Table Of ContentThurston’s Work on Surfaces
A. Fathi, F. Laudenbach, V. Po´enaru, et al.
Translation by Djun Kim and Dan Margalit
This translation is copyright c 2009 Djun M. Kim and Dan Margalit
(cid:13)
Contents
Expos´e 1. Thurston’s Theorems on Surfaces 1
1.1 Introduction 1
1.2 The space of simple closed curves 1
1.3 Measured foliations 4
1.4 Teichmu¨ller space 6
1.5 Pseudo-Anosov diffeomorphisms 8
1.6 The case of the torus 11
Expos´e 2. Surface Diffeomorphisms 19
2.1 The space of homotopy equivalences of a surface 19
2.2 The braid groups 20
2.3 Diffeomorphisms of the pair of pants 26
Expos´e 3. Hyperbolic Geometry in Dimension 2 33
3.1 A little hyperbolic geometry 33
3.2 The Teichmu¨ller space of the pair of pants 37
3.3 Generalities on intersection number 46
3.4 Systems of simple closed curves and hyperbolic isometries 55
Expos´e 4. The Space of Simple Closed Curves in a Surface 57
4.1 The weak topology on the space of simple closed curves 57
4.2 The space of multicurves 59
4.3 An explicit parametrization of the space of multicurves 61
Appendix A. Pair of Pants Decompositions 69
Expos´e 5. Measured Foliations 73
5.1 Measured foliations and the Euler–Poincar´e Formula 73
5.2 Poincar´e Recurrence and the Stability Lemma 77
5.3 Measured foliations and simple closed curves 82
5.4 Curves as measured foliations 92
Appendix B. Spines of Surfaces 97
Expos´e 6. The Classification of Measured Foliations 101
6.1 Foliations of the annulus 102
6.2 Foliations of the pair of pants 104
6.3 The pants seam 110
6.4 The normal form of a foliation 113
6.5 Classification of measured foliations 119
6.6 Enlarged curves as functionals 126
6.7 Minimality of the action of the mapping class group on PMF 128
6.8 Complementary measured foliations 130
Appendix C. Explicit Formulas for Measured Foliations 131
Expos´e 7. Teichmu¨ller Space 139
Expos´e 8. The Thurston Compactification 153
8.1 Preliminaries 153
8.2 The Fundamental Lemma 157
8.3 The manifold T 162
Appendix D. Estimates of Hyperbolic Distances 167
D.1 The hyperbolic distance from i to a point z 167
0
D.2 Relations between the sides of a right hyperbolic hexagon 169
D.3 Bounding distances in pairs of pants 171
Expos´e 9. The Classification of Surface Diffeomorphisms 175
9.1 Preliminaries 175
9.2 Rational foliations (the reducible case) 177
9.3 Arational measured foliations 177
9.4 Arational foliations with λ=1 (the finite order case) 182
9.5 Arational foliations with λ6=1 (the pseudo-Anosov case) 183
9.6 Some properties of pseudo-Anosov diffeomorphisms 194
Expos´e 10. Dynamics of Pseudo-Anosov Diffeomorphisms 201
10.1 Topological entropy 201
10.2 The fundamental group and entropy 205
10.3 Subshifts of finite type 212
10.4 The entropy of pseudo-Anosov diffeomorphisms 216
10.5 ConstructingMarkovpartitionsforpseudo-Anosovdiffeomorphisms 223
10.6 Pseudo-Anosov diffeomorphisms are Bernoulli 227
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Expos´e 11. Diffeomorphisms of Surfaces with Boundary 231
11.1 The space of curves and measured foliations 231
11.2 Teichmu¨ller space and its compactification 234
11.3 A sketch of the classification of diffeomorphisms 235
11.4 Thurston’s classification and Nielsen’s theorem 241
11.5 The spectral theorem 245
Expos´e 12. Uniqueness Theorems for Pseudo-Anosovs 249
12.1 Statement of results 249
12.2 The Perron–Frobenius Theorem and Markov partitions 251
12.3 Unique ergodicity 253
12.4 The action of pseudo-Anosovs on PMF 256
12.5 Uniqueness of pseudo-Anosov maps 266
Expos´e 13. Construction of pseudo-Anosov maps 271
13.1 Generalized pseudo-Anosov diffeomorphisms 271
13.2 A construction by ramified covers 272
13.3 A construction by Dehn twists 274
1
Expos´e 14. Fibrations of S 281
Expos´e 15. Presentation of the mapping class group 303
15.1 Preliminaries 303
15.2 A method for presenting the mapping class group 304
15.3 The cell complex of marked functions 307
15.4 The marking complex 312
15.5 The case of the torus 316
References 328
Index 329
i
FOREWORD TO FIRST EDITION
This book contains an exposition of results of William Thurston
on the theory of surfaces: measured foliations, the natural compact-
ification of Teichmu¨ller space, and the classification of surface dif-
feomorphisms. Our scope is essentially that outlined in the research
announcement of Thurston, and in the notes of his Princeton course,
as written up by M. Handel and W. Floyd.
Part of this work, most notably the classification of curves and of
measured foliations, is an elaboration of lectures made in the Semi-
naire d’Orsay in 1976–1977. But we were not able to write the proofs
for the remaining portions of the theory until much later. In the
Spring of 1978, at Plans-Sur-Bex, Thurston explained to us how to
see the projectivization of the space of measured foliations as the
boundary of Teichmu¨ller space.
The first expos´e enumerates the principal results, the proofs of
which follow in Expos´es 2 through 13. The last two expos´es present
work somewhat marginal to the theme of the classification of surface
diffeomorphisms. Expos´e14,orallypresentedbyD.FriedandD.Sulli-
van, discusses nonsingular closed 1-forms on 3-dimensional manifolds,
following Thurston; in particular it treats fiber bundles over S1 for
which the monodromy diffeomorphism is pseudo-Anosov. Expos´e 15,
presented orally by A. Marin, gives a finite presentation of the map-
ping class group, following Hatcher and Thurston.
The seminar consisted also of expos´es of an analytical nature (holo-
morphic quadratic differentials, quasiconformal mappings) presented
by W. Abikoff, L. Bers, and J. Hubbard. In the end, the two points of
view were found to be more independent of each other than was ini-
tially believed. The analytic point of view is the subject of a separate
text written by W. Abikoff; see [Abi80].
We thank all of the active participants in the seminar; all have
contributed assistance in various sections: A. Douady, who, after the
oral presentations, helped us to capture the content of the lectures;
M.Shub,whodiscussedwithustheergodicpointofview; D.Sullivan,
who besides giving much advice and encouragement, strove to make
us understand how the image of a curve under iteration of a pseudo-
ii
Anosovdiffeomorphism“approaches”afoliationofthesurface(ittook
many more months to fully understand this “mixing”).
Finally, we thank Mme. B. Barbichon (typography) and S. Berberi
(illustrations)forthecarethattheytookinpreparingthemanuscript.
FOREWORD TO SECOND EDITION
Theresearchannouncement“Onthegeometryanddynamicsofdif-
feomorphismsofsurfaces”,byWilliamThurston([Thu88]),hasfinally
appeared in Bulletin of Amer. Math. Soc. 19 (1988), 417–431. One
can find there a list of references to later work and to the first edition
of this book. We also point out the book ([CB88]) by S. Bleiler and
A. Casson, “Automorphisms of surfaces after Nielsen and Thurston,”
Cambridge University Press, 1988.
We limit ourselves to a few corrections that one can find assembled
in an errata at the end of this volume.1
Orsay, May 27, 1991
1Translators’ note: We have incorporated these corrections into the main text
of the translated edition.
iii
TRANSLATORS’ NOTES
We are very happy to present this translation of the now-classic text
Travaux de Thurston sur les surfaces ([FLP79]), commonly referred
to as FLP. We have attempted to stay as faithful to the original as
possible, making small modifications only as necessary.
Inthethreedecadessinceitsoriginalpublication,FLP wastheonly
sourceformanyofthedetailsinvolvedinthemeasuredfoliationspoint
of view on Thurston’s work. However, several other books and papers
have appeared that elucidate other aspects of Thurston’s theory, for
instancethoseofgeodesiclaminations, traintracks, geodesiccurrents,
and quadratic differentials. The following works give an overview of
these perspectives.
An Extremal Problem for Quasiconformal Mappings and a Theorem
by Thurston, Lipman Bers, Acta Mathematica, 1978. ([Ber78])
The Real Analytic Theory of Teichmu¨ller Space, William Abikoff,
Springer, 1980. ([Abi80])
New Proofs of Some Results of Nielsen, Michael Handel and William
P. Thurston, Advances in Mathematics, 1985. (HanThu85)
Onthegeometryanddynamicsofdiffeomorphismsofsurfaces,William
P. Thurston, AMS Bulletin, 1988. ([Thu88])
Automorphisms of Surfaces After Nielsen and Thurston, Andrew J.
Casson and Steven A. Bleiler, Cambridge, 1988. ([CB88])
The Geometry of Teichmu¨ller Space Via Geodesic Currents, Francis
Bonahon, Inventiones Mathematicae, 1988. ([Bon88])
Subgroups of Teichmu¨ller Modular Groups, Nikolai V. Ivanov, AMS,
1992. ([Iva92])
Train Tracks for Surface Homeomorphisms, Mladen Bestvina and
Michael Handel, Topology, 1995. ([BH95])
Teichmu¨ller Theory and Applications to Geometry, Topology, and Dy-
namics, Volume I: Teichmu¨ller Theory, John H. Hubbard, Matrix
Editions, 2006. ([Hub06])
A Primer on Mapping Class Groups, Benson Farb and Dan Margalit,
Princeton University Press, to appear.
iv
ACKNOWLEDGEMENTS
We would like to extend our sincerest thanks to Jayadev Athreya,
Matt Bainbridge, Jessica Banks, Jason Behrstock, Aaron Brown,
Indira Chatterji, Matt Clay, Moon Duchin, Justin Malestein,
Ben McReynolds, Erica Meucci, Dylan Thurston, Liam Watson, and
the referees for reading early drafts and offering corrections and help-
ful suggestions. We are especially grateful to Chris Leininger and
Lee Mosher for several in-depth conversations.
Djun would like to thank Dale Rolfsen, for providing support, en-
couragement and many fruitful discussions; the Mathematics depart-
ments at Berkeley and McGill, for providing library and computer fa-
cilitiesduringvisitsin1990and1996respectively; andElisabethKim,
who read early drafts of some of the expos´es and provided corrections
and helpful suggestions. He is grateful for the encouragement and ad-
viceofBillCasselman,CurtMcMullen,KevinPilgrim,LarryRoberts,
and Jonathan Walden. Last, but certainly not least, he thanks his
wife, Pamela Richardson, for her wisdom, strength and love.
Dan would like to thank the University of Chicago, the University
of Utah, Tufts University, and the National Science Foundation for
providing resources and pleasant working environments. He is also
grateful to Benson Farb, Mladen Bestvina, and Kevin Wortman for
their invaluable mathematical and moral support. He would like to
thank his family for their endless encouragement, and most of all his
wife, Kathleen, for her constant love, support, and inspiration.
Finally, we would like to thank William Thurston, as well as Albert
Fathi, Franc¸ois Laudenbach, Valentin Po´enaru, and all others who
contributed to the beautiful mathematics contained in this volume.
Djun Kim Dan Margalit
Vancouver, B.C., Canada Medford, MA
[email protected] [email protected]
