Collected Works of William P. Thurston with Commentary: I. Foliations, Surfaces and Differential Geometry 1470451646, 9781470451646

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COLLECTED WORKS OF

WILLIAM P. THURSTON WITH COMMENTARY

I

Foliations, Surfaces and Differential Geometry Benson Farb David Gabai Steven P. Kerckhoff Editors

COLLECTED WORKS OF

WILLIAM P. THURSTON WITH COMMENTARY

COLLECTED WORKS OF

WILLIAM P. THURSTON WITH COMMENTARY

I

Foliations, Surfaces and Differential Geometry Benson Farb David Gabai Steven P. Kerckhoff Editors

Providence, Rhode Island

Editorial Board Jane Gilman (Chair) Joseph Silverman Andras Vasy 2020 Mathematics Subject Classification. Primary 57K20, 57R30, 57K32, 53C15.

Library of Congress Cataloging-in-Publication Data Names: Farb, Benson, editor. | Gabai, David, editor. | Kerckhoff, Steve, editor. Title: Collected works of William P. Thurston with commentary / Benson Farb, David Gabai, Steven P. Kerckhoff, editors. Description: Providence, Rhode Island : American Mathematical Society, [2022] | Includes bibliographical references. | Contents: Volume I. Foliations, surfaces and differential geometry – Volume II. 3-manifolds, complexity and geometric group theory – Volume III. Dynamics, computer science and general interest. Identifiers: LCCN 2021037684 | ISBN 9781470451646 (hardcover-set) | ISBN 9781470463885 (hardcover–vol. I) | ISBN 9781470463892 (hardcover–vol. II) | ISBN 9781470463908 (hardcover– vol. III) | ISBN 9781470468330 (ebook–vol. I) | ISBN 9781470468347 (ebook–vol. II) | ISBN 9781470468354 (ebook–vol. III) Subjects: LCSH: Thurston, William P., 1946–2012. | Differential topology. | Geometry, Differential. | Dynamics. | AMS: Manifolds and cell complexes – Low-dimensional topology in specific dimensions – 2-dimensional topology (including mapping class groups of surfaces, Teichm¨ uller theory, curve complexes, etc.). | Manifolds and cell complexes – Low-dimensional topology in specific dimensions – Foliations in differential topology; geometric theory. | Manifolds and cell complexes – Low-dimensional topology in specific dimensions – Hyperbolic 3-manifolds. | Differential geometry – Global differential geometry – General geometric structures on manifolds (almost complex, almost product structures, etc.). | Group theory and generalizations – Special aspects of infinite or finite groups – Geometric group theory. | Group theory and generalizations – Special aspects of infinite or finite groups – Word problems, other decision problems, connections with logic and automata (group-theoretic aspects). | Dynamical systems and ergodic theory – Dynamical systems over complex numbers – Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets. | Computer science – Theory of data – Data structures. Classification: LCC QA611 .C645 2022 | DDC 514/.22–dc23 LC record available at https://lccn.loc.gov/2021037684

c 2022 by the American Mathematical Society. All rights reserved.  Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines  established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

27 26 25 24 23 22

Contents

Volume I Preface

xiii

Acknowledgements

xv

Part 1: Foliations

1

Commentary: Foliations

3

“Foliations of three-manifolds which are circle bundles,” Ph.D. Thesis, University of California, Berkeley, 1972.

13

(with J. F. Plante) “Anosov flows and the fundamental group,” Topology 11 (1972), 147–150.

81

“Noncobordant foliations of S 3 ,” Bull. Amer. Math. Soc. 78, no. 4, (1972), 511–514.

85

(with H. Rosenberg) “Some remarks on foliations,” Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 463–478. Academic Press, New York, 1973. 89 “Foliations and groups of diffeomorphisms,” Bull. Amer. Math. Soc. 80, no. 2, (1974), 304–307. 105 “A generalization of the Reeb stability theorem,” Topology 13 (1974), 347–352.

109

“The theory of foliations of codimension greater than one,” Comment. Math. Helv. 49 (1974), 214–231. 115 (with Morris W. Hirsch) “Foliated bundles, invariant measures and flat manifolds,” Ann. Math. (2) 101 (1975), 369–390.

133

“The theory of foliations of codimension greater than one,” Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, p. 321. Amer. Math. Soc., Providence, R.I., 1975. 155 (with H. E. Winkelnkemper) “On the existence of contact forms,” Proc. Amer. Math. Soc. 52 (1975), 345–347. 157

WILLIAM P. THURSTON

v

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CONTENTS

“A local construction of foliations for three-manifolds,” Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 315–319. Amer. Math. Soc., Providence, R.I., 1975. 161 “On the construction and classification of foliations,” Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 547–549. Canad. Math. Congress, Montreal, Que., 1975. 167 “Existence of codimension-one foliations,” Ann. of Math. (2) 104 (1976), no. 2, 249–268. 171 (with J. F. Plante) “Polynomial growth in holonomy groups of foliations,” Comment. Math. Helv. 51 (1976), no. 4, 567–584.

191

(with Michael Handel) “Anosov flows on new three manifolds,” Invent. Math. 59 (1980), no. 2, 95–103. 209 “A norm for the homology of 3-manifolds,” Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130. 219 (with Yakov M. Eliashberg) “Contact structures and foliations on 3-manifolds,” Turkish J. Math. 20 (1996), no. 1, 19–35. 257 (with Yakov M. Eliashberg) Confoliations, University Lecture Series 13, American Mathematical Society, Providence, RI, 1998.

275

“Three-manifolds, foliations and circles, I,” December 1997 eprint.

353

“Three-manifolds, foliations and circles, II: the Transverse Asymptotic Geometry of Foliations,” January 1998 preprint

413

Part 2: Surfaces and Mapping Class Groups

451

Commentary: Surfaces and Mapping Class Groups

453

(with A. Hatcher) “A presentation for the mapping class group of a closed orientable surface,” Topology 19 (1980), no. 3, 221–237.

457

(with Michael Handel) “New proofs of some results of Nielsen,” Adv. in Math. 56 (1985), no. 2, 173–191. 475 “On the geometry and dynamics of diffeomorphisms of surfaces,” Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. 495 “Earthquakes in 2-dimensional hyperbolic geometry,” Low-dimensional topology and Kleinian groups, (Coventry/Durham, 1984), 269–289, London Math. Soc. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge, 1986.

511

“Minimal stretch maps between hyperbolic surfaces,” 1986 preprint, 1998 eprint.

533

(with Steven P. Kerckhoff) “Non-continuity of the action of the modular group at Bers’ boundary of Teichmuller space,” Invent. Math. 100 (1990), no. 1, 25–47. 587

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CONTENTS

vii

Part 3. Differential Geometry

611

Commentary: Differential Geometry

613

“Some simple examples of symplectic manifolds,” Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. 617 (with J. Milnor) “Characteristic numbers of 3-manifolds,” Enseign. Math. (2) 23 (1977), no. 3-4, 249–254. 619 (with D. B. A. Epstein) “Transformation groups and natural bundles,” Proc. London Math. Soc. (3) 38 (1979), no. 2, 219–236. 625 (with Dennis Sullivan) “Manifolds with canonical coordinate charts: some examples,” Enseign. Math. (2) 29 (1983), no. 1–2, 15–25.

643

(with M. Gromov) “Pinching constants for hyperbolic manifolds,” Invent. Math. 89 (1987), no. 1, 1–12.

655

(with M. Gromov and H. B. Lawson, Jr.) “Hyperbolic 4-manifolds and ´ conformally flat 3-manifolds,” Inst. Hautes Etudes Sci. Publ. Math. 68 (1988), 27–45 (1989). 667 “Shapes of polyhedra and triangulations of the sphere,” The Epstein birthday schrift, 511–549, Geom. Topol. Monogr., 1 Geom. Topol. Publ., Coventry, 1998. 687 (with John H. Conway, Olaf Delgado Friedrichs and Daniel H. Huson) “On three-dimensional space groups,” Beitr¨ age Algebra Geom. 42 (2001), no. 2, 475–507.

727

Volume II Preface

xiii

Acknowledgements

xv

Part 1. Three-Dimensional Manifolds

1

Commentary: Three-Dimensional Manifolds

3

“Three dimensional manifolds, Kleinian groups and hyperbolic geometry,” Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381.

9

“Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds,” Ann. of Math. (2) 124 (1986), no. 2, 203–246.

35

“Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle,” 1986 preprint, 1998 eprint.

79

“Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary,” 1986 preprint, 1998 eprint. 111 “Hyperbolic structures on 3-manifolds: Overall logic,” 1980 preprint.

WILLIAM P. THURSTON

131

vii

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CONTENTS

“Three-manifolds with symmetry,” 1982 preprint.

147

“Hyperbolic geometry and 3-manifolds,” Low-dimensional topology (Bangor, 1979), pp. 9–25, London Math. Soc. Lecture Note Ser., 48, Cambridge Univ. Press, Cambridge-New York, 1982. 153 (with A. Hatcher) “Incompressible surfaces in 2-bridge knot complements,” Invent. Math. 79 (1985), no. 2, 225–246.

171

(with D. Cooper) “Triangulating 3-manifolds using 5 vertex link types,” Topology 27 (1988), no. 1, 23–25.

193

(with Nathan M. Dunfield) “The virtual Haken conjecture: experiments and examples,” Geom. Topol. 7 (2003), 399–441. 197 (with Nathan M. Dunfield) “Finite covers of random 3-manifolds,” Invent. Math. 166 (2006), no. 3, 457–521.

241

(with James W. Cannon) “Group invariant Peano curves,” Geom. Topol. 11 (2007), 1315–1355. 307 (with Ian Agol and Peter A. Storm) “Lower bounds on volumes of hyperbolic Haken 3-manifolds. With an appendix by Nathan Dunfield,” J. Amer. Math. Soc. 20 (2007), no. 4, 1053–1077. 349 (with Joel Hass and Abigail Thompson) “Stabilization of Heegaard splittings,” Geom. Topol. 13 (2009), no. 4, 2029–2050. 375 Part 2. Complexity, Constructions and Computers

397

Commentary: Complexity, Constructions and Computers

399

(with Frederick J. Almgren, Jr.) “Examples of unknotted curves which bound only surfaces of high genus within their convex hulls,” Ann. of Math. (2) 105 (1977), no. 3, 527–538. 401 (with Joel Hass and Jack Snoeyink) “The size of spanning disks for polygonal curves,” Discrete Comput. Geom. 29 (2003) no. 1, 1–17. 413 (with Joel Hass and Jeffrey C. Lagarias) “Area inequalities for embedded disks spanning unknotted curves,” J. Differential Geom. 68 (2004), no. 1, 1–29. 431 (with Ian Agol and Joel Hass)“3-manifold knot genus is NP-complete,” Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, 761–766,ACM, New York, 2002.

461

(with Ian Agol and Joel Hass) “The computational complexity of knot genus and spanning area,” Trans. Amer. Math. Soc. 358 (2006), no. 9, 3821–3850.

467

Part 3. Geometric Group Theory

497

Commentary: Geometric Group Theory

499

“Finite state algorithms for the braid groups,” February 1988 preprint.

501

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(with D. B. A. Epstein) “Combable groups,” Conference on Differential Geometry and Topology (Sardinia, 1988). Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), suppl., 423–439. 525 (with J. W. Cannon, W. J. Floyd and M. A. Grayson) “Solvgroups are not almost convex.” Geom. Dedicata 31 (1989), no. 3, 291–300.

543

“Groups, Tilings and Finite State Automata,” Summer 1989 AMS Colloquium Lectures, Research Report GCG 1, 1989 preprint. 553 “Conway’s tiling groups,” Amer. Math. Monthly 97 (1990), no. 8, 757–773.

603

(with T. R. Riley) “The absence of efficient dual pairs of spanning trees in planar graphs,” Electronic Journal of Combinatorics, 13 2006, #N13.

621

Volume III Preface

xiii

Acknowledgements

xv

Part 1: Dynamics and Complex Analysis

1

Commentary: Dynamics

3

(with John Milnor) “On iterated maps of the interval,” Dynamical systems, (College Park, MD, 198687), 465–563, Lecture Notes in Math. 1342, Springer, Berlin, 1988.

7

“On the dynamics of iterated rational maps,” February 1984 preprint.

107

“Entropy in dimension one,” Frontiers in complex dynamics: In Celebration of John Milnor’s 80th Birthday, Araceli Bonifant, Misha Lyubich, and Scott Sutherland, 339–384, Princeton Math. Ser., 51, Princeton Univ. Press, Princeton, NJ, 2014. 177 (with Hyungryul Baik, Gao Yan, John H. Hubbard, Tan Lei, Kathryn A. Lindsey and Dylan P. Thurston) “Degree-d-invariant laminations,” What’s Next?: The Mathematical Legacy of William P. Thurston, Dylan Thurston (ed.). Annals of Mathematics Studies 205, 259–325, Princeton University Press, Princeton, NJ, 2020.

223

(with Ethan M. Coven, William Geller and Sylvia Silberger) “The symbolic dynamics of tiling the integers,” Israel J. Math. 130 (2002), 21–27.

291

(with Dennis P. Sullivan) “Extending holomorphic motions,” Acta Math. 157 (1986), no. 3-4, 243–257. 299

WILLIAM P. THURSTON

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x

CONTENTS

“Zippers and univalent functions. The Bieberbach conjecture,” (West Lafayette, Ind., 1985), 185–197, Math. Surveys Monogr., 21, Amer. Math. Soc., Providence, RI, 1986.

315

Part 2: Computer Science

329

Commentary: Computer Science

331

(with James K. Park and Kenneth Steiglitz) “Soliton-like behavior in automata,” Phys. D 19 (1986), no. 3, 423–432.

333

(with Daniel D. Sleator and Robert E. Tarjan) “Rotation distance, triangulations, and hyperbolic geometry,” J. Amer. Math. Soc. 1 (1988), no. 3, 647–681. 343 (with Daniel D. Sleator and Robert E. Tarjan) “Short encodings of evolving structures,” SIAM J. Discrete Math. 5 (1992), no. 3, 428–450. 379 (with Gary L. Miller, Shang-Hua Teng, and Stephen A. Vavasis) “Automatic mesh partitioning. Graph theory and sparse matrix computation,” 57–84, IMA Vol. Math. Appl., 56, Springer, New York, 1993. 403 (with Gary L. Miller, Shang-Hua Teng, and Stephen A. Vavasis) “Separators for sphere-packings and nearest neighbor graphs,” J. ACM 44 (1997), no. 1, 1–29. 431 (with Gary L. Miller, Shang-Hua Teng, and Stephen A. Vavasis) “Geometric separators for finite-element meshes,” SIAM J. Sci. Comput. 19 (1998), no. 2, 364–386. 461 Papers for General Audiences

485

Commentary: General Audience

487

“On proof and progress in mathematics,” Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 161–177. 489 (with Jean-Pierre Bourguignon) “Interview de William Thurston,” Gaz. Math. No. 65 (1995), 11–18.

507

“How to see 3-manifolds”; Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity 15 (1998), no. 9, 2545–2571.

515

“The Eightfold Way: a mathematical sculpture by Helaman Ferguson,” The Eightfold Way, 1–7, Math. Sci. Res. Inst. Publ. 35, Cambridge Univ. Press, Cambridge, 1998.

543

Miscellaneous

551

Commentary: Miscellaneous Papers

553

“A Constructive Foundation for Topology,” Senior Thesis, New College, June 14, 1967. 555

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COLLECTED WORKS WITH COMMENTARY

CONTENTS

xi

(with D. M. Kan) “Every connected space has the homology of a K(π, 1),” Topology 15 (1976), no. 3, 253–258.

595

(with L. Vaserstein) “On K1 -theory of the Euclidean space,” Topology Appl. 23 (1986), no. 2, 145–148. 601 Volume IV–The Geometry and Topology of Three-Manifolds Publisher’s Note

ix

Editor’s Preface

xi

Introduction

xvii

Chapter 1. Geometry and three-manifolds

1

Chapter 2. Elliptic and hyperbolic geometry

7

Chapter 3. Geometric structures on manifolds

23

Chapter 4. Hyperbolic Dehn surgery

37

Chapter 5. Flexibility and rigidity of geometric structures

71

Chapter 6. Gromov’s invariant and the volume of a hyperbolic manifold

105

Chapter 7. Computation of volume

135

Chapter 8. Kleinian groups

149

Chapter 9. Algebraic convergence

195

NOTE

249

Chapter 11. Deforming Kleinian manifolds by homeomorphisms of the sphere at infinity 251 Chapter 13. Orbifolds

261

Index

313

WILLIAM P. THURSTON

xi

PREFACE

William Paul Thurston was born on October 30, 1946 and died on August 21, 2012 at the age of 65. During his lifetime Thurston changed the landscape of mathematics in at least two ways. First, his original ideas changed and connected whole subjects in mathematics, from low-dimensional topology to the theory of rational maps to hyperbolic geometry and far beyond. But, just as importantly, through both his written and non-written work Thurston changed the way we think about and encounter mathematics. One hope in bringing (almost) all of Thurston’s written work together in one place is that it might shed light on the long intellectual journey of a unique thinker: how Thurston developed his viewpoint; what it brought to the subjects he wrote about; and how he applied insights gained in one topic to understand others. Just as important, perhaps, are the countless gems contained in these papers, many wellknown but perhaps some still undiscovered by the general mathematical community. A central theme running through all of Thurston’s work is his emphasis on understanding and imagination. We invite and challenge the reader to find others. Contents. Thurston’s holistic approach to mathematics makes it difficult to organize his papers in a way that does not seem to erect artificial dividing lines between different topics. Of course one must pick some ordering, and hence some groupings. We have done our best. We have organized Thurston’s collected work into three volumes, with a fourth consisting of his famous and highly influential 1977-8 Princeton Course notes. Volume I contains Thurston’s papers on foliations, on surfaces and mapping class groups, and on differential geometry. Volume II contains Thurston’s papers on the geometry and topology of 3-manifolds; on complexity, constructions and computers; and on geometric group theory. Volume III contains Thurstons papers on dynamics and on computer science; it also contains his papers written for general audiences, as well as a few miscellaneous papers, including his 1967 New College undergraduate thesis, a fascinating document that foreshadows Thurston’s broad view of mathematics. At the start of each grouping of Thurston’s papers we give an introduction, both as a warmup discussion and as a means of placing the papers in a broader context. We have tried to abide by the philosophy that “less is more”, as Thurston’s papers truly stand on their own. Acknowledgements. We would like to thank Joan Birman and Bill Veech for initiating this project. We thank Eriko Hironaka and the American Mathematical Society for their support and help. Finally, we are extremely grateful to Julian Thurston for allowing both preprints and published papers to be used for these volumes. Without her this project would not have been possible.

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xiii

Permissions & Acknowledgments The American Mathematical Society gratefully acknowledges the kindness of the following individuals and institutions in granting permission to reprint material in this volume: Ian Agol and Joel Hass (with Ian Agol and Joel Hass ) “The computational complexity of knot genus and spanning area,” Trans. Amer. Math. Soc. 358 (2006), no. 9, 3821–3850. American Mathematical Society “Noncobordant foliations of S 3 ,” Bull. Amer. Math. Soc. 78, no. 4, (1972), c 511–514; 1972, American Mathematical Society. “Foliations and groups of diffeomorphisms,” Bull. Amer. Math. Soc. 80, no. c 2, (1974), 304–307; 1974, American Mathematical Society. “The theory of foliations of codimension greater than one,” Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., c 1973), Part 1, p. 321. Amer. Math. Soc., Providence, R.I., 1975; 1975, American Mathematical Society. “A local construction of foliations for three-manifolds,” Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), c Part 1, pp. 315–319. Amer. Math. Soc., Providence, R.I., 1975; 1975, American Mathematical Society. (with H. E. Winkelnkemper) “On the existence of contact forms,” Proc. Amer. c Math. Soc. 52 (1975), 345–347; 1975, American Mathematical Society. “Some simple examples of symplectic manifolds,” Proc. Amer. Math. Soc. 55 c (1976), no. 2, 467–468; 1976, American Mathematical Society. “Three dimensional manifolds, Kleinian groups and hyperbolic geometry,” Bull. c Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381; 1982, American Mathematical Society. “A norm for the homology of 3-manifolds,” Mem. Amer. Math. Soc. 59 c (1986), no. 339, i–vi and 99–130; 1986, American Mathematical Society. “Zippers and univalent functions. The Bieberbach conjecture,” (West Lafayette, Ind., 1985), 185–197, Math. Surveys Monogr., 21, Amer. Math. Soc., Providence, c RI, 1986; 1986, American Mathematical Society. xv

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PERMISSIONS & ACKNOWLEDGMENTS

(with Daniel D. Sleator and Robert E. Tarjan) “Rotation distance, triangulations, and hyperbolic geometry,” J. Amer. Math. Soc. 1 (1988), no. 3, 647–681; c 1988, American Mathematical Society. “On the geometry and dynamics of diffeomorphisms of surfaces,” Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. “On proof and progress in mathematics,” Bull. Amer. Math. Soc. (N.S.) 30 c (1994), no. 2, 161–177; 1994, American Mathematical Society. (with Yakov M. Eliashberg) Confoliations, University Lecture Series 13, Amerc ican Mathematical Society, Providence, RI, 1998; 1998, American Mathematical Society. (with Ian Agol and Peter A. Storm) “Lower bounds on volumes of hyperbolic Haken 3-manifolds. With an appendix by Nathan Dunfield,” J. Amer. Math. c Soc. 20 (2007), no. 4, 1053–1077; 2007, Ian Agol, Peter A. Storm, and William Thurston. Annals of Mathematics (with Morris W. Hirsch) “Foliated bundles, invariant measures and flat manifolds,” Ann. Math. (2) 101 (1975), 369–390. “Existence of codimension-one foliations,” Ann. of Math. (2) 104 (1976), no. 2, 249–268. (with Frederick J. Almgren, Jr.) “Examples of unknotted curves which bound only surfaces of high genus within their convex hulls,” Ann. of Math. (2) 105 (1977), no. 3, 527–538. “Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds,” Ann. of Math. (2) 124 (1986), no. 2, 203–246. Association for Computing Machinery Republished with permission of the Association for Computing Machinery, “Separators for sphere-packings and nearest neighbor graphs,” with Gary L. Miller, Shang-Hua Teng, and Stephen A. Vavasis, J. ACM 44 (1997), no. 1, 1–29; permission conveyed through Copyright Clearance Center, Inc. Republished with permission of the Association for Computing Machinery, “3-manifold knot genus is NP-complete,” with Ian Agol and Joel Hass, Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, 761– 766,ACM, New York, 2002; permission conveyed through Copyright Clearance Center, Inc. Cambridge University Press “Hyperbolic geometry and 3-manifolds,” Low-dimensional topology (Bangor, 1979), pp. 9–25, London Math. Soc. Lecture Note Ser., 48, Cambridge Univ. c Press, Cambridge-New York, 1982. 1979 Cambridge University Press and reproduced with permission.

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PERMISSIONS & ACKNOWLEDGMENTS

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Canadian Mathematical Society “On the construction and classification of foliations,” Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 547–549. Canad. Math. Congress, Montreal, Que., 1975. James W. Cannon (with James W. Cannon) “Group invariant Peano curves,” Geom. Topol. 11 (2007), 1315–1355. Nathan M. Dunfield (with Nathan M. Dunfield) “The virtual Haken conjecture: experiments and examples,” Geom. Topol. 7 (2003), 399–441. Electronic Library of Mathematics (with John H. Conway, Olaf Delgado Friedrichs and Daniel H. Huson) “On three-dimensional space groups,” Beitr¨ age Algebra Geom. 42 (2001), no. 2, 475– 507. Elsevier Reprinted from “Anosov flows and the fundamental group,” with J. F. Plante, c Topology 11 (1972), 147–150; 1972 with permission from Elsevier. Reprinted from “Some remarks on foliations,” with H. Rosenberg, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pp. 463–478. Academic c Press, New York, 1973 with permission from Elsevier. Reprinted from “A generalization of the Reeb stability theorem,” Topology 13 c (1974), 347–352; 1974 with permission from Elsevier. Reprinted from “Every connected space has the homology of a K(π, 1),” with c D. M. Kan, Topology 15 (1976), no. 3, 253–258; 1976 with permission from Elsevier. Reprinted from “A presentation for the mapping class group of a closed oric entable surface,” with A. Hatcher, Topology 19 (1980), no. 3, 221–237; 1980 with permission from Elsevier. Reprinted from “New proofs of some results of Nielsen,” with Michael Handel, c Adv. in Math. 56 (1985), no. 2, 173–191; 1985 with permission from Elsevier. Reprinted from “Soliton-like behavior in automata,” with James K. Park and c Kenneth Steiglitz, Phys. D 19 (1986), no. 3, 423–432; 1986 with permission from Elsevier. Reprinted from “On K1 -theory of the Euclidean space,” with L. Vaserstein, c Topology Appl. 23 (1986), no. 2, 145–148; 1986 with permission from Elsevier. Reprinted from “Triangulating 3-manifolds using 5 vertex link types,” with D. c Cooper, Topology 27 (1988), no. 1, 23–25; 1988 with permission from Elsevier. Fondation L’Enseignement Math´ ematique (with J. Milnor) “Characteristic numbers of 3-manifolds,” Enseign. Math. (2) 23 (1977), no. 3-4, 249–254.

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(with Dennis Sullivan) “Manifolds with canonical coordinate charts: some examples,” Enseign. Math. (2) 29 (1983), no. 1–2, 15–25. Joel Hass and Abigail Thompson (with Joel Hass and Abigail Thompson) “Stabilization of Heegaard splittings,” Geom. Topol. 13 (2009), no. 4, 2029–2050 ´ Institut des Hautes Etudes Scientifiques (with M. Gromov and H. B. Lawson, Jr.) “Hyperbolic 4-manifolds and confor´ mally flat 3-manifolds,” Inst. Hautes Etudes Sci. Publ. Math. 68 (1988), 27–45 (1989). International Press of Boston, Inc. (with Joel Hass and Jeffrey C. Lagarias) “Area inequalities for embedded disks spanning unknotted curves,” J. Differential Geom. 68 (2004), no. 1, 1–29. Courtesy of International Press of Boston, Inc. IOP Publishing, Ltd. Republished with permission of IOP Publishing Ltd., from “How to see 3manifolds”; Topology of the Universe Conference (Cleveland, OH, 1997), Classic cal Quantum Gravity 15 (1998), no. 9, 2545–2571, 1998; permission conveyed through the Copyright Clearance Center, Inc. John Wiley and Sons (with D. B. A. Epstein) “Transformation groups and natural bundles,” Proc. c London Math. Soc. (3) 38 (1979), no. 2, 219–236. 1979 John Wiley and Sons, all rights reserved. Mathematical Sciences Research Institute “The Eightfold Way: a mathematical sculpture by Helaman Ferguson,” The Eightfold Way, 1–7, Math. Sci. Res. Inst. Publ. 35, Cambridge Univ. Press, Cambridge, 1998. Princeton University Press “Entropy in dimension one,” Frontiers in complex dynamics: In Celebration of John Milnor’s 80th Birthday, Araceli Bonifant, Misha Lyubich, and Scott Sutherland, 339–384, Princeton Math. Ser., 51, Princeton Univ. Press, Princeton, NJ, 2014. Reprinted with permission of Princeton University Press; permission conveyed through Copyright Clearance Center, Inc. “Degree-d-invariant laminations,” What’s Next?: The Mathematical Legacy of William P. Thurston, with Hyungryul Baik, Gao Yan, John H. Hubbard, Tan Lei, Kathryn A. Lindsey and Dylan P. Thurston, Dylan Thurston (ed.), Annals of Mathematics Studies 205, 259–325, Princeton University Press, Princeton, NJ, c 2020; 2020. Republished with permission of Princeton University Press; permission conveyed through the Copyright Clearance Center, Inc.

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Tim Riley (with T. R. Riley) “The absence of efficient dual pairs of spanning trees in planar graphs,” Electronic Journal of Combinatorics, 13 2006, #N13. Society for Industrial and Applied Mathematics (with Daniel D. Sleator and Robert E. Tarjan) “Short encodings of evolving structures,” SIAM J. Discrete Math. 5 (1992), no. 3, 428–450. (with Gary L. Miller, Shang-Hua Teng, and Stephen A. Vavasis) “Geometric separators for finite-element meshes,” SIAM J. Sci. Comput. 19 (1998), no. 2, 364–386. Soci´ et´ e Math´ ematique de francede Gazette des Mathematiciens (with Jean-Pierre Bourguignon) “Interview de William Thurston,” Gaz. Math. No. 65 (1995), 11–18. Springer Nature Reprinted by permission from Springer Nature, “Anosov flows on new three c manifolds,” with Michael Handel, Invent. Math. 59 (1980), no. 2, 95–103; 1980. Reprinted by permission from Springer Nature, “Incompressible surfaces in 2bridge knot complements,” with A. Hatcher, Invent. Math. 79 (1985), no. 2, c 225–246; 1985. Reprinted by permission from Springer Nature, “Extending holomorphic moc tions,” with Dennis P. Sullivan, Acta Math. 157 (1986), no. 3-4, 243–257; 1986. Reprinted by permission from Springer Nature, “Pinching constants for hyperbolic manifolds,” with M. Gromov, Invent. Math. 89 (1987), no. 1, 1–12. Reprinted by permission from Springer Nature, “Solvgroups are not almost convex.” with J. W. Cannon, W. J. Floyd and M. A. Grayson, Geom. Dedicata 31 c (1989), no. 3, 291–300; 1989. Reprinted by permission from Springer Nature, (with John Milnor) “On iterated maps of the interval,” with John Milnor, Dynamical systems, (College Park, MD, 198687), 465–563, Lecture Notes in Math. 1342, Springer, Berlin, 1988; c 1988. Reprinted by permission from Springer Nature, “Non-continuity of the action of the modular group at Bers’ boundary of Teichmuller space,” with Steven P. c Kerckhoff, Invent. Math. 100 (1990), no. 1, 25–47; 1990. Reprinted by permission from Springer Nature, “Automatic mesh partitioning. Graph theory and sparse matrix computation,” with Gary L. Miller, Shang-Hua Teng, and Stephen A. Vavasis, 57–84, IMA Vol. Math. Appl., 56, Springer, New c York, 1993; 1993. Reprinted by permission from Springer Nature, “The symbolic dynamics of tiling the integers,” with Ethan M. Coven, William Geller and Sylvia Silberger, c Israel J. Math. 130 (2002), 21–27; 2002.

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Reprinted by permission from Springer Nature, “The size of spanning disks for polygonal curves,” with Joel Hass and Jack Snoeyink, Discrete Comput. Geom. 29 c (2003) no. 1, 1–17; 2003. Reprinted by permission from Springer Nature, (with Nathan M. Dunfield) “Finite covers of random 3-manifolds,” with Nathan M. Dunfield, Invent. Math. c 166 (2006), no. 3, 457–5212006. Swiss Mathematical Society “The theory of foliations of codimension greater than one,” Comment. Math. Helv. 49 (1974), 214–231. (with J. F. Plante) “Polynomial growth in holonomy groups of foliations,” Comment. Math. Helv. 51 (1976), no. 4, 567–584. Taylor & Francis “Conway’s tiling groups,” Amer. Math. Monthly 97 (1990), no. 8, 757–773. Julian Thurston “Three-manifolds, foliations and circles, I,” December 1997 preprint. “Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle,” 1986 preprint, 1998 eprint. “Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary,” 1986 preprint, 1998 eprint. “Hyperbolic structures on 3-manifolds: Overall logic,” 1980 preprint. “Three-manifolds with symmetry,” 1982 preprint. “Minimal stretch maps between hyperbolic surfaces,” 1986 preprint, 1998 eprint. “Groups, Tilings and Finite State Automata,” Summer 1989 AMS Colloquium Lectures, Research Report GCG 1, 1989 preprint. “Three-manifolds, foliations and circles, II: the Transverse Asymptotc Geometry of Foliations,” January 1998 preprint. “A Constructive Foundation for Topology,” Senior Thesis, New College, June 14, 1967. “Foliations of three-manifolds which are circle bundles,” Ph.D. Thesis, University of California, Berkeley, 1972. “On the dynamics of iterated rational maps,” February 1984 preprint. “Earthquakes in 2-dimensional hyperbolic geometry,” Low-dimensional topology and Kleinian groups, (Coventry/Durham, 1984), 269–289, London Math. Soc. Lecture Note Ser., 112, Cambridge Univ. Press, Cambridge, 1986. “Finite state algorithms for the braid groups,” February 1988 preprint. “Shapes of polyhedra and triangulations of the sphere,” The Epstein birthday schrift, 511–549, Geom. Topol. Monogr., 1 Geom. Topol. Publ., Coventry, 1998.

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Nathaniel Thurston Figure 1. Lines of the form nx + my = 1/2 where n and m are integers. Any convex polygon in this network which is symmetric in the origin is the unit sphere in H2 (M ), for some 3-manifold M . This computer drawn picture was prepared by Nathaniel Thurston. Appeared in : Thurston, William P., A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130. Turkish Journal of Mathematics (with Yakov M. Eliashberg) “Contact structures and foliations on 3-manifolds,” Turkish J. Math. 20 (1996), no. 1, 19–35. Universit` a degli Studi di Cagliari (with D. B. A. Epstein) “Combable groups,” Conference on Differential Geometry and Topology (Sardinia, 1988). Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), suppl., 423–439.

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Part 1.

Foliations

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Quoting Thurston’s definition of foliation [Th8]. “Given a large supply of some sort of fabric, what kinds of manifolds can be made from it, in a way that the patterns match up along the seams? This is a very general question, which has been studied by diverse means in differential topology and differential geometry [. . . ]. A foliation is a manifold made out of striped fabric - with infintely thin stripes, having no space between them. The complete stripes, or leaves, of the foliation are submanifolds; if the leaves have codimension k, the foliation is called a codimension-k foliation. In order that a manifold admit a codimension-k foliation, it must have a plane field of dimension (n − k).” Such a foliation is called an (n − k)-dimensional foliation. The first definitive result in the subject, the so called Frobenius integrability theorem [Fr], concerns a necessary and sufficient condition for a plane field to be the tangent field of a foliation. See [Spi] Chapter 6 for a modern treatment. As Frobenius himself notes [Sa], a first proof was given by Deahna [De]. While this work was published in 1840, it took another hundred years before a geometric/topological theory of foliations was introduced. This was pioneered by Ehresmann and Reeb in a series of Comptes Rendus papers starting with [ER] that was quickly followed by Reeb’s foundational 1948 thesis [Re1]. See Haefliger [Ha4] for a detailed account of developments in this period. Reeb [Re1] himself notes that the 1-dimensional theory had already undergone considerable development through the work of Poincare [Po], Bendixson [Be], Kaplan [Ka] and others. In addition there was the well known extension to the Poincare - Hopf index theorem: a closed manifold has Euler characteristic 0 if and only if it has a nowhere vanishing smooth vector field if and only if it has a 1-dimensional foliation. Another impetus was Hopf’s question as to whether or not the 3-sphere has a codimension-1 foliation [Re2]. The foliation exhibited in Reeb’s thesis, now known as the Reeb foliation gave a positive answer to Hopf’s question. At the outset Reeb asks the following fundamental and far-reaching generalization to Hopf’s question [Re1]: Si la vari´et´e Vn admet un champ Eq de classe C 1 admet-t-elle aussi un champ E q de classe C 1 compl`etement integrale? which experts soon after expressed as follows: Is every q-plane field homotopic to the tangent plane field of a foliation? In a series of three papers [Th5], [Th7], [Th9] Thurston obtained the following results. Theorem 1 ([Th9] [Th7]). Let M be a smooth manifold without boundary. Every codimension-1 plane field on M is homotopic to the tangent plane field of a C ∞ codimension-1 foliation. Corollary 2 ([Th9]). Every closed, connected, smooth manifold with Euler characteristic 0 has a C ∞ codimension-1 foliation. Theorem 3 ([Th5]). Every smooth k-plane field on a closed n-manifold is homotopic to the tangent plane field of a Lipschitz foliation with C ∞ -leaves. 1

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The Lipschitz condition was improved to C 1 by Tsuboi [Ts1]. Thurston uses the Bott vanishing theorem [Bo] in [Th3] to show that there cannot be a C 2 -version of this theorem and further that the dimension obstruction given by Bott is sharp. See [Mo] for an explicit example. For 2-plane fields we have the following result. Theorem 4 ([Th5]). Every C ∞ 2-plane field on a manifold is homotopic to a completely integrable C ∞ plane field. Remark 5. In [Th9], after mentioning various results of Lickorish, NovikovZieschang, Wood, Lawson, Durfee, A’Campo and Tamura, Thurston states, “My method, on the other hand, is local in nature: one cannot see the whole manifold. A disadvantage is that it is hard to picture the foliations so constructed. For this reason, I think that further work on the geometric methods of constructing foliations is called for.” The proofs of these results required generalizations of deep results of Mather [Ma3], [Ma1] on the group of compactly supported diffeormorphisms of R with the discrete topology and relations with Haefliger’s classifying spaces. In particular Thurston proved the following. Theorem 6 ([Th3]). If M is a closed manifold, then Diff ∞ 0 (M ) is a simple group. Theorem 7 ([Th3]). For all r ≥ 1 and p ≥ 1, r, p ∈ N, there is a map ¯ Diff r (Rp ) → Ωp (B Γ ¯ rp ) B c that induces an isomorphism on integer homology. ∞ Here Diff ∞ 0 denotes the connected component of the identity of the group of C r r ¯ diffeomorphisms, Diff c denotes compactly supported C diffeomorphisms and B Γrp ¯ Diff rc (Rp ) is also a homotopy fiber. Theorem is defined in Remark 13. Note that B 7 is due to John Mather for p = 1. The paper [Th3] is a research announcement with few hints of proofs. Thurston lectured on this at Harvard in 1974 and Mather wrote a proof of the above theorem in [Ma2]. See also [Sar], [McD], [La], and [Ts2]. One of Thurston’s proofs introduced the far reaching technique now known as fragmentation, see [Na]. Forty years later Gael Meigniez [Me] proved Theorem 1 for n ≥ 4 without using the Mather - Thurston theory. In addition his foliations are minimal, i.e. every leaf is dense. This includes simply-connected manifolds and connect sums, in contrast to Novikov’s closed leaf theorem [No], which disallows minimal foliations in such 3-manifolds. There are also relative versions of Thurston’s theorems, e.g. given a manifold M with boundary, then under suitable circumstances a plane field defined on M that is tangent to a foliation near ∂M , is homotopic rel a neighborhood of ∂M to a completely integral plane field. Reeb’s stability theorem precludes this holding in complete generality, for if ∂M is simply connected, then any foliation on M with ∂M a leaf would be foliated by leaves covered by ∂M . However, if each component N of ∂M satisfies H 1 (N, R) = 0, then extension theorems hold. Consult [Th5], [Th9] for exact statements. Thurston proved the following generalization of the Reeb stability theorem.

Theorem 8 ([Th4]). (a) Let F be a codimension-1 C 1 transversely oriented foliation on a compact manifold M whose (possibly empty) boundary is a union of

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leaves. Suppose Ln−1 is a compact leaf of F such that H 1 (L, R) = 0. Then every leaf of F is diffeomorphic with Ln−1 and M fibers over S 1 or [0, 1] with fiber Ln−1 . (b) Let F be a C 1 codimension-k foliation. If L is a compact leaf of F with trivial linear holonomy and H 1 (L, R) = 0, then L has trivial holonomy and hence L has a tubular neighborhood which fibers over Dk with leaves as fibers. Remark 9. Using hyperbolic geometry Thurston demonstrated a C 0 counterexample to (a), nevertheless he states, “It would be interesting to have a characterization of compact leaves L for which Reeb’s [stability] conclusion holds in the C 0 case.” Remark 10. Haefliger [Ha1] showed that the 3-sphere does not support an analytic codimension-1 foliation, thus Theorems 1, 2, and 4 are not applicable to analytic foliations. Thurston notes in [Th9] that “[Haefliger’s] class of counterexamples was somewhat enlarged by Novikov [No] and Thurston [Th1], but the theory of analytic foliations still has many unanswered questions.” We now present background to Thurston’s work on Haefliger structures and homotoping smooth plane fields to smooth foliations. Let M by an n-dimensional manifold and F a C r codimension-q foliation. Let ν denote the normal bundle to F, an Rq -bundle over M . The exponential map gives a submersion from a neighborhood of the 0-section in ν to M . Pulling back F gives a codimension-q foliation G of ν transverse to the fibers. As the zero section s : M → ν is transverse to G we can recover F as the intersection of s(M ) with G. A Haefliger structure [Ha2], [Ha3], [Ha4] is a Rq -bundle E over M together with a section s : M → E and a germ near s(M ) of a C r codimension-q foliation G of E transverse to the fibers of E. Two Haefliger structures are concordant if they cobound a Haefliger structure on M × [0, 1]. Unlike foliations, Haefliger structures pull back under continuous maps and may be studied with the tools of algebraic topology. Indeed, Haefliger showed that C r codimension-q Haefliger structures on M , up to concordance, correspond bijectively with homotopy classes of maps from M to the classifying space BΓrq , where Γrq is the groupoid of germs of C r diffeomorphisms of Rq with a certain topology. This work was motivated by Bott’s vanishing theorem and work of [Ph1], [Ph2], that grew out of the Smale - Hirsch immersion theory. Using the work of Gromov [Gr] and Phillips [Ph3], Haefliger further showed [Ha3] that if M is open, then every Haefliger structure is concordant to one arising from a foliation. One of Thurston’s major accomplishments was to prove similar results for closed manifolds and certain relative cases. Recall that codimension-k foliations F0 and F1 are concordant if there exists a codimension-k foliation on M × I which restricts to F0 , F1 respectively on M × 0 and M × 1. Theorem 11 ([Th5], [Th9]). Concordance classes of foliations on the closed manifold M correspond 1-1 with homotopy classes of Haefliger structures H together with concordance classes of bundle monomorphisms i : vH → T (M ). Theorem 12 ([Th5], [Th9], see also P. 347 [Ha5]). If K is a smooth p-plane field of codimension-q ≥ 1 on the closed manifold M , then K is homotopic to a smooth completely integrable p-plane field if and only if the map of M into BGlq classifying the normal bundle to K can be lifted to a map into BΓrq , where D : BΓrq → BGlq is induced by the differential. See [Th5] and [Th9] for more results in this direction and other interesting applications. These last two theorems were subsequently proved for codimension-q ≥ 2

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using wrinkled mappings by Eliashberg and Mishachev with extensions to families of foliations [EM1], [EM2]. Remark 13. Haefliger remarks ([Ha5] p.348) that despite the spectacular and profound results of the 70’s, much remains to be understood about the homotopy ¯ rq , the homotopy fibers of D. type of the spaces B Γ ¯ rq connected for q > 1? Question 14 (Haefliger [Ha5]). a) Is B Γ ¯ rq ) = 1? b) What is the first p for which πp (B Γ ¯ ∞ ; Z) = 0 is Conjecture 15 (Thurston [Th3]). The smallest k for which Hk+p (B Γ p p + 1. Remark 16 ([Th3]). When p = 1, then k > 1 by Mather [Ma3] and k ≤ p + 1 using a generalized Godbillon - Vey invariant. A central question is to understand how to distinguish different classes of foliations on a given manifold M . The Godbillon - Vey invariant [GV] provides an invariant for smooth foliations on 3-manifolds, defined as follows. If T (F) denotes the tangent plane field of F, then it is the kernal of a 1-form α. The Frobenius theorem implies that α ∧ dα = 0 so that dα = α ∧ θ for some θ. The Godbillion - Vey form is the closed 3-form θ ∧ dθ. It is a concordance invariant of foliations, indeed of Haefliger structures. In [Th2] Thurston cryptically states “The form θ ∧ dθ [the Godbillon - Vey class] may be interpreted as a measure of the helical wobble of the leaves of F...” . An interpretation, attributed to him is as follows. The form θ can be viewed as the logarithmic derivative of the rate at which leaves spread apart under the holonomy; it is dual to a vector field to F in the direction of maximal contraction. As one moves transverse to the foliation the Godbillion - Vey form measures infinitesimally the algebraic area of the swept by the vectors under this motion. See [Pi] for more details. Thurston proves the following result using the geodesic flow on the unit tangent bundle of hyperbolic 2-space. Theorem 17 ([Th2]). There are uncountably many noncobordant codimension-1 C ∞ -foliations of S 3 . The Godbillon-Vey invariant induces a surjective homomorphism of π3 (BΓr1 ) onto R for 2 ≤ r ≤ ∞. In contrast to Theorem 4 there is Thurston’s unpublished 1971 Ph. D. thesis [Th1]. Theorem 18 ([Th1]). Let M 3 be a circle bundle over a closed surface. Any C 2 foliation of M either has a compact leaf or can be isotoped to be transverse to the fibers. As a corollary, using the Milnor - Wood inequality, it follows that if M is a circle bundle over a surface S and the Euler class of the bundle is > |χ(S)|, then any C 2 -foliation of M has a compact leaf, thereby giving the first closed aspherical 3-manifolds with that property. Thurston shows that both the theorem and corollary are false for C 0 -foliations. Here is how he constructs a foliation without compact leaves on any circle bundle M over a surface S of genus ≥ 2. Start with a transversely orientable geodesic lamination λ ⊂ S without compact leaves and only 4-prong disc complementary regions. Extend to a foliation of M by first suspending λ in the S 1 direction and then filling in the complementary regions with bundles

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of saddles. Note that geodesic laminations play a central role in his future work on homeomorphisms of surfaces as well as geometrization. The thesis also gives a counterexample to an assertion of Novikov that a certain partial order within Novikov components of a foliation takes on a minimum. An important technical result in [Th1], proved independently by Roussarie [Rou], is that an embedded incompressible surface in a C 2 -taut foliation can be isotoped so that each component is either a leaf or transverse to the foliation except for finitely many saddle tangencies. By taut we mean a transversely orientable foliation such that every leaf intersects a closed transversal. Roussarie and Thurston proved a version for manifolds with boundary and Roussarie a version for transversely orientable foliations without generalized Reeb components. Implicit in [Rou] and stated in [Th10] is the generalization to Reebless foliations, i.e. foliations having no Reeb components. Here, the isotoped surface may also have finitely many circle tangencies. Using [Ca2], this technical result and its proof extend to C 0 -foliations. It also holds for immersed π1 -injective surfaces [Ga2]. More distinctions between C 0 and C 2 foliations are given in the paper [RosT] with H. Rosenberg. Here is one. Theorem 19 ([RosT]). There exist smooth foliations F0 , F2 on a closed 3-manifold such that F0 , F2 are C 0 -concordant but not C 2 -cobordant. Thurston’s two papers with Plante are about foliations and growth rates of groups. The paper [PlTh1] shows that the fundamental group of a compact Riemannian manifold with a codimension-1 Anosov flow has exponential growth. Paper [PlTh2] shows that if the holonomy of a leaf of a codimension-1 foliation has polynomial growth, then it is virtually nilpotent. While the main result is true for all groups by the celebrated result of Gromov [Gr2], the paper is full of interesting observations. For example, Theorem 20 ([PlTh2]). If G is a group of C 2 -diffeomorphisms of [0, ∞) of polynomial growth, then G is free abelian. Thurston’s (unique) paper with his thesis advisor Morris Hirsch [HiT] was motivated by a paper by Plante [Pl] that in turn was a generalization of the Poincare - Bendixson theorem. As an application of their main result they prove: Theorem 21 ([HiT]). If M is a compact flat manifold whose fundamental group is obtained by taking free products and finite extensions of solvable groups, then χ(M ) = 0. This implies a special case of the so called Chern conjecture which asserts that χ(M ) = 0 for all closed affine manifolds. The unpublished 1997 ArXiv preprint, Three-manifolds, Foliations and Circles, I [Th11] introduced the notion of a manifold slithering over another and in particular that of a 3-manifold M slithering over the circle. In that case the universal cover ˜ of M fibers over S 1 so that the deck transformations are bundle automorphisms M ˜ where F is a taut foliation of M . Thurston and the fibers are unions of leaves of F, ˜ shows that F has space of leaves R, i.e. F is R-covered, and any two leaves of F˜ are at bounded Hausdorff distance, i.e. F˜ is uniform. Furthermore he shows that if F˜ has this property then M slithers over S 1 . He conjectured that R-covered foliations

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are uniform, but a counterexample was found in Calegari [Ca1]. Thurston states that non Haken 3-manifolds that have such slitherings, e.g., arising from Fenley’s R-covered Anosov foliations [Fe]. He shows that a slithered foliation F possesses much structure analogous to that of a fibration. For example, M supports a genuine lamination transverse to F and the ends of the leaves of F˜ organize to a single circle on which π1 (M ) acts. The abstract of [Th11] and the unfinished paper [Th12] assert that the first property and an analogue of the second holds for any taut foliation F on any closed atoroidal 3-manifold M . In particular, Thurston proves the following. Theorem 22 (Thurston, [CD]). Let F be a taut foliation of an orientable 3˜ . Then manifold M with hyperbolic leaves with F˜ denoting its preimage in M 1 there exists a universal circle Suniv for F. I.e. there exists an action of π1 (M ) 1 ˜ there exists a quotient map on Suniv with the property that for each leaf L of F, 1 1 1 pL : Suniv → S∞ (L) natural with respect to the action of π1 (M ) on Suniv . 1 (See 6.1 [CD] for the precise definition of Suniv .)

Theorem 23 (Calegari - Dunfield [CD]). If M is atoroidal, then the action of 1 π1 (M ) on Suniv is faithful. In Spring 1997, Thurston gave several three-hour lectures on the construction of the universal circle at the Very Informal Foliations Seminar at the MSRI and 1 most of [Th12] is concerned with a construction on Suniv . Based on these lectures, Danny Calegari and Nathan Dunfield wrote a careful account of Theorem 22, see §5 - §6 of [CD]. The key technical lemma, Thurston’s leaf pocket theorem proved in §5 roughly asserts that for any leaf L of F, the holonomy is defined along most rays in L. (By Candel [Can], there exists a Riemannian metric on M for which ˜ is naturally identified with a each leaf has constant negative curvature, hence ∂ L circle.) The proof of the leaf pocket theorem in [CD] is topological as opposed to Thurston’s original proposed proof using various properties of Lucy Garnett’s harmonic measures [Gar] and is extended to essential laminations. It is used to show that the fundamental group of many manifolds including the Weeks manifold do not act faithfully on the circle, hence do not support a taut foliation. Earlier, Roberts, Shareshian and Stein [RobSS] found examples of closed hyperbolic 3-manifolds that do not support taut foliations. Inspired by [Th11], D. Calegari proved the following. Theorem 24 (Calegari [Ca3]). A taut foliation of a closed, orientable,algebraically atoroidal 3-manifold is either the weak-stable foliation of an Anosov flow, or else there are a pair of very full genuine laminations transverse to the foliation. Thurston’s A norm for the homology of 3-manifolds was a 1976 preprint published 10 years later. For a 3-manifold M with (possibly empty) boundary, this seminal paper introduced a norm on H2 (M, ∂M ; R) that generalizes the notion of genus of a knot. It is non-degenerate if M is atoroidal. The unit ball is a finite sided polyhedron and the set of homology classes realized by fibers of a fibration over S 1 corresponds to a finite (possibly empty) set of open top dimensional faces i.e., all the lattice points lying in rays through these faces are representable by fibers and conversely. Thurston further proves that leaves of taut (and more generally Reebless) foliations are homologically norm minimizing. In contrast, about a year later Sullivan [Su] proved that leaves of a C 2 taut foliation F are homologically area

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minimizing. In fact F can be calibrated [HarL]. Homologically area minimizing means that if L is a leaf of F and R ⊂ L is a smooth compact subsurface, then area(R) ≤ area(S) for all compact surfaces S with ∂S = ∂R and [S] = [R] rel ∂R. While stated for compact leaves, the analogous subsurface property also holds for homological tautness. Theorem 25 ([Th10]). Let M be a compact orientable 3-manifold with a transversely orientable Reebless foliation F transverse to ∂M . If L is a compact leaf of F, then L is norm minimizing in its homology class as an element of H2 (M, ∂M ). Theorem 26 (Sullivan [Su]). If F is a C 2 -taut foliation of the 3-manifold M, then there exists a Riemannian metric such that every leaf is homologically area minimizing. Thurston proved Theorem 25 by observing that the Euler class χ(F) of a foliation F evaluated on a compact leaf L gives up to sign |χ(L)|. On the other hand, if F is taut and T is an incompressible surface, then using Rousserie - Thurston general position [Rou], [Th1], T can be isotoped to be transverse to F except at saddle tangencies. Thus if T is homologous to L, evaluating the Euler class on T via obstruction theory shows that |χ(L)| ≤ |χ(T )|. Thurston conjectured a similar Euler class type inequality for contact structures (compare Theorem 3.8 [ETh1]) that was proved by Bennequin [Ben]. If k is a knot in R3 transverse to the contact structure τ and F is a Seifert surface for k, then τ |F is trivial. Pushing k off itself using a non vanishing section of that bundle gives a knot k ′ . Define the self-linking number sl(k) to be the linking number of k ′ with k. Theorem 27 (Bennequin). Let τ denote the standard contact structure on R3 and k a knot transverse to τ . If F is an oriented Seifert surface for k, then sl(k) ≤ −χ(S). Bennequin used this result to show that there exist non-standard contact structures on R3 , giving the first example of what are now called overtwisted contact structures. Eliashberg introduced the dichotomy of tight and overtwisted contact structures [El] and proved Theorem 27 for null homologous knots transverse to tight contact structures in general 3-manifolds as well as a version for null homologous Legendrian knots in tight contact structures. To bridge the theory of foliations with contact structures in oriented 3-manifolds, Thurston and Eliashberg [ETh2] introduced the theory of positive (resp. negative) confoliations, i.e. those having plane fields annihilated by 1-forms α, such that α ∧ dα ≥ 0 (resp. α ∧ dα ≤ 0). They proved the following foundational result. Theorem 28. Any C 2 -confoliation ψ on an oriented 3-manifold 6= S 2 × S 1 can be C 0 -approximated by contact structures. If ψ is a foliation, then it can be approximated by both positive and negative contact structures. Contact structures C 0 -close to C 2 -taut foliations are symplectically fillable and hence tight. This theorem for C 0 taut foliations was independently proved by Bowden [Bo] and Kazez-Roberts [KR]. The following converse to Theorem 25 was proved by Gabai. Theorem 29 ([Ga2]). If M is a closed, orientable, irreducible, atoroidal 3-manifold and S is a Thurston norm minimizing surface, then there exists a C ∞ taut foliation on M having S as a leaf.

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See [Ga2], [Ga3] for finite depth versions as well as ones for manifolds with boundary. Thurston’s conjecture [Th10] that the norm based on embedded surfaces is equal to the norm based on singular (i.e. mapped) surfaces was proved in [Ga1]. This led to a proof that the Gromov norm on H2 (M, ∂M, R) is equal to twice the Thurston norm as well as a generalization of the loop and sphere theorems to higher genus surfaces. Efforts to prove these theorems led to Gabai’s theory of sutured manifold hierarchies [Ga2], [Ga1] and a proof of a strong form of the Property R conjecture, that for a knot in the 3-sphere a minimal genus surface extends to a Thurston norm minimizing surface under 0-frame surgery. A counterexample to Conjecture 3 of [Th10] is given in [Ya] and [GY], though it follows from [Ga2] that there is a positive solution for vertices of the unit ball of the dual Thurston norm. See Theorem 1.4 [GY]. The Thurston norm and its relation to fibrations and more generally taut foliations has become fundamental to knot theory, low dimensional topology, theory of contact structures, foliation and lamination theory, geometric group theory, symplectic topology, dynamical systems, as well as gauge theory and Heegaard Floer homology and their variants. Acknowledgements. We thank Danny Calegari for his many constructive comments. We thank an anonymous reader for his/her ideas on introducing Haefliger structures, insights into Thurston’s cryptic interpretation of the Godbillon - Vey invariant as well as other suggestions. We thank Sam Nariman for his comments on Thurston’s fragmentation technique. References [Be] [Ben] [Bo] [Bow] [Ca1] [Ca2] [Ca3] [Can] [CaCo] [CD] [De] [ER] [El] [EM1] [EM2] [ETh1]

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I. Bendixson, Sur les courbes definies par des equations differentialles, Acta Math. 24 (1901) 1–88. D. Bennequin, Entrelacements et equations de Pfaff, Asterisque, 107-108(1983), 83–61. R. Bott, On a topological obstruction to integrability, Proc. Symp. Pure Math., AMS, 10 (1970), 127–131. J Bowden, Approximating C 0 -foliations by contact structures, Geom. Funct. Anal. 26(2016) 1255–1296. D. Calegari, R-covered foliations of hyperbolic 3-manifolds, Geom. Topol. 3 (1999), 137– 153. D. Calegari, Leafwise smoothing laminations, Algebr. Geom. Topol. 1(2001), 579–585. D. Calegari, Promoting essential laminations, Invent. Math. 166 (2006), 583–643. A. Candel, Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. (4) 26 (1993), no. 4, 489–516. A. Candel & L. Conlon, Foliations II, Graduate Studies in Math., 60, AMS 1999. D. Calegari & N. Dunfield, Laminations and groups of homeomorphisms of the circle, Invent. Math. 152 (2003), 149–204. F. Deahna, Ueber die Bedingungen der Integrabilitat, J. Reine Angew. Math. 20 (1840) 340–350. C. Ehresmann & G. Reeb, Sur les champs d’elements de contact completement integrables dans un variete continuement differentiable, C. R. Acad. Sci. Paris, 218 (1944), 955–957. Y. Eliashberg, Contact 3-manifolds twenty years after J. Martinet’s work, Annales de l’Inst. Fourier, 42 (1992), 165–192. Y. Eliashberg & N. Mishachev, Surgery of singularities of foliations, English translation: Functional Anal. Appl. 11 (1977), no. 3, 197–206 (1978). Y. Eliashberg & N. Mishachev, Wrinkling of smooth mappings. III. Foliations of codimension greater than one, Topol. Methods Nonlinear Anal. 11 (1998), 321–350. Y. Eliashbert & W. Thurston, Contact structures and foliations on 3-manifolds, Turkish. J. of Math. 20 (1996) , 19 – 35.

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[ETh2] Y. Eliashberg & W. Thurston, Confoliations, University Lecture Series 13, Amer.Math. Soc., Providence, RI (1998). [Fe] S. Fenley, Anosov flows in 3-manifolds, Ann. of Math. (2) 139 (1994), 79–115. [Fr] G. Frobenius, Uber das Pfaffsche Problem, J. Reine Angew. Math. (1875) 230–315. [Ga1] D. Gabai, Foliations and genera of links, Topology 23 (1984), 381–394. [Ga2] D. Gabai, Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983), 445–503. [Ga3] D. Gabai, Foliations and the topology of 3-manifolds III, J. Differential Geom. 26 (1987), 479–536. [GY] D. Gabai & M. Yazdi, The fully marked surface theorem, Acta Math. 225 (2021), 369–413. [Gar] L. Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983), 285–311. [GV] C. Godbillon & J. Vey, Un invariant des feuilletages de codimension 1, C. R. Acad. Sci. Series A-B 273 (1971), A92. [Gr] M. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR 33, (1969), 707–734. [Gr2] M. Gromov, Groups of polynomial growth and expanding maps, IHES Publ. Math. No. 53 (1981), 53–73. [Ha1] A. Haefliger, Structures Feuilletees et cohomologie a valeur dans un faisceau de groupoids, Comment. Math. Helv. 32 (1958), 249–329. [Ha2] A. Haefliger, Groupoides d’holonomie et classifiants, Asterisque 116 (1984), 70–97. [Ha3] A. Haefliger, Feuilletages sur les varietes ouvertes, Topology 9 (1970), 183–194. [Ha4] A. Haefliger, Homotopy and integrability, Springer Lecture Notes in Math., 197 (1971), 133–163. [Ha5] A. Haefliger, Naissance des feuilletages, 335–353, Geometrie au XXe siecle : Histoire et horizons 2005, Hermann. [HarL] R. Harvey & B. Lawson, Calibrated foliations (foliations and mass-minimizing currents, Amer. J. Math. 104(1982), 607–633. [HiT] M. W. Hirsch & W. P. Thurston, Foliated bundles, invariant measures and flat manifolds, Ann. Math. (2) 101 (1975), 369–390. [Ka] W. Kaplan, Regular curves filling the plane, I, Duke Math. J. 7 (1940) 153–185. [KR] W. Kazez & R. Roberts, C 0 approximations of foliations, Geom. Top. 21 (2017) 3601– 3657. [La] H. B. Lawson Jr., Quantitative Theory of Foliations, AMS CBMS 27, 1977. [Ma1] J. Mather, Integrability in codimension 1, Comment. Math. Helv. 48 (1973), 195–233. [Ma2] J. Mather, On the homology of Haefliger’s classifying spaces, C.I.M.E. Varenna 1976, Liguori 1979, 73-116. Reprinted in Differential Topology, CIME Summer Schools, Springer 74, 2010. [Ma3] J. Mather, On Haefliger’s classifying space. I, Bull. AMS 77 (1971), 1111–1115. [McD] D. McDuff, The lattice of normal subgroups of the group of diffeomorphisms or homeomorphisms of an open manifold, J. London Math. Soc. (2) 18 (1978), 353–364. [Me] G. Meigniez, Regularization and minimization of Haefliger structures of codimension one, J. Diff. Geom. 107 (2017), 157–202. [Mo] I. Moskowitz A note on the Bott vanishing theorem, Proc. Amer. Math. Soc. 94 (1985), 529–530. [Na] S. Nariman, Thurston’s fragmentation, non-abelian Poincare duality and c-principles, arXiv:2011.04156. [No] S. P. Novikov, The topology of foliations, Trudy Moskov. Mat. Obshch 14 (1965), 248–278. [Ph1] A. Phillips, Foliations on open manifolds I, Comm. Math. Helv. 43 (1968), 204–211. [Ph2] A. Phillips, Foliations on open manifolds II, Comm. Math. Helv. 44, (1969), 367–370. [Ph3] A. Phillips, Smooth maps transverse to a foliation, Bull. AMS 76 (1970), 183–194. [Pi] H. Pittie, Characteristic classes of foliations, Research Notes in Math. 10. Pitman Publishing, London-San Francisco, Calif.-Melbourne, 1976. v+107 pp. [Pl] J. Plante, A generalization of the Poincare - Bendixson theorem for foliations of codimension one, Topology 12 (1973), 177–181. [PlTh1] J. F. Plante & W. P. Thurston, Polynomial growth in holonomy groups of foliations, Comment. Math. Helv. 51 (1976), 567–584.

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[PlTh2] J. F. Plante & W. P. Thurston, Anosov flows and the fundamental group Topology 11 (1972), 147–150. [Po] H. Poincare, Sur les courbes definies par les equations differentielles, Oeuvres, tome 1. [Re1] G. Reeb, Sur certaines proprietes topologiques des varietes feuilletees, Actualites Scientifiques et Industrielles 1183 (1952), Hermann. [Re2] G. Reeb, Structures feuilletees, Differential Topology, Foliations and Gelfand-Fuks cohomology, Rio de Janeiro, 1976, Springer Lecture Notes in Math. 652 (1978), 104–113. [RobSS] R. Roberts, J. Shareshian, & M. Stein, Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation, J. Amer. Math. Soc. 16 (2003), 639–679. [Rou] R. Roussarie, Plongements dans les varietes feuilletees et classification de feuilletages sans holonomie, IHES Publ. Math. No. 43 (1974), 101–141. [RosT] H. Rosenberg & W. P. Thurston, Some remarks on foliations. Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), 463–478. Academic Press, New York, 1973. [Sa] H. Samelson, Differential forms, the early days; or the stories of Deahna’s theorem and Volterra’s Theorem, Amer. Math. Monthly 108 (2001), 522–530. [Sar] F. Sergeraert, BΓ (d’apres Mather et Thurston), Sem. Bourbaki 524 (1977), 300–314. [Sm] J. Smillie, Flat manifolds with non-zero Euler characteristics, Comment. Math. Helv. 52 (1977) 453–455. [Spi] M. Spivak A comprehensive introduction to differential geometry. Vol. I., Second edition, Publish or Perish, Inc., 1979. [Su] D. Sullivan, A homological characterization of foliations consisting of minimal surfaces, Comment. Math. Helv. 54 (1979) 218–223. [Th1] W. P. Thurston, Foliations of three-manifolds which are circle bundles, Thesis (Ph.D.) University of California, Berkeley. 1972. [Th2] W. P. Thurston, Noncobordant foliations of S 3 , Bull. Amer. Math. Soc. 78 (1972), 511– 514. [Th3] W. P. Thurston, Foliations and groups of diffeomorphisms, Bull. Amer. Math. Soc. 80 (1974), 304–307. [Th4] W. P. Thurston, A generalization of the Reeb stability theorem, Topology 13 (1974), 347–352. [Th5] W. P. Thurston, The theory of foliations of codimension greater than one, Comment. Math. Helv. 49 (1974), 214–231. [Th6] W. P. Thurston, “The theory of foliations of codimension greater than one”, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 321. Amer. Math. Soc., Providence, R.I., 1975. [Th7] W. P. Thurston, “A local construction of foliations for three-manifolds”, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 1, pp. 315–319. Amer. Math. Soc., Providence, R.I., 1975. [Th8] W. P. Thurston, “On the construction and classification of foliations”, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 547–549. Canad. Math. Congress, Montreal, Que., 1975. [Th9] W. P. Thurston, Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268. [Th10] W. P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339. [Th11] W. P. Thurston, Three-manifolds, Foliations and Circles, I, ArXiv preprint math/9712268. [Th12] W. P. Thurston, Three-manifolds, Foliations and Circles, II, unfinished 1998 manuscript. [Ts1] T. Tsuboi, On the foliated products of class C 1 , Ann. Math., (2) 130 (1989), 227–271. [Ts2] T. Tsuboi, Classifying spaces for Groupoid structures, Contemp. Math. Studies, AMS 498 (2009), 67–82. [Ya] M. Yazdi, On Thurston’s Euler class one conjecture, Acta Math. 225 (2021), 313–368.

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ś ś   0, /z > 0 such that for all X, E E”, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA t > 0, llrp, * CXJII

A similar

condition

I

Cexp(-~~N~,/1.

(1)

holds for E” but with t > 0. qr is said to be codimension-one

Anosov

if E” or E” is one-dimensional. Let z,(M) denote the fundamental group of M. A finitely generated group is said to have exponential growth if given a finite set of generators the function

T(n) = (number

of distinct group elements of word-length

0, a > 0. This definition is independent of the set of generators, and is equivalent to the function p(r) = (number of homotopically distinct paths in M of length Ir) dominating B exp(br) for some real numbers B > 0, b > 0, in the case the group is the fundamental following

group

of a manifold

M. The object

of this note is to prove the

THEOREM. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Ifrp, :M ---*M is a codimension-one Anosov~¶ow then q(M) has exponential growth. In fact, P(r) 2 B exp(J.r/2a),

where o = sup, E MII@‘,(x)jl and B is some real number.

This result is given by Margulis in the appendix of [2] in the case dim(M) natural to conjecture that this result is true for all Anosov flows.

= 3. It is

$2. PROOF OF THEOREM We begin by fixing some notation.

Assume,

without

loss of generality,

that dim(E’) = 1.

(Otherwise, reverse the direction of the flow.) E” is not necessarily C’ [S], but is uniquely integrable [I, 61. For x E M we denote by W=(x) the integral manifold of E” through x. Let qr denote the unit-speed continuous flow along W”“. The invariant bundle E” @ ET is, in this case, C’ [j] and we denote by W”(x)(x E M) its integral manifolds. Let N c M be a minimal set for the flow qt and take x0 EN to be the basepoint. Let D c W’(x,) be a smoothly embedded closed disk containing x,, as an interior point. There is a smallest 147

WILLIAM P. THURSTON

81

148

J. F. PLANTE and w. P. THURSTON

positive t, such that v~,(x,,) E D. Inductively, such that q,(.Q E D.

given t,_,,

define t, to be the smallest t > t,_,

LEMMA 1. There exists L > 0 such that t, - t,_, < L for all n E Z’. Proof. Since ~(1 N is minimal, given E > 0 there e?tists L, such that for each z E N the set (V,(Z) IO I t I L,} intersects the c-neighborhood of every .YE N [7, p. 3751. Choose E so that if n(x, x0) 0. L = L, + 26 satisfies the requirements of the lemma. Now define loops yn = a, * /I, where TX,goes from x0 to v~,(_Y~)along the qt orbit and the image of /I,, lies in D and goes from qr,(xO) to x0). (We assume that the length of /?, is bounded by a constant independent of n.) LEMMA2. y, 2: y,,, on/y if n = m. Proof. j?, N 0. X,-l

Suppose,

on the contrary,

* LY,is homotopic

that y” 2: y,,, where n > m. Thus, p,,,-l * %,,,-I * c(, *

(with endpoints

fixed) to the path r*,_,,, along the q,-orbit from

I,_, to qr,(x,). Now p,,,-’ * a,,,-r * Bfl * U, 1: p,,,-l * Q,_, * /?, and the latter is homotopic to a closed transversal to the IV”-foliation. Since this closed transversal is nullhomotopic it spans an immersed disk which may be put in general position with respect to the IV”foliation

by arguments

in [4]. (See the proof of (5.1).) By a Poincare-Bendixson

argument

(as in (4.2) of [5]) we can find a one-sided limit cycle in the foliation (with singularities) induced on the disk. This implies that the IV-foliation has a non-trivial element of holonomy which corresponds to a zero element of linear holonomy. But this cannot happen since any non-trivial element of holonomy in the IV-foliation is conjugate to (~~1 W”(p) for some periodic

point p of period T (i.e. (P=(P) = p) and the Jacobian

from 1. This contradiction

completes

the proof of Lemma

of qT] IV(p)

at p is different

2.

We denote by 1ct1 the length of the path a and by E its homotopy

class, if it is a closed

loop. LEMMA

homotopic

3. Let u: [0, l] + M represent (with endpoints heldfixed)

any orbit segment

of the j?obv q,.

Th

% is

to a path a’ such that

Ia’1 5 C+ 2(o//l)log]r(. Proof. Let 6,. f be the path given by 6,. *(s) = cp,,,(x). We homotope E by letting it flow under (Pi, while reeling out tie-lines from the end-points of r to its respective ends-formally, c( 2: a,,, t * cpt0 c( * Sa;fi for every t. Now set t = log I a 1/d and denote the curve on the right by a’. Then by (l), 151’15 a(loglrl/J.) + C]cc]exp(-I I c + 2(a/l)log 1%I.

log]a]/%)

+ o(log(~]/~~)

This proves the lemma. LEMMA

4. Let G be anyfinite

nian manifold. For CcE q(M)

82

set of generators for n,(M)

define II(g) to be the minimum

COLLECTED WORKS WITH COMMENTARY

where M is a compact Rieman-

length of a word in G representing

A SO SO V

FLOWS A>D THE FIJKDAMENTAL

149

GROUP

5 and let I,(E) be the minimum of the lengths of loops in 2. Then there exist k,, kl(O < k, 5 kz) such that for erery 5 # e,

First note that 1*(E) > 0 for 5 # e. If Cc = gilel .. g,.,‘” with ei = f 1, then We have I,(?)ll,(Z) I li2. 12(sOI Il(gi,) + ’ ’ + rI(gi,). Let k, = max,(fl(g)). Proof:

To find k, we look at the universal Riemannian covering manifold M. Let K be a compact fundamental domain, let the diameter of K be D, and let the diameter of M be d. The ball B of radius 3d + D about x0 has finite volume. so there can be at most a finite number of images of K contained entirely in B. But any image of K intersecting the ball of radius 3d about x0 lies entirely in B. Thus, there are finitely many homotopy classes of loops max It(E). Given a loop r we may partition it (with paths of 12(i)_ 0. For c( + O,I z( + rf_
-

The theorem

now follows from the Lemmas.

By Lemmas

P(C + 2(o/A)log(nL)) Thus, for values of r which are arguments

I, 2, and 3,

2 n.

of the function

P for some II,

P(r) 2 B exp(riJ2a). Now, by adjusting the constant B, we can make this inequality hold for all r, thus proving the second statement of the theorem. The first statement follows immediately by Lemma 4.

$3. FURTHER

REMARKS

In [3], Bowen shows that the number of closed orbits of an Anosov (more generally Axiom A) flow grows exponentially with the period. There are examples in which distinct closed orbits are in the same free homotopy class; but if an upper bound, or even a low growth rate, for the number of distinct closed orbits in a given free homotopy class could be found, this would show that the number of conjugacy classes of x1(M) grows exponentially with the minimum length of a representative. An improved

estimate

can be found for the case the non-wandering

set of the Anosov

WILLIAM P. THURSTON

83

J. F. PLANTE and W. P. THURSTOS

150 flow is the entire foliation :

manifold,

by estimating

are the constants tively.

for the rate of expansion

P(r) 2 B exp(i,(n

the growth

- 2)/o + &/20),

rate of disks on leaves of the W”where d, and &

of E” and the rate of contraction

of E’, respec-

REFERENES 1. D. V. Axosov: Geodesic flows on closed Riemannian manifolds with negative curvature, Proceedinqs of the Steklou Institute of Math. 90 (1967); Am. math. Sot. translation (1969). 2. D. V. ANOSOVand Ya. G. SINAI: Some smooth ergodic svstems, Usoekhi Math. Naauk22 (1967). 107-172; Rtissian Math. Surceys 22 (1967). 103-168. _ . 3. R. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA BOW EN: Periodic orbits for hyperbolic flows (to appear). 4. J. FRANKS: Anosov diffeomorphisms, Proceedings of Symposia in Pure Marh., Vol. I4 (1970). pp. 61-93. 5. A. HAEFLIGER:Varitttes feuille&s, Ann. Scuola Norm. Sup. Piss (3) 16 (1962), 367-397. 6. M. W. HIRSCH,C. C. PUGH and M. SHUB: Invariant manifolds (to appear). 7. V. V. NEMY-~SKII and V. V. STEPANOV:Qualitative theory of differential equations, Princeton (1960). 8. J. PLANE: Anosov flows (to appear). 9. S. SMALE:Differentiable dynamical systems, Bull. Am. math. Sot. 73 (1967). 747-817.

Unicersity of Wisconsin cd University nf Cdfornifl

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WILLIAM P. THURSTON

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 %$)#%p 3O;p 0p *.%)*$# +8%A+5BI *+I %C0,>C19F+I "61G+@1D,I *+I &;I @%HI :6)+I (*I #61G+@1DHI 7.I %41/=1%I  0 be sufficiently

+ 1, E’) = 1 +f(r,

with constant

(2)

E’)(l

+

E’).

term r.

small that E’ .f(l, E’) _< E

and let 6’ > 0 be sufficiently

ljdH,(x) Finally,

(3)

small that if llxjl I 6’, and r* E B’, then

let 0 < 6 < 6’ be sufficiently

L E B’. This, we claim, is a 6 suitable x, for some a. First, we will give an upper

- 1115 small

(4)

E’.

that

if jjxjl I 6, then

llH,(.~)jl 5 S’ for

for Lemma 2. Let x be such that //x/j 2 6, and H,(X) #

bound

for y,(a), when a E B’, r I I: zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ

il~,(4ll If(r,

0

0)

This is true for r = 0 or r = 1. Assuming inductively that (5) is true for r = k, let u be an arbitrary element of B ‘+’ . Write 0 = psc, with r E Bk, /I E B’. Then 1 lIHa)Il = - IIH,(.u) - XII m

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A GESERALIZATIOS

OF THE REEB STABILITY

351 zyxwvutsrqp

THEOREM

NOW

1 H.(r) =- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE dH, - I). m ix Thus, L zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA IIH,(H,(x)) - H,(x)I/ < max[ldHP - 111 k (H,(x) - x + 1 < &‘(f(k, E’) -t 1 fn II II by (A), (5), for r = k, and (1). So liy,(a)Ii I 1 + (1 + e’)f(k,

E’) (x) - x -

zz - ;

= --

[H,,(H&))

1

- H&r)

HP, ,&)

- (HP,+)

+ x]

- (x))]

HbZ(X) (dHb, - 1)

m iX

so

II

II

IlhJA, Pdll 2 ; (H,,(x) - x) . ,,;;“o IldHp,(y) by (5), (4) and (3). Thus, y, is a normal Theorem 2 is concluded. Modifications

be a compact

necessary

neighborhood

(B’, s)-cocycle

inversion

Ill I

E

with values in Wk, and the proof of

to proce Theorem 3. In the proof

of 1, closed under

-

of Theorem

and generating

3. we take B to G. We denote

the

representation of C by H. If p is a fixed point of H, then for x near p (and in some Euclidean neighborhood, which we identify with Euclidean space by a fixed chart) such that x is not fixed by H, a continuous function y, on a subset of G containing B is constructed by the same formula

(1).

The proof of Lemma 2 works verbatim: by the continuity of Hand the compactness of B’, there are no problems with findin g sufficiently small E’S and 6’s. Lemma 1 must be helped along, here and there. In the definition of normal (B’, &)-cocycles, for a topological group G, we do not add any requirement that they be continuous. However, a normal (B’, E)-cocycle is bounded by I + (I - I)&. Then, in the topology of pointwise convergence,

WILLIAM P. THURSTON

113

352

WILLIAM

the space of normal conclude Theorem

P. THURSTON

(B’, &)-cocccles is compact.

Assuming

the hypotheses

of Lemma

1, we

that there is a normal (G, O’kcocqcle with talues in R”. For the purposes 01 3. we need a continuous cocycle. However. a cocycle 7 which is bounded (say, by 11

on B is automatically

continuous.

We need only see it is continuous

at 1, because of the

cocycle condition 87 = 0. But for every n. there is a neighborhood L. of 1 such that 5” 3 B Then for 2 E L’, q(r) = ;‘(Y) I 1. so ‘/ is bounded by I/n on C,‘: therefore. 7 is continuous This shows that if the linear action at p is trivial, and if there are no non-trivial continuou: representations

of G in R. then a neighborhood

H for which the linear action so it is all of V+.

is trivial

of p is fixed by H. The set of fixed points 01

is a closed set; the above argument

shows it is open

REFERENCES I. M. \V. HIRSCH and W. THURSTOS: Foliated Annals). 2. G. REEB: Sur certaines proprietes topologiques

Paris (1952). 3. W. THURSTOS: Foliations Berkeley

invariant

* Current

Cmlbricige,

address:

which

are circle

Massachusetts*

Princeton

University,

measures.

des variCt& feuilletees,

(1972).

M.I.T.,

114

of 3-manifolds

bundles,

Princeton,

COLLECTED WORKS WITH COMMENTARY

N.J.

bundles.

and flat manifolds

(to appear

Acrual Sci. Ind. No. 1183. Hermann Thesis,

University

of California

WILLIAM P. THURSTON

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http://dx.doi.org/10.1090/pspum/027.1/0375345

Proceedings of Symposia in Pure Mathematics Volume 27, 1975

THE THEORY OF FOLIATIONS OF CODIMENSION GREATER THAN ONE WILLIAM P. THURSTON*

A method is developed to circumvent the Gromov-Phillips theorem and to extend Haefliger's classifying theorem for foliations [1] to the case of foliations of codimension greater than one on compact manifolds. In particular, a plane-field zn~k on Mn, k ^ 2, is homotopic to a foliation iff TMn\zn~k admits a Haefliger structure. Every plane field of codimension greater than one is homotopic to a C° foliation with C°° leaves. If Sn has a &-plane field, k f£w/2, then Sn has a C°°, codimension k foliation. Every 2-dimensional plane-field is homotopic to a C°° foliation. (Cf. [3] for the codimension one case.) ADDED IN PROOF. These results have now been extended to codimension 1. REFERENCES 1. A. Haefliger, Feuilletages sur les varietes ouvertes, Topology 9(1970), 183-194. MR41#7709. 2. W. Thurston, The theory of foliations of codimension greater than one, Comment. Math. Helv. 49(1974), 214-231. 3. , A local construction of foliations for three-manifolds, these PROCEEDINGS. MASSACHUSETTS INSTITUTE OF TECHNOLOGY

AMS (MOS) subject classifications (1970). Primary 57D30. *Current address. Princeton University. © 1975, American Mathematical Society

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WILLIAM P. THURSTON

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