riemannian metric without singularities. Now the curve c cuts its image '.c/, which we suppose to intersect c transversally, into finitely many arcs, i.c; '.c// in number. Let a  '.c/ be a shortest arc of that type, and let b  c be the shorter arc joining the two endpoints of a along c. We estimate lq .c/  lq .a [ b/ 

lq .c/ lq .c/ lq .c/ lq .'.c// C  C ; i.c; '.c// 2 i.c; '.c// 2

from which we conclude     n i.c; '.c//  log : log./  log 2 2



Proof of Proposition 22. According to the previous lemma, a pseudo-Anosov diffeomorphism 'W † ! † with stretch factor  < 32 admits a simple closed curve c  † with i.c; '.c//  2: We claim that there exists a periodic diffeomorphism P W † ! † with P .c/ D '.c/. From this, we obtain ' D PR with P periodic and R D P 1 ' reducible. We are left to prove the claim about curves a D c; b D '.c/ 2 S.†/ with i.a; b/  2.

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12 Pseudo-Anosov maps with small stretch factor

Case (1): i.a; b/ D 0. In this case, there exists P W † ! † of finite order dividing g with P .a/ D b, as shown at the top left of Figure 12.2, for the two cases of non-separating and separating pairs of curves a; b. Case (2): i.a; b/ D 1. In this case, both curves a; b are non-separating, and there exists P W † ! † of order g with P .a/ D b, as shown at the top right of Figure 12.2.

Figure 12.2. Periodic maps.

Case (3): i.a; b/ D 2. In this case, there are three configurations to consider: 

† n .a [ b/ has four connected components,



† n .a [ b/ has two connected components,



† n .a [ b/ is connected.

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In all three cases, there exists P W † ! † of finite order dividing g with P .a/ D b, as shown at the bottom of Figure 12.2.  One might wonder if the above decomposition theorem admits an analogue in the context of integer matrices. p Question 3. Let A 2 SL.n; Z/ be a matrix with small spectral radius, say   2. Does there exist a periodic matrix P 2 SL.n; Z/, so that P 1 A has a non-zero fixed point in Zn ? The answer is “yes” for non-negative matrices, since every non-negative vector v 2 Zn with small sum norm (say 1 or 2) can be complemented to a periodic matrix.

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Index

.9g 9/-theorem, 23, 45 84.g 1/-theorem, 21 MCG.†/, 1 MF .†/, 40 PF .†/, 47 PMF .†/, 41 PSL.2; R/, 7, 19 R6g 6 , 23 S6g 7 , 47 SL.2; Z/, 3, 5 SL.n; Z/, 4, 69 T .†/, 19

Kronecker, 58

Alexander trick, 29 algebraic integer, 57

pair of pants, 13, 19, 45 Penner construction, 56 periodic, 2, 4, 55, 66 Perron–Frobenius matrix, 58 projective measured foliation, 41, 47 pseudo-Anosov, 2, 55, 65, 66

Basmajian identity, 10 bigon criterion, 26 braid relation, 27 center, 29 collar theorem, 16 compactification, 49 curve graph, 32 Dehn twist, 25, 32 Fenchel–Nielsen coordinates, 23 geodesic, 7, 14, 26 homological criterion, 56 horizontal rectangle, 61 Hurwitz, 21, 38 hyperbolic plane, 7 hyperbolic structure, 13, 19 index formula, 38 index of a singular point, 37 intersection number, 25 irreducible, 56 isometry, 7, 20

Lambert quadrilateral, 8 leaf, 37 Lehmer problem, 5 Möbius transformation, 7 Mahler measure, 5 mapping class group, 1, 31, 55 measured foliation, 37 orthogeodesic, 10 orthospectrum, 10

quasi-transverse arc, 41 reducible, 2, 4, 55, 66 right-angled hexagon, 17, 20 simple closed curve, 1, 3, 14, 25 spectral radius, 4, 69 spectrum, 65 stretch factor, 55, 57, 65, 66 systole, 15, 17, 49 Teichmüller space, 19 Thurston classification, 55 torus, 3 transverse arc, 41 transverse measure, 40 transverse measured foliation, 55 uniformization theorem, 7 universal covering, 13 upper half-plane, 7 Whitehead operation, 40

Z U R I C H L E CT U R E S I N A DVA N C E D M AT H E M AT I C S

Sebastian Baader

Geometry and Topology of ­Surfaces These lecture notes cover the classification of hyperbolic structures and measured foliations on surfaces in a minimalist way. While the inspiration is obviously taken from the excellent books Primer on mapping class groups and Travaux de ­Thurston sur les surfaces, we tried to include a little bit more of hyperbolic trigonometry, including a proof of Basmajian’s identity on the orthogeodesic spectrum, while keeping the rest short.

https://ems.press ISBN 978-3-98547-000-6