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Publisher:

Princeton University Press

Publication Date:

2012

Number of Pages:

255

Format:

Paperback

Series:

Mathematical Notes

Price:

60.00

ISBN:

9780691147352

Category:

Monograph

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by , on ]

Scott Taylor

03/7/2013

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

If

When *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

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

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

*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.

**References:**

[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

171

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 S^{1} 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

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