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Thurston's Work on Surfaces

Albert Fathi, François Laudenbach, and Valentin Poénaru
Princeton University Press
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Mathematical Notes
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Scott Taylor
, on

William Thurston made many beautiful and fundamental contributions to mathematics. Perhaps best known, in recent years, has been his Geometrization Conjecture (proven by Grigori Perelman). One of Thurston’s important earlier contributions (and one closely related to the Geometrization Conjecture) was a classification of surface diffeomorphisms. Although many of the details are quite technical and require machinery from hyperbolic geometry, measure theory, and dynamical systems, the statement of the classification is both simple and beautiful.

Thurston himself never wrote down the details of his arguments, but he presented many of them in a seminar in Orsay in 1976–1977. Thurston’s lectures were expanded into a collection of “exposés” by Fathi, Laudenbach, and Poénaru and published in 1979 as Travaux de Thurton sur les surfaces. It rapidly became a classic and much of the material became standard for mathematicians working in low-dimensional topology and geometry, Teichmüller theory, and Kleinian groups. This new English translation by Kim and Margalit remains faithful to the original while incorporating a few corrections and making a limited number of comments on subsequent developments. The book remains essential reading and hopefully this translation will make Thurston’s ideas even better known outside the research communities already well-acquainted with them.

Before commenting on the text itself, I can’t resist the opportunity to share the essence of Thurston’s classification. For our purposes, we will consider closed (= compact, without boundary), orientable surfaces. Recall that such surfaces are classified by their genus. Let S be such a surface and g(S) its genus and consider a homeomorphism h:SS. For simplicity, assume also that the homeomorphism is orientation preserving.

If g(S)=0, then S is a sphere and, by classical results from algebraic topology, h has a fixed point and h can be deformed (by “isotopy”) so that it is the identity. This is the classification for the case when S is a sphere (a result which considerably predates Thurston).

When g(S)=1, S is a torus and the homeomorphisms are more interesting. We can build the torus by identifying opposite sides of a parallelogram of, say, unit area. The universal cover of the torus is the plane R2 and the covering map can be used to give the torus a Euclidean geometry. Every homeomorphism of the torus can be deformed so that the images of geodesics (locally length-minimizing paths) are geodesics. A result of Nielsen classifies the homeomorphisms of the torus into those that are reducible, periodic, or Anosov. The reducible homeomorphisms are those that (up to deformation) leave a simple closed curve invariant.

For example, cut the torus open along a simple closed curve, twist by 2π, and then reglue. The periodic homeomorphisms are those which have a non-trivial power which can be deformed to the identity map. If the torus is obtained by identifying opposite sides of a square, rotation of the square by π/2 is an example of a periodic homeomorphism. Anosov homeomorphisms stretch the torus by a factor of λ>1 in one direction and shrink it by a factor of 1/λ in the other direction. Anosov homeomorphisms do not leave any simple closed curve invariant but (up to deformation) they do leave invariant two transverse geodesic foliations on the torus. For example, let p:R2S be the covering map which identifies all points of R2 whose coordinates differ by integers. Let h~:R2R2 be the linear map h~(x,y)=(1213)(x,y). Let h=ph~p1. It is not hard to check that h defines a homeomorphism of the torus. The map h~ stretches R2 by a factor of λ=(2+3) along the line L1 defined by y=(13)x. It shrinks R2 by a factor of 1/λ=23 along the line L2 defined by y=(13)x. The covering map p projects the sets of lines parallel to L1 and L2 to transverse foliations on S. Finally, whether a homeomorphism is reducible, periodic, or Anosov can be determined with the trace of the induced isomorphism on the first homology group of the torus.

Thurston’s achievement is to show that Nielsen’s classification of homeomorphisms also applies, with suitable adaptations, to surfaces of higher genus. The homeomorphisms are classified as reducible, periodic, and pseudo-Anosov. Reducible and periodic homeomorphisms are defined the same way as for homeomorphisms of the torus. Pseudo-Anosov homeomorphisms, like Anosov homeomorphisms of the torus, stretch and shrink the surface in transverse directions. When g(S)2, hyperbolic geometry is used to define geodesics and measure the stretching factor along singular measured foliations.

The book under review thoroughly investigates all three types of homeomorphisms of surfaces and proves Thurston’s classification theorem. In particular, the book carefully explains (singular, measured) foliations, their relationship to surface homeomorphisms, and how they can be used to construct “Thurston’s compactification of Teichmüller space.” Several chapters are devoted to applying the classification inductively by cutting open the surface along a simple closed curve left invariant by a reducible homeomorphism. Later chapters also elucidate some of the dynamics of pseudo-Anosov maps and construct them in different ways. The penultimate chapter gives an account of Thurston’s norm on the first cohomology group of a 3-manifold and its relation to 3-manifolds which are obtained by gluing the boundary components of S×[0,1] together using a pseudo-Anosov homeomorphism S×{0}S×{1}. The final chapter explains Hatcher and Thurston’s presentation of the mapping class group (i.e. group of homeomorphisms modulo isotopy) of S.

Thurston’s Work on Surfaces is very clear and very well-written, although there are many arguments left to the dedicated reader. The prerequisites for the book are a bit of a hodgepodge: certainly some differential topology and geometry is required. Reading [CB] and [FM] simultaneously is also suggested. The first also proves Thurston’s classification theorem, but in a somewhat different and somewhat more linear manner. It, however, does not discuss the connections to Teichmüller space or dynamics. The second is a comprehensive introduction to mapping class groups and gives an expanded explanation of some of the hyperbolic geometry techniques.

In summary, Thurston’s Work on Surfaces is a pleasure to study, for it contains beautiful, challenging, and exceedingly useful mathematics developed by one of the great mathematicians of the late twentieth century.


[CB] Andrew J. Casson and Steven A. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, Cambridge, 1988.

[FM] Benson Farb and Dan Margalit, A Primer on Mapping Class Groups, Princeton, 2012.

Scott Taylor is Assistant Professor of Mathematics at Colby College.

Preface ix
Foreword to the First Edition ix
Foreword to the Second Edition x
Translators' Notes xi
Acknowledgments xii
Abstract xiii

Chapter 1 An Overview of Thurston's Theorems on Surfaces 1
Valentin Poénaru
1.1 Introduction 1
1.2 The Space of Simple Closed Curves 2
1.3 Measured Foliations 3
1.4 Teichmüller Space 5
1.5 Pseudo-Anosov Diffeomorphisms 6
1.6 The Case of the Torus 8

Chapter 2 Some Reminders about the Theory of Surface Diffeomorphisms 14
Valentin Poénaru
2.1 The Space of Homotopy Equivalences of a Surface 14
2.2 The Braid Groups 15
2.3 Diffeomorphisms of the Pair of Pants 19

Chapter 3 Review of Hyperbolic Geometry in Dimension 2 25
Valentin Poénaru
3.1 A Little Hyperbolic Geometry 25
3.2 The Teichmüller Space of the Pair of Pants 27
3.3 Generalities on the Geometric Intersection of Simple Closed Curves 35
3.4 Systems of Simple Closed Curves and Hyperbolic Isometries 42
V4 The Space of Simple Closed Curves in a Surface 44
Valentin Poénaru
4.1 The Weak Topology on the Space of Simple Closed Curves 44
4.2 The Space of Multicurves 46
4.3 An Explicit Parametrization of the Space of Multicurves 47
A Pair of Pants Decompositions of a Surface 53
Albert Fathi

Chapter 5 Measured Foliations 56
Albert Fathi and François Laudenbach
5.1 Measured Foliations and the Euler-Poincaré Formula 56
5.2 Poincaré Recurrence and the Stability Lemma 59
5.3 Measured Foliations and Simple Closed Curves 62
5.4 Curves as Measured Foliations 71
B Spines of Surfaces 74
Valentin Poénaru

Chapter 6 The Classification of Measured Foliations 77
Albert Fathi
6.1 Foliations of the Annulus 78
6.2 Foliations of the Pair of Pants 79
6.3 The Pants Seam 84
6.4 The Normal Form of a Foliation 87
6.5 Classification of Measured Foliations 92
6.6 Enlarged Curves as Functionals 97
6.7 Minimality of the Action of the Mapping Class Group on PMF 98
6.8 Complementary Measured Foliations 100
C Explicit Formulas for Measured Foliations 101
Albert Fathi

Chapter 7 Teichmüller Space 107
Adrien Douady; notes by François Laudenbach

Chapter 8 The Thurston Compactification of Teichmüller Space 118
Albert Fathi and François Laudenbach
8.1 Preliminaries 118
8.2 The Fundamental Lemma 121
8.3 The Manifold T 125
D Estimates of Hyperbolic Distances 128
Albert Fathi
D.1 The Hyperbolic Distance from i to a Point z0 128
D.2 Relations between the Sides of a Right Hyperbolic Hexagon 129
D.3 Bounding Distances in Pairs of Pants 131

Chapter 9 The Classification of Surface Diffeomorphisms 135
Valentin Poénaru
9.1 Preliminaries 135
9.2 Rational Foliations (the Reducible Case) 136
9.3 Arational Measured Foliations 137
9.4 Arational Foliations with ? = 1 (the Finite Order Case) 140
9.5 Arational Foliations with ? 6= 1 (the Pseudo-Anosov Case) 141
9.6 Some Properties of Pseudo-Anosov Diffeomorphisms 150

Chapter 10 Some Dynamics of Pseudo-Anosov Diffeomorphisms 154
Albert Fathi and Michael Shub
10.1 Topological Entropy 154
10.2 The Fundamental Group and Entropy 157
10.3 Subshifts of Finite Type 162
10.4 The Entropy of Pseudo-Anosov Diffeomorphisms 165
10.5 Constructing Markov Partitions for Pseudo-Anosov Diffeomorphisms
10.6 Pseudo-Anosov Diffeomorphisms are Bernoulli 173

Chapter 11 Thurston's Theory for Surfaces with Boundary 177
François Laudenbach
11.1 The Spaces of Curves and Measured Foliations 177
11.2 Teichmüller Space and Its Compactification 179
11.3 A Sketch of the Classification of Diffeomorphisms 180
11.4 Thurston's Classification and Nielsen's Theorem 184
11.5 The Spectral Theorem 188

Chapter 12 Uniqueness Theorems for Pseudo-Anosov Diffeomorphisms 191
Albert Fathi and Valentin Poénaru
12.1 Statement of Results 191
12.2 The Perron-Frobenius Theorem and Markov Partitions 192
12.3 Unique Ergodicity 194
12.4 The Action of Pseudo-Anosovs on PMF 196
12.5 Uniqueness of Pseudo-Anosov Maps 204

Chapter 13 Constructing Pseudo-Anosov Diffeomorphisms 208
François Laudenbach
13.1 Generalized Pseudo-Anosov Diffeomorphisms 208
13.2 A Construction by Ramified Covers 209
13.3 A Construction by Dehn Twists 210

Chapter 14 Fibrations over S1 with Pseudo-Anosov Monodromy 215
David Fried
14.1 The Thurston Norm 216
14.2 The Cone C of Nonsingular Classes 218
14.3 Cross Sections to Flows 224

Chapter 15 Presentation of the Mapping Class Group 231
François Laudenbach and Alexis Marin
15.1 Preliminaries 231
15.2 A Method for Presenting the Mapping Class Group 232
15.3 The Cell Complex of Marked Functions 234
15.4 The Marking Complex 238
15.5 The Case of the Torus 241

Bibliography 243
Index 251