Collected Works of William P. Thurston with Commentary II. 3-Manifolds, Complexity and Geometric Group Theory 147046389X, 9781470463892

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COLLECTED WORKS OF

WILLIAM P. THURSTON WITH COMMENTARY

COLLECTED WORKS OF

WILLIAM P. THURSTON WITH COMMENTARY II

3-Manifolds, Complexity and Geometric Group Theory Benson Farh David Gahai Steven P. Kerckhoff Editors

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AMS

AMERICAN MATHEMATICAL SOCIETY

Providence, Rhode Island

Editorial Board Jane Gilman (Chair) Joseph Silverman Andras Vasy

2020 Mathematics Subject Classification. Primary 20F65, 57K32, 53C15.

Library of Congress Cataloging-in-Publication Data Names: Farb, Benson, editor. I Cabai, David, editor. I Kerckhoff, Steve, editor. Title: Collected works of William P. Thurston with commentary I Benson Farb, David Cabai, Steven P. Kerckhoff, editors. Description: Providence, Rhode Island American Mathematical Society, [2022] I Includes bibliographical references. I 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 I ISBN 9781470451646 (hardcover-set) I ISBN 9781470463885 (hardcover-vol. I) I ISBN 9781470463892 (hardcover-vol. II) I ISBN 9781470463908 (hardcovervol. III) I ISBN 9781470468330 (ebook-vol. I) I ISBN 9781470468347 (ebook-vol. II) I ISBN 9781470468354 (ebook-vol. III) Subjects: LCSH: Thurston, William P., 1946-2012. I Differential topology. I Geometry, Differential. I Dynamics. I AMS: Manifolds and cell complexes - Low-dimensional topology in specific dimensions - 2-dimensional topology (including mapping class groups of suriaces, Teichrnilller theory, curve complexes, etc.). I Manifolds and cell complexes - Low-dimensional topology in specific dimensions - Foliations in differential topology; geometric theory. I Manifolds and cell complexes - Low-dimensional topology in specific dimensions - Hyperbolic 3-manifolds. I Differential geometry - Global differential geometry - General geometric structures on manifolds (almost complex, almost product structures, etc.). I Group theory and generalizations - Special aspects of infinite or finite groups - Geometric group theory. I Group theory and generalizations Special aspects of infinite or finite groups - Word problems, other decision problems, connections with logic and automata (group-theoretic aspects). I Dynamical systems and ergodic theory Dynamical systems over complex mnnbers - Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets. I Computer science - Theory of data Data structures. Classification: LCC QA611 .C645 2021 I DDC 514j.22--dc23 LC record available at https://lccn.loc.gov/2021037684

©

o

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:llwww.ams.org/ 109 8 7 6 5 4 3 2 1

27 26 25 24 23 22

Contents

Volume II Preface

Xlll

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.

131

"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 SeL, 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

WILLIAM P. THURSTON

v

CONTENTS

vi

(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," 1. A mer. Math. 349 Soc. 20 (2007), no. 4, 1053-1077. (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 413 curves," Discrete Comput. Geom. 29 (2003) no. 1, 1-17. (with Joel Hass and Jeffrey C. Lagarias) "Area inequalities for embedded disks spanning unknotted curves," 1. 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

(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.

vi

COLLECTED WORKS WITH COMMENTARY

603

CONTENTS

vii

(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 I Preface

Xlll

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 S3," 511-514.

85

Bull. Amer. Math. Soc. 78, no. 4, (1972),

(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 "A local construction of foliations for three-manifolds," Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 161 1973), Part 1, pp. 315-319. Amer. Math. Soc., Providence, R.I., 1975. "On the construction and classification of foliations," Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 167 547-549. Canad. Math. Congress, Montreal, Que., 1975.

WILLIAM P. THURSTON

vii

CONTENTS

viii

"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), 219 no. 339, i-vi and 99-130. (with Yakov M. Eliashberg) "Contact structures and foliations on 3-manifolds," Turkish 1. 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 eprin!.

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 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

viii

COLLECTED WORKS WITH COMMENTARY

CONTENTS

ix

(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," Froc. 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," Beitrage Algebra Geom. 42 (2001), no. 2, 475-507.

727

Volume III Preface

Xlll

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," VVhat's Next?: The Mathematical Legacy of William P. Thurston, Dylan Thurston (ed.). Annals of Mathematics Studies 205, 259-325, Princeton University Press, Princeton, N J, 2020.

223

(with Ethan M. Coven, William Geller and Sylvia Silberger) "The symbolic dynamics of tiling the integers," Israel 1. Math. 130 (2002), 21-27.

291

WILLIAM P. THURSTON

ix

CONTENTS

x

(with Dennis P. Sullivan) "Extending holomorphic motions," Acta Math. 157 299 (1986), no. 3-4, 243-257. "Zippers and univalent functions. The Bieberbach conjecture," (West Lafayette, Ind., 1985), 185-197, Math. Surveys Managr., 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," 1. Amer. Math. Soc. 1 (1988), no. 3, 647-681. 343 (with Daniel D. Sleator and Robert E. Tarjan) "Short encodings of evolving structures," SIAM 1. 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," 1. 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 1. Sci. Camput. 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. Pub!. 35, Cambridge Univ. Press, Cambridge, 1998.

543

Miscellaneous

551

Commentary: Miscellaneous Papers

553

x

COLLECTED WORKS WITH COMMENTARY

CONTENTS

xi

"A Constructive Foundation for Topology," Senior Thesis, New College, June 14, 1967. 555 (with D. M. Kan) "Every connected space has the homology of a K(7r, 1)," Topology 15 (1976), no. 3, 253-258.

595

(with L. Vaserstein) "On K,-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

Xl

Introduction

XVll

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.

WILLIAM P. THURSTON

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 S3," Bull. Amer. Math. Soc. 78, no. 4, (1972), 511-514; ©1972, American Mathematical Society. "Foliations and groups of diffeomorphisms," Bull. Amer. Math. Soc. 80, no. 2, (1974),304-307; ©1974, American Mathematical Society. "The theory of foliations of codimension greater than one," Differential geom-

etry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 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), 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," Froc. Amer. Math. Soc. 52 (1975), 345-347; ©1975, American Mathematical Society. "Some simple examples of symplectic manifolds," Proc. Amer. Math. Soc. 55 (1976), no. 2, 467-468; ©1976, American Mathematical Society. "Three dimensional manifolds, Kleinian groups and hyperbolic geometry," Bull. 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 (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 Managr., 21, Amer. Math. Soc., Providence, RI, 1986; ©1986, American Mathematical Society. xv

WILLIA M P. THURSTON

xv

xvi

PERMISSIONS & ACKNOWLEDGMENTS

(with Daniel D. Sleator and Robert E. Tarjan) "Rotation distance, triangulations, and hyperbolic geometry," 1. Amer. Math. Soc. 1 (1988), no. 3, 647-681; ©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 (1994), no. 2,161-177; ©1994, American Mathematical Society. (with Yakov M. Eliashberg) Confoliations, University Lecture Series 13, American 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," 1. Amer. Math. 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 1. Miller, Shang-Hua Teng, and Stephen A. Vavasis, 1. 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, 761766,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 SeL, 48, Cambridge Univ. Press, Cambridge-New York, 1982. ©1979 Cambridge University Press and reproduced with permission.

xvi

COLLECTED WORKS WITH COMMENTARY

PERMISSIONS & ACKNOWLEDGMENTS

xvii

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," Beitriige Algebra Geom. 42 (2001), no. 2,475507.

Elsevier Reprinted from "Anosov flows and the fundamental group," with J. F. Plante, 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 Press, New York, ©1973 with permission from Elsevier. Reprinted from "A generalization of the Reeb stability theorem," Topology 13 (1974),347-352; ©1974 with permission from Elsevier. Reprinted from "Every connected space has the homology of a Kerr, 1)," with 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 orientable 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, 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 Kenneth Steiglitz, Phys. D 19 (1986), no. 3, 423-432; ©1986 with permission from Elsevier. Reprinted from "On Kl-theory of the Euclidean space," with 1. Vaserstein, 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. Cooper, Topology 27 (1988), no. 1, 23-25; ©1988 with permission from Elsevier.

Fondation L'Enseignement Mathematique (with J. Milnor) "Characteristic numbers of 3-manifolds," Enseign. Math. (2) 23 (1977), no. 3-4, 249-254.

WILLIAM P. THURSTON

xvii

xviii

PERMISSIONS & ACKNOWLEDGMENTS

(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 conformally 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.

lOP Publishing, Ltd. Republished with permission of lOP Publishing Ltd., from "How to see 3manifolds"; Topology of the Universe Conference (Cleveland, OH, 1997), Classical 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," Froc. 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. Pub!. 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," VVhat'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, 2020; ©2020. Republished with permission of Princeton University Press; permission conveyed through the Copyright Clearance Center, Inc.

xviii

COLLECTED WORKS WITH COMMENTARY

PERMISSIONS & ACKNOWLEDGMEN TS

xix

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 1. Discrete Math. 5 (1992), no. 3,428-450. (with Gary 1. Miller, Shang-Hua Teng, and Stephen A. Vavasis) "Geometric separators for finite-element meshes," SIAM 1. Sci. Comput. 19 (1998), no. 2, 364-386.

Societe Mathematique 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 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, 225-246; ©1985. Reprinted by permission from Springer Nature, "Extending holomorphic motions," 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 (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; ©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. 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 1. Miller, Shang-Hua Teng, and Stephen A. Vavasis, 57-84, IMA Vol. Math. Appl., 56, Springer, New 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, Israel 1. 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 (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. 166 (2006), no. 3, 457-521©2006. 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. Topo!. Pub!., Coventry, 1998.

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Nathaniel Thurston Figure 1. Lines of the form nx + my = 1/2 where nand m are integers. Any convex polygon in this network which is symmetric in the origin is the unit sphere in H 2 (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 1. Math. 20 (1996), no. 1, 19-35.

Universita degli Studi di Cagliari (with D. B. A. Epstein) "Combable groups," Conference on Differential Geometry and Topology (Sardinia, 1988). Rend. Sem. Fac. Sci. Univ. Gagliari 58 (1988), suppl., 423-439.

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Part 1.

Three-Dimensional Manifolds

THREE-DIMENSIONAL MA N IFOLDS

Bill Thurston revolutionized t he field of 3-dimensio nal topology. By utilizing geometric structures he WM able t o solve old topological problems and reorient the fo cus of t he area" providing a new and vast scope to the discipline . The classificatio n of compact 2-dimensional manifolds had b een k nown for m any years and they were all known to admit constant curvature metrics. Before Thurston, the idea that something similar was t rue in dimension 3 - that homogeneous geD metric manifolds could b e the building blocks for any 3-rnanifold - was b eyond imaginatio n. In part t his was because so few explicit examples of 3-rnanifolds, outside a few special families, were known. Thursto n's hands-on and co mputational approach provided many new examples, p art icularly of hyperbolic 3-manifolds, making it clear that geometric strllct ures were ubiquitous. Thurston's geo metrization co nj ecture states that a closed , irreducible 3-rnanifold , after p ossibly cutting a long a canonical co llection of disjoint tori , splits into pieces each of which carries one of eight homo geneous metric geometries. Although it is formulated in geometric t erms, it remarkably implies m any long-standing t opological conjectures, of which the Poincare Conjecture is the most famous. Thursto n was led t o his conjecture after proving it for a lru:ge class of manifolds. More than t wo decades later the fu ll conjecture was solved by Perelman [PI . A brief, informal discussion of the history of the geometrization conjecture can b e fo und in [GKI. In his 1976-1977 course at Princeton, Thurston fust conjectur ed and ultimately proved t hat the geometrization conjecture waB t r ue for Haken manifolds . A 3manifold M is irreducible if every embedded 2-sphere in M bounds a 3-dimensional b all. It is called sufficiently larg e if it contains a n embedded, orientable surface of genus bigger than zero whose fundamental group injects into 7rl M. This includes 3-manifolds with positive genus boundary. A co mpact, orientable, irreducible, suffi ciently large 3-manifold is called a Hak en manifold. T he work of J aco-Shalen a nd Johannson ([JSI, [J]) provides any closed, or ientable, irreducible 3-manifold with a canonical (up to isotopy) decomp osition along disjoint incompressible tori. The pieces of this deco mposition (called t he JSJ decomp osition) are either Seifert fibered or atoroidaJ (all incompressible t or i ru:e b oundary parallel). When t he JS J decomposition of a Haken manifold is non-trivial, Thursto n's theorem a mounts t o saying that the atoroidal p ieces have finite volume , complete hyp erbolic structures; p utting a geometric structure on t he Seifert fib ered pieces is straightforward. But more specifically it says t hat when the JS J decomposition is trivial and the entire manifold is ator oidal, a Haken manifold has a hyperbolic structure . In [HI and [W] it is proved that Haken manifo lds can b e successively cut along a fi nite number of incompressible surfaces into simpler pieces, ultimately ending up with 3-balls. T his makes it natural to try to prove t heorems about t hem inductively. Indeed, t his is the strategy of T hurston's proof. But that means t hat, even if M 3 is closed, one needs to consider manifolds with b oundary t hat typically have genus b igger than one . A complete hyperbolic 3-manifold with a higher genus boundary necessarily has infinite volume with a large parameter space of str uctures. Th show t hat a n atoroidal Haken manifold has a hyperbolic structure, one shows inductively

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t hat the simpler p ieces have hyperb olic structures a nd that t hey can be defor med in a way t hat allows them to be pieced together to give a more complicated hyperbolic m anifold. In t he inductive step one has a Haken manifold N with t wo incompressible b oundary comp onents of genus 9 obtained by cut ting a larger manifold M along a n incompr essible surface. N has a n infinite volume geometr ically fin ite structure. The classical t heory of Kleinian groups, develop ed by Ahlfors , Bers a nd others, implies t hat the space GF(N ) of s uch s tructures is parametrized by the 'Thichmuller spaces T (8 N ) of conformal structures o n t he b oundary surfaces . This is a product Tg x Tg of 'Thichmuller spaces of a surface of genus g . The goal is to choose t hese c onformal structur es so that N can b e glued back to give a hyperbolic structure on M . Because the boundary components are both convex this is not just a matter of choosing the conformal structur es to mat ch. Instead, o ne considers the covering space of N corresp onding to the fundamental group of each boundary co mp onent . The cover ing space is diffeomorphic to a surface of genus 9 times an interval and it has an induced geo met r ically fi nite str ucture, called a quasi-Fu chsian st r ucture . Quasi-Fuchsian st ructur es are parametr ized by conformal structures o n t he two b oundary comp onents . In the c urr ent s ituatio n, for each component of 8N one of t he co nfor mal str uctures comes from 8N . The other is d et ermined by the geomet r ically finite str ucture o n N. We refer t o t his other structure as being "inside" of N , tho ugh it is not actually e mbedded. For each b oundary comp onent t his determines a map Tg -+ T g taking the outside confor mal structur e to the insid e one. In t urn this defines a continuous map UN : T (8N ) -+ T (8N ) that Thursto n calls the skinning map of N . The goal is to match up the quasi-Fuchsian groups correspo nding t o t he two b oundary comp onents in a way t hat the outside conformal structure of o ne agrees with the inside st ructure of the ot her and vice versa. If t his is accomplished, the Maskit combination theorem ([Mal) implies that the hyperbolic structure on N can be used to determine a geometrically fin ite structure on M . Thursto n defines a gluing map T O UN T (8N ) -+ T (8N ) with t he property t hat a fixed point corresponds to a solution of this matching problem W hen N is not (virt ually) a s urface t imes an interval, the gluing m ap is a strict co nt r action . However , since 'Thichm uller space is no n-compact this does not provide a fixed p oint. The bulk of Thurst on's proof is dedicated to pr oving that, for any geometrically finite str ucture mE G F( N ), t he sequence (TO UN ) k (m ) is bounded. T he existe nce of a (unique) fixed p oint of t he gluing map fo llows easily. Thurston provided many details of his proof in t hree r emarkable papers ([Th1l , [Th2J, [Th3l). In the introduction to the first of these papers, he outlined his intention to wr ite three more pap ers in t he series. Specifically, t he fourt h paper was intended to p ut together the material in the first thr ee, along with preliminary m aterial fr om Chapters 8 and 9 of his Princeton Notes ([T h7]), to provide a co mplete proof. These latter papers did not app ear but Thursto n provided details of the remaining pieces of the proof in a numb er of lectures, some of which app ear in written form here. First, from t he pro ceedings of a 1979 conference in Bangor , Wales, is an account of Thurston 's lecture ([Th6]) written by Peter Scott . Secondly, in the summer of 1980 t here was a mult i-week co nference at Bowdoin College at which Thurston gave his most extensive exposition of his proof. He pr ovided a document ([Th5]) to the confer ence participants t hat we include here. The short

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description of the skinning and gluing maps ab ove are based on t hat d ocument. J ohn Morgan [Mo] wrote a detailed account of Thurston's pr oof, assuming geometric results that roughly correspond to [Th 1,2,3]. Given the central role of Thurston's three ma,in papers in the proof of t he geometrization of Haken manifolds , we provide a brief overview of their contents and explain how they relate to geo metrization. For a compact, oriented , irreducible 3-rnanifold M 3 , T hurston considers co mplete hyperbolic 3-manifolds N, together with a homotopy equivalence f : M -+ N , where t wo structures (N, !) and (N', f ') are equivalent when they are isometric via a map respect ing the ho motopy equivalences. Such a structure determines a holonomy representation ?'r1M -+ G t o the group G of isometries of hyperbolic 3-spaC€, modulo conjugation in G . The space of equivalence classes of such representations with the topology of convergence on generators induces a t opology on this space of hyperbolic struct ures which is denoted by AH (M ) . In [Th1] Thurston s tudies AH(M) in t he special case where M has incompressible boundary and is acylindrical. The latter condition means t hat any map of a cylinder C = S1 X I , f: (C , 8C) -+ (M, 8M ) that t akes the boundary co mponents of C t o non-trival homotopy clas.ses in 8M is homotopic reI boundary into 8M. The ma,in result of the paper is that in this case AH (M) is compact. The p aper [Th3] is t he natur al mathematical sequel to [Th1]. AH(M ) is studied in the case when M has incompressible b oundary b ut is allowed to have non-trivial cylinders. The extreme case when M is a surface times a n interval, hence is "all cylinders" , is not allowed (and is considered in [Th2]) . In this case a relative compactness result for AH (M ) is proved. In t he context of the gluing pr oblem discussed above, denote by N the manifold on which the gluing map is defined and assume it is not a product . When N has a geomet r ically fi nite hyperbolic structure we have that GF(N) C AH(N ) is a n open subset.Then these compactness results imply that , for any m E GF( N ) the sequence (To uN)k(m) has a subsequence that co nverges in AH(N ) and t hat the closure of such limits is compact. Using results fr om Chapters 8 and 9 of [Th7] one can show that these po ints are actually in GF(N ). Thus the sequence (T OGH )k(m) is bounded in GF(N ) as desired . The case when M is a s urface bundle over S 1 is handled in [Th2]. Such an M can be described as S x / , a surface of genus at least t wo times an interval, modulo the identification of (x,D) with (4)( x), 1), where 4> : S -+ S is an orientation preserving homeomorphism. M is atoroidal and Haken if and only if 4> is pseudo-A noso v. Thurston proves t hat in t his case M has a hyperbolic structure. The surface fib er provides a n inco mpressible surface, but cutting a long it gives the product N = S x /, t he case not covered in [Th1] and [Th3]. In particular, the iterates (T OUN)k (m) are not bo unded in G F (N) , a nd the previo us arguments don't apply. Instead, a more direct argument is used . Geo metrically finite (quasi-Fuchsian) structures on S x / ar e parametrized by a pair of conformal structures (X , Y), X , Y E T (S) . Thurston considers the sequence of quasi-Fuchsian structures determined by (4) -k X ,4>k X) for any choice of X E T(S ). He shows that it converges in AH(S x 1) to a discrete faithful representation of ]f1S and that t he limit s et of t his representation is the entire 2-sphere at infinity. T he limiting structur e is a hyperb olic str uct ure on S x JR . It has has a quasi-conformal self-map induced by a primitive covering t ransfor mation of the covering S x.IR -+ M. Beca use the limit set is the entire

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T HRE8-DIMENS ION AL M ANI FOLD S

2-spher e at infinity, Sulllivan's r igidity t heorem ( [S]) implies the map is hom otopic t o an iso metry. Dividing out by t his isometry gives a hyperbolic structure on M. The m ain step of the proof is t o show that there actually is a limiting str ucture . Thurst on shows t his b y proving t hat if a pair of sequences in T(S) converge in t he Thurston compactification of 'Thichmuller s pace to points suffici ently far fro m t he diago nal, t hen the corresponding sequence of quasi-Fuchsian groups converges . (This r esult is called the "double limit t heorem" ) It is interesting that the proo f is so differ ent in t he non-fibered Haken CMe than t hefibered case, which WM actually the final case that Thurston s olved. Apparently, t his WM the CMe that Thursto n was most unsure might be hyperbolizable. He realized t hat if such a surface bundle had a hyperbolic structure, a ny surface fiber would lift to lEV M an embedded disk that would accumulate densely on the sphere at infinit y, a very surprising phenomeno n. In a p ap er that is a striking combinatio n of geo metry and p oint-set topology, Cannon and Thurston [Ca'TJ s howed t hat not only was t his t r ue but t he map of the universal cover of a fiber, viewed as a n open disk, extends continuously to the boundary circle as a space-filling curve in the sp here at infinity. Thurst on [Th4] wrote an exp ository article for t he Bulletin of the AMS in which he discussed the Geometrizatio n Conj ect ure and t he ro le of geo metric structures in 3-dimensional t opology. It's a fascinating glimpse at his appr oach to the field . But its most influential section was the list of problems and questions at t he end of the paper. They included what came t o be called the Density Conjecture, the 'Th.meness Conjecture, the Subgroup Separability Conj ect ure, t he Virutal Haken Conjecture, the V irtual F ibering Conject ure, and t he Geometrization of Orbifolds Conjecture. These questions, all of which have ult imately been answered , were t o drive r esearch in t he field for the next several de cades. The fo cus of this note has b een on the p apers t hat are most directly connected with geometrization. However, this section contains a number of other papers by Thurst on and coauthors t hat demonstrate the breadth of his work on 3-manifolds . Some are largely t opolo gical and combinat orial ([H 'TJ, [Co'TJ ) while others use geom etric methods t o attack topological problems ([HTfJ , [AS'T1) . What t hey a ll share is his special insight and creativity.

REFERENCES

[AST] I. Agol, p , St orm an d W . Thur ston, Lower bounds on volumes of hyperbolic Haken 3man ifolds, with an ap pendix by N . Dunfield, J . Amer'. Math S oc. 20 (2 00'r), no, 4, 105 310" [CaT] J C annon a nd W . T hurs ton, G rou p invariant P ea no curves, Ge om. To po L 11 (2 00'r ), 131 51355 [CoT] D , C ooper and W . T hurston, Triangulat ing 3-manifolds usi ng 5 vertex li nk typ es, Topolog y 27 (1988), no, 1, 23-25 [G K] D , Gabai a nd S. Kerckhoff, Thurst on's geometrization conjecture, Clay M athematia; Inst itut e, 2009 Annu al Rep or t, 32-38 [H) W . Haken, Ube r das homoomor phie problem der 3-man nigfal tigkeiten I, M ath. Z 80 (1962), 89- 120. [HT] A, Hatcher a nd W . T hurs ton, Incom pressible surfaces in 2-bricige knot com plements, Invent M ath 79 ( 1985). no. 2, 225-246 [HIT] J , Hass, A, T homp son, and W, Thurston, Stabilization of Heegaard splitting;, Geom TopoL 13 (2009), no. 4, 2029-2050 [JS] W , J aco and p, Shalen, Seifert fi bered spaces in 3-manifolds, M emoir s A MS, 220 (19'r9)

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K. Johannson, Homo topy equival ences of a- manifolds with b ounda ry, Lecture Nates in Mathematic s, 761 , Springer-Ve rl ag (19'/'9) [1v1a) B , M askit, On Klein 's combination theore m III, in "Advances in th e t heory of R iemann su rfac es", A nnals of Math. Stu.die s 66, 29'T- a16, Princeton Univ. Press (19'71). [1v1o) J. Morgan, On Thurst on's uni formizatio n t heorem for three-ma nifolds, i n "The Sm ith co njecture (New York, 197"9 )" , Pu.re A ppl. M ath. 11 2 , 37"- 125, Academic Press (1984) [P) G , Pe relman, R icci flow with s urge ry on t hre e- manifolds, arX iv: math.DG 0303 109 [S) D , Sullivan, O n the e rgodic theore m at infin ity of an ar bitra ry discre te gr oup of hyperbolic motions, i n "Riemann sur faces and related topics (St ony Brook, 19'T8 )" , A nnals of Math S t ti.die s, 97, 465-496, Pri nceton Univ. P ress (1981) [T hl) W , Thurston, H ype rb olic structu res on 3- manifo lds I : D eformations of acylindric al manifol ds, An 1'1llLs of Math , (2) 124 (1986), no , 2, 203-246 [Th2) W , Thurs t on, Hyp e rbolic str uct ur es on 3- mani folds II: S urface groups and 3-m anifolds which fiber over the ci rcl e, prep fi Td . [Th3) W , Thurston, Hyp er bolic structu res on 3- m anifo lds III: Defor matio ns of 3-manifolds wi th incom pressi ble boundary, preprird [T h4] W , Thurston, Three-di mensio nal manifo lds, Klein ian groups an d hyp erbolic ge ome tr y, B u.J.letin A MS (N.s.) 6 ( 198 2), no. 3, a5'7- 38 1 [Th5) W , Thurs ton, Hyper bolic structu res on 3- manifo lds: Ove rall logic, prep riTd. [T h6) W , T hurst on, Hyper bolic geo m etr y and 3-manifold s in "Low-di m ensio nal topolog y (Bango r, 19'T 9) , " London Math. S oc. Lect we Serie s 48 ,9-25, C ambr idge Univ. P ress ( 1982) . [Th(7) W , Thurston, The g eometry a nd topology of 3-mani folds, " The collected wor ks of Willi am P. Thurston" , AMS. [W] F , W aldhaus en, On ir red ucible 3- m anifol ds which are sufficiently large, An 1'1llLs of M ath. 8 7 ( 1968), 56-88

[J)

WI LLIAM p , T HURSTON

BULLETIN (New Series) OF THE AMERICAN MA THEMA TICAL SOCIETY Volume 6, Number 3, J\.fuy 1982

THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY BY WILLIAM P. THURSTON

I. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformal structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than I) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or morel-except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the 1.1. CONJECTURE. The interior of every compact 3-manifold has a canonical decomposition into pieces which have geometric structures.

In §2, I will describe some theorems which support the conjecture, but first some explanation of its meaning is in order. For the purpose of conservation of words, we will henceforth discuss only oriented 3-manifolds. The general case is quite similar to the orientable case. 1. The decomposition referred to really has two stages. The first stage is the prime decomposition, obtained by repeatedly cutting a 3-manifold M3 along 2-spheres embedded in M3 so that they separate the manifold into two parts neither of which is a 3-ball, and then gluing 3-balls to the resulting boundary components, thus obtaining closed 3-rrtanifolds which are "simpler". Kneser [Kn] proved that this process terminates after a finite number of steps. The resulting pieces, called the prime summands of M 3 , are uniquely determined by M3 up to homeomorphism; cf. Milnor, [Mill]. The second stage of the decomposition involves cutting along tori. This was discovered much more recently, by Iohannson [Joh] and Iaco and Shalen [Ja, Sh], even though the elementary theory of the torus decomposition does not Presented to the Symposium on the Mathematical Heritage of Henri Poincare, April 7-10, 1980; received by the editors July 20, 1981. 1980 Mathematics Subject Classification. Primary 57M99, 30F40, 57S30; Secondary 57M25,

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require any deep techniques. In the torus decomposition, one cuts along certain tori embedded non trivially in M', thus obtaining a 3-manifold whose boundary consists of tori. There is no canonical procedure to close off the boundary components, so they are left alone. The interested reader can learn the details elsewhere.

2. One way to think of a geometric structure on a manifold M is that it is given by a complete, locally homogeneous Riemannian metric. It is better, however, to define a geometric structure to be a space modelled on a homogeneous space (X, G), where X is a manifold and G is a group of diffeomorphisms of X such that the stabilizer of any point x E X is a compact subgroup of G. For example, X might be Euclidean space and G the group of Euclidean isometries. M is equipped with a family of "coordinate maps" into X which differ only up to elements of G. We make the assumption that M is complete. If X is simply-connected, this condition says that M must be of the form x/r, where r is a discrete subgroup of G without fixed points. There are precisely eight homogeneous spaces (X, G) which are needed for geometric structures on 3-manifolds. These eight homogeneous spaces are determined by the following conditions: (a) The space X is simply-connected. A manifold modelled on a non-simplyconnected homogeneous space is also modelled on its universal cover. (b) The group G is unimodular, that is, there is a measure on G invariant by multiplication on the right or the left. Otherwise, X would possess a vector field invariant by G which expands volume, so there could be no (X, G)manifolds which are compact or which even have finite volume. (c) G is a maximal group of homeomorphisms of X with compact stabilizers. If G were contained in a larger group G', then any (X, G)-manifold would be an (X, G')-manifold, so (X, G) would be redundant. We will describe these eight geometries in §4. For the moment, it will suffice to say that of these eight, hyperbolic geometry is by far the most interesting, the most complex, and the most useful. The other seven come into play only in exceptional cases. Conjecture 1.1 subsumes the Poincare conjecture, which was posed by Poincare not as a conjecture but as a question: Is every 3-manifold with trivial fundamental group homeomorphic to the 3-sphere? Poincare raised the question in [Poio], but he did not pursue it, for as he said, "cette question nous entrainerait trop loin". Conjecture 1.1, just as the Poincare conjecture, is likely not to be resolved quickly, but I hope it will be a more productive guide to research on 3-manifolds than Poincare's question has proven to be. My hope is based on the fact that it applies to a1l3-manifolds, so there are many examples which arise. To find a geometric structure for a particular manifold is a great help in understanding that manifold. 2. Supporting evidence. As I hinted earlier, Conjecture 1.1 can be proven in many cases. For instance, I will state a simple necessary and sufficient topological condition for the interior of a 3-manifold with nonempty boundary to have a hyperbolic structure. This theorem will imply Conjecture 1.1 for prime manifolds with nonempty boundary. There is a similar theorem for

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closed manifolds which satisfy a technical hypothesis (which is not necessary but frequently true). A surface N' embedded in a 3-manifold M ' is two-sided if N' cuts a regular neighborhood of N' into two pieces, i.e., the normal bundle to N' is oriented. Since we are assuming that M 3 is oriented, this is equivalent to the condition that N 2 is oriented. A two-sid ed surface is in compressible if every simple curve on N ' which bounds a disk in M' with interior disjoint from N ' also bounds a disk on N2. A 3-manifold M 3 is geometrically atoroidal if every incompressible torus embedded in M ' is isotopic to a boundary component. A 3-manifold M ' is homotopically atoroidal if every map of T' to M ' which acts injectively on the fundamental group is homotopic to aM. Homotopically atoroidal implies geometrically atoroidal; the converse is no t quite true. The condition that M is homotopically atoroidal can be rephrased in terms of the fundamental group. Assuming that aM is incompressible, the condition that M is homotopically atoroidal is equivalent to the condition that, first, "I(M) is not expressible as a free product, (this is guaranteed if M is prime), and second, that any subgroup of "I(M) isomorphic to Z 2 is conjugate to a subgro up of the fundamental gro up of some torus boundary component. Here are two simple examples of homotopically atoroidal manifolds. They are nonrepresentative because their interiors have Euclidean structures. 2.1. EXAMPLE. M ' = T ' X I. The interior of M ' is homeomorphic to E ' /r, where E3 is Euclidean 3-space and r is generated by (x, y, z) ~ ( x

+

I , y, z)

and

(x , y , z) ~ (x, y

+

I, z ).

2.2. EXAMPLE. M ' = T ' X J/(Z/2), where the Z/ 2 action flips I and acts as a covering transformation of T2 over th e Klein bottle. Its interior is the quotient of E' by the group r generated by ( x,y ,z) ~ {x + I ,y, z)

and

( x , y,z )~(-x,y + l,-z).

2.3. THEOREM [Th 2]. The interior of a compact 3-manifold M ' with nonempty boundary has a hyperbolic structure iff M ' is prime, homotopically atoroidal and not homeomorphic to Example 2.2. A hyperbolic structure on the interior of a compact manifold M3 has finite volume iff aM' consists of tori, with the single exception of Example 2.1, which has no hyperbolic structure of finite volume.

2.4. COROLLAR Y. Conj ecture 1.1 is true f or all compact, prime 3-manifolds witli nonempty boundary.

A good class of examples arises from knots in S3. (A kn ot is an embedding of Sl in S 3.) The complement of a knot is homeomorphic to the interior of a manifold whose boundary is a torus. A torus knot is a knot which can be placed on an ordinary torus in S 3.

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I. A torus knot of type (3,8). It can be placed on a torus so that it winds 3 times around the short way while going 8 times around the long way.

FIGURE

With any nontrivial knot K there is associated a whole collection of other knots, known as satellites of K; these are knots which are obtained by a nontrivial embedding of a circle in a small solid torus neighborhood of K. Here, "nontrivial" means that the embedding is not isotopic to K itself and is not contained within a ball inside the solid torus. A knot is a satellite knot if it is a satellite of a nontrivial knot.

FIGURE

2. A knot and a satellite of it.

2.5. COROLLARY. If K C S3 is a knot, S3 - K has a geometric structure iff K is not a satellite knot. It has a hyperbolic structure iff, in addition, K is not a torus knot. Indeed, the complement of a knot is always prime, and the torus decomposition is nontrivial exactly when K is a satellite. Corollary 2.5 was first conjectured by R. Riley [Ri 1] based on his construction of a number of beautiful examples, with the aid of the computer. His work gave a big impetus to me to prove Theorem 2.3. In order to give the statement for closed manifolds, we need some more terminology.

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A 3-manifold M3 is called a Haken manifold if it is prime and it contains a 2-sided incompressible surface (whose boundary, if any, is on aM) which is not a 2-sphere. A prime 3-manifold whose boundary is not empty is always Haken (with the trivial exception of the 3-ball, which is often considered to be Haken anyway). Any prime 3-manifold whose first homology has positive rank is Haken. 2.5. THEOREM [Th 2]. Conjecture 1.1 is true for Haken manifolds. A closed Haken manifold has a hyperbolic structure iff it is homotopically atoroidal. It is hard to say how general the class of Haken manifolds is. There are many closed manifolds which are Haken and many which are not. Haken manifolds can be analyzed by inductive processes, because as Haken proved [Hak], a Haken manifold can be cut successively along incompressible surfaces until one is left with a collection of 3-balls. The condition that a 3-manifold has an incompressible surface is useful in proving that it has a hyperbolic structure (when it does), but intuitively it really seems to have little to do with the question of existence of a hyperbolic structure. A link L in a 3-manifold M3 is a I-dimensional compact submanifold. A 3-manifold N 3 is said to be obtained from M3 by Dehn surgery along L if N 3 is obtained by removing a regular neighborhood of L, and gluing it back in by some new identification. The new identification is determined by choosing a diffeomorphism of the torus for each component of L. Two choices of diffeomorphisms et> and", give rise to diffeomorphic manifolds if ",-let> extends to a diffeomorphism of a solid torus (the regular neighborhood of the component of L).

~ Pluggi!! by I/J ~ Plugging by t/J

l f

*=>

t/J-l I/J extends to} { the solid torus 0

a\

-

~~2A I

?

I

, ,-,,/ a ----:-----1_

SOLI D TORUS

t/J"'o I/J

3. Modifying a 3-manifold by Dehn surgery. Plugging in a solid torus by cp gives a result diffeomorphic to plugging by '" iff the diffeomorphism of the torus '" -\ 0 cp extends to a diffeomorphism of the solid torus.

FIGURE

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There remains, for each component of L, a countably infinite set of essentially distinct choices. 2.6. THEOREM [Th 1]. Suppose L C M3 is a link such that M - L has a hyperbolic structure. Then most manifolds obtained from M by Dehn surgery along L have hyperbolic structures. In fact, if we exclude, for each component of L, a finite set of choices of identification maps (up to the appropriate equivalence relation as mentioned above), all the remaining Dehn surgeries yield hyperbolic manifolds. It has been proven that for many knots or links, most Dehn surgeries give rise to non-Haken manifolds (see Thurston [Th 1], Hatcher and Thurston [H, T], Culler, Jaco and Rubinstein [C, J, R], Floyd and Hatcher [F, H], Hatcher [HatD, so Theorem 2.6 yields many hyperbolic manifolds not covered by Theorem 2.5. Every closed 3-manifold is obtained from the three-sphere S3 by Dehn surgery along some link whose complement is hyperbolic, so in some sense Theorem 2.6 says that most 3-manifolds are hyperbolic. The most promising approach to Conjecture 1.1 seems to be to try to eliminate the vagueness from Theorem 2.6, and analyze exactly what happens under all Dehn surgeries. This can be studied by analytic continuation through families of geometric structures on M - L which become singular at L. For certain examples (see [Th 1]) this has been successfully accomplished, so that here we have geometric structures for all manifolds obtained by Dehn surgeries along L. It is quite feasible to use computers to study Conjecture 1.1 for classes of manifolds obtained by Dehn surgery. In fact, I have undertaken such a project in order to gain more feeling for the evolution of geometric structures on 3-manifolds. The examples I studied were arbitrary torus bundles over the circle, with Dehn surgery performed along a section of this bundle. Troels Jorgensen was the first to prove that the complement of such a section has a hyperbolic structure, in most cases. I wrote a computer program which found hyperbolic structures for particular Dehn surgeries on particular torus bundles over the circle. With the aid of the program, I found the pattern of which Dehn surgeries give rise to hyperbolic manifolds, and I could then show by hand that those examples for which the program was not producing hyperbolic structures in fact had other geometric structures. For sufficiently complicated torus bundles, the empirical observation is that all but the trivial Dehn surgery gives a hyperbolic manifold. The geometric structures turn out to be very beautiful when you learn to see them. Often, the information which determines a geometric structure can be expressed in terms of some construction in plane Euclidean geometry. For instance, the output from my computer program which performs Dehn surgery on torus bundles over the circle is a tesselation of the plane minus the origin by triangles. The combinatorial pattern is predetermined, together with rules that certain triangles are similar. If such a pattern exists, then the manifold in question has a hyperbolic structure. l lA dded in proof. I can now prove that 1.1 is true for all prime 3-manifolds with a symmetry having fixed point set of dimension ;;0. 1. This includes the examples above.

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FIGURE 4.

363

Three o'clock sky .

3. Applications. Theorems 2.3 and 2.5, or better yet, the conviction that Conjecture 1.1 is true, have many applications to the understanding of 3manifolds. First, we should point out the well-known Mostow rigidity theorem, which asserts that hyperbolic structure on closed manifolds, or more generally, any hyperbolic structure of finite volume, is canonical, provided the dimension is at least 3. (The original theorem, which does not include the case of noncom pact manifolds, was proven by Mostow [Mos]. The proof for the noncompact case was completed by Prasad [Pra].) 3.1. MOSTOW RIGIDITY THEOREM. Suppose M and M2 are hyperbolic manifolds of finite volume for which there is an isomorphism J

:

(M J )

'1T 1

--> '1T J

(M2 ) ·

Then there is an isometry F: M J --> M2 which induces the isomorphism (up to conjugacy) between the fundamental groups.

The homotopy type of a hyperbolic manifold is determined by its fundamental group. Theorem 3.1 says that any invariant of the geometry of a hyperbolic manifold is actually an invariant of its homotopy type. This gives a powerful tool for distinguishing 3-manifolds. 3.2. COROLLARY. Let M be a homotopically atoroidal Haken manifold. Then there are only a finite number of isotopy classes of homeomorphisms : M .... M. The group of isotopy classes of self-homeomorphisms of M lifts to a group of actual homeomorphisms. OUTLINE OF PROOF. The group of isometries of any Riemannian manifold of finite volume is compact. An isometry of a hyperbolic manifold of finite

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volume which is near the identity actually equals the identity, so the group of isometries is finite. An isometry homotopic to the identity is equal to the identity, so the group of isometries is isomorphic to the group of automorphisms of the fundamental group. Waldhausen (Wa IJ showed that a homeomorphism inducing the identity on '1T1 (up to conjugacy) is isotopic to the identity, so the group of isometries is also isomorphic to the group of isotopy classes of homeomorphisms. REMARK. When M is not geometrically atoroidal, there are selfhomeomorphisms of M supported in a neighborhood of a torus which have infinite order up to isotopy. A group G is called residually finite if for each g E G there is some finite quotient map f: G ~ F such that f(g) "" I. If G = "I(M), an equivalent condition is that every compact subset of the universal covering space of M maps homeomorphically to some finite sheeted covering space. 3.3. THEOREM. The fundamental group of a Haken manifold, or of any manifold for which Conjecture 1.1 holds, is residually finite. It is a standard fact that a finitely generated subgroup of GL"(C) is residually finite. Using this, one easily sees that the fundamental group of any geometric 3-manifold is residually finite. After a certain amount of fussing, one can assemble finite quotients of the fundamental groups of pieces of a geometric decomposition of a 3-manifold to obtain finite quotients of the fundamental group of the entire manifold. One difficulty in the study of 3-manifolds has been the lack of any good invariants, such as the Euler characteristic for 2-manifolds. (The Euler characteristic of a closed 3-manifold is always 0.) Just as the area of a complete hyperbolic 2-manifold is a topological invariant (proportional to the Euler characteristic), so the volume of a hyperbolic 3-manifold is a topological invariant, which in some sense is a single measure of complexity of the manifold. One indication of this is the following theorem, whose basic core is due to Gromov, and which was sharpened in Thurston (Th IJ to give parts (b) and (c).

3.4. THEOREM. (a) Suppose f: M ~ N is a map of nonzero degree between closed hyperbolic 3-manifolds. Then volume(M);"1 degree(f) 1volume(N). (b) If equality holds in (a), then f is homotopic to a covering map which is a local isometry. (c) Suppose M is a hyperbolic manifold of finite volume, and N is a hyperbolic manifold containing a (nonempty) link L such that M '" N - L. Then volume(N) < volume(M).

The situation in part (c) is not unusual. 3.5. THEOREM (J0RGENSEN). For any constant C, let Xc denote the set of hyperbolic 3-manifolds of volume .;; C. Then there is a finite subset GJlC c Xc such that any element N E Xc contains a link LeN whose components consist of short geodesics such that N - L is homeomorphic to some element ME GJlC.

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Intuitively, there is a finite set of great-grandmothers whose offspring are all the hyperbolic manifolds of volume,;;; C. If the link L has a short total length, one can show that the volumes of M and N are close. With this information, Theorems 3.4 and 3.5 imply 3.6. THEOREM. The set of volumes of hyperbolic manifolds is a well-ordered subset ofR. The set of manifolds with any given volume is finite.

Using the Existence Theorem, 2.6, for hyperbolic Dehn surgeries, one deduces 3.7. THEOREM (SEE [TIt 1]). The order type of the set of all volumes of hyperbolic manifolds is wW.

Note. The notation wW is that of ordinals, not cardinals: wW is the countable ordinal which describes the order type of polynomials (of finite degree) in the symbol w with natural number coefficients, ordered by the limiting order of the values when higher and higher integers are substituted for w. In other words, there is some lowest volume VI; some second lowest volume v 2 ; a third lowest volume v 3 , and so on until the first accumulation point of volumes vW ' which is the smallest volume of a hyperbolic manifold with one cusp. There is a next highest volume vw + I' and so on until the second accumulation point v 2w ' The first accumulation point of accumulation points is called V w 2, and it is the smallest volume of a hyperbolic manifold with two cusps. And so on. The structure of the set of volumes is significant, but it seems impractical to actually compute the real numbers v,,, for a a typical ordinal, and probably silly to try for very many ordinals. It would be interesting to know a few simple cases, like VI and vW ' however. Based on a not at all exhaustive examination I made of a few examples and some computer computations of likely classes of examples by Bob Meyerhoff, it seems plausible that Vw could be the volume of the complement of the figure eight knot, 2.0 ... ,

Is Vw = 2.029883 the volume of S3 - ~ FIGURE

5.

The best candidate for VI so far is the volume of a manifold obtained by Dehn surgery along the figure eight knot; its volume is about .98. There is another promising invariant of oriented hyperbolic 3-manifolds, about which practically nothing has been known until recently. This is the eta invariant, or the Chern-Simons invariant which is obtained by reducing the eta invariant mod a constant. The eta invariant should probably be thought of as the imaginary part of a complex constant, of which the volume is the real part. The eta invariant changes sign under reversal of orientation, while the volume is fixed.

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Unfortunately, the eta invariant, unlike the volume, has been very difficult to compute, basically because in its definition some very arbitrary choices are involved, and these choices are difficult to make in any reasonable way. Recently, however, Bob Meyerhoff [Mey] has developed some good formulas for the Chern-Simons invariant. Among other things, he has shown that the set of values of the Chern-Simons invariant is dense in the circle-in fact, the manifolds obtained by Dehn surgery on any link with hyperbolic complement have Chern-Simons invariants whose values are dense. The solution of the Smith conjecture is a development in which several mathematicians were fairly directly involved and in which Theorem 2.3 played an essential part, along with the equivariant loop theorem of Meeks and Yau [M,Y]. 3.4. THEOREM. Suppose : S3 - S3 is an orientation-preserving diffeonwrphism such that " = I and for some x, x = x. Then the fixed point set of is an unknotted circle, and 4> is conjugate to an isometry. 2

For an explanation of the various parts of the proof and the somewhat complex way they fit together, see the proceedings of the Smith conjecture conference [Smi]. Conjecture 1.1 leads to a very concrete method of analyzing 3-manifolds. Eventually, it should be practical to begin with any of the usual descriptions of a 3-manifold M (e.g., a description of Musing Dehn surgery along a link) and by a routine procedure on the computer, calculate its geometric decomposition (assuming that it has one). The first case to implement should be the case that M is hyperbolic. Robert Riley has, in fact, found hyperbolic structures for a variety of knot complements by computer, but his calculations are not routine except in special cases. Riley's work makes it clear that there is a rigorous, but not generally practical, algorithm for computing hyperbolic structures. The first step in this algorithm is to calculate representations of ,,\(M) in PSLz{C), which is the group of isometries of H3 One' wants a discrete, faithful representation; such a representation is unique up to conjugacy if it exists. Conjugacy classes of representations correspond to solutions of a system of polynomials in a number of variables; one need check only isolated solutions, of which there are only a finite number. An algorithm certainly exists for finding them, but unfortunately, it is not practical for a complicated 3-manifold since the number of variables becomes large and the degree of the polynomials grows exponentially large with the complexity of the 3-manifold. For many less complicated 3-manifolds, however, the set of all representations can be computed practically. The second step is the part which at first sounds dubious. However, it works theoretically and Riley has written a computer program which handles it quite nicely and practically. This step is to check whether a given representation is discrete and faithful. Riley's method (which originates from Poincare) is to 2Added in proof. The result alluded to in the previous footnote generalizes this and gives a new, purely geometric proof. It implies for instance, that a diffeomorphism of finite order of a hyperbolic manifold has a fixed point set isotopic to a finite union of geodesics,

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find a fundamental domain for the group action, or else find enough group elements close to the identity to imply the group is not discrete. If a fundamental domain is obtained, that implies that the image of the representation is discrete; if all the relations which come from the combinatorics of the fundamental domain are implied by the relations in the original group, the representation is faithful.

FIGURE 6. A fundamental domain for a discrete group of hyperbolic motions with compact quotient, generated by R. Riley's computer program. This is a projectively correct picture, using the Klein model for hyperbolic space.

To make the calculation of hyperbolic structures routine, what remains to be done is to find a practical general method of computing a likely candidate for the right representation of '1T](M) in PSL 2C. An approach using analytic continuation, depending on the theory of hyperbolic Dehn surgery, seems promising, but we will know whether it is successful only after it has been implemented on the computer. There is an alternate approach to the second step, provided for the first step one calculates all representations of '1T](M) in PSL 2 C and provided also that one knows ahead of time that M has a hyperbolic structure. There is a characteristic number associated to a representation, fairly readily computable, which equals the volume of the quotient space when the representation is discrete and faithful. This characteristic number takes its unique maximal value for a discrete, faithful representation. Successful completion of the two steps outlined gives a hyperbolic manifold whose fundamental group is '1T](M). Conjecture 1.1, or simply the validity of Conjecture 1.1 for M, would imply that the hyperbolic manifold obtained is homeomorphic to M. If one doubts Conjecture 1.1, then, in general, a third step remains: to check whether M is homeomorphic to the hyperbolic manifold. The method of hyperbolic Dehn surgery often guarantees this. There are

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also very tedious algorithms which will eventually construct a homeomorphism provided it exists, but unfortunately there is no known algorithm which will show that the two manifolds are not homeomorphic. There are indications that computations of other kinds of geometric structures may come as a byproduct of a method of computing hyperbolic structures: often there is a way to interpret these structures as degenerate hyperbolic structures.

4. Eight 3-dimensional geometries. As we promised in §l, we will now describe briefly the eight 3-dimensional homogeneous spaces, or geometries, which are needed to give geometric structures for 3-manifolds. We will in each case give a description of a simply-connected 3-manifold X, and a group G of orientation-preserving diffeomorphisms of X with compact stabilizers Gx . We will group them by the identity component of Gx : it is isomorphic either to SO(3), to SO(2), or to the trivial group SO(l). For the first three geometries, Gx is SO(3). 1. SPHERICAL GEOMETRY. Xis the 3-sphere S3 and G is SO(4). All 3-dimensional spherical manifolds have been classified. Cf. Seifert [Sei], for a complete enumeration. The three-sphere S3 has the structure of a group: the group of unit .

(g.h)

quatermons. Therefore S3 X S3 acts on S3 by the formula x ---,) gxh- l • The kernel of this action is Z /2, generated by (-I, I) in quaternionic notation, and this produces an incredible isomorphism

S3 X S3/Z/2 = SO(4). It is with the aid of this isomorphism that the structure of 3-dimensional spherical manifolds can be analyzed. The most beautiful is the Poincare dodecahedral space, obtained from a dodecahedron by identifying opposite faces with a 1/10 right-handed rotation. This identification pattern forces the edges to be identified in triples. A regular dodecahedron in Euclidean space has dihedral angles somewhat less than 120°. Regular convex polyhedra in S3 have angles anywhere between the Euclidean value and 180°. To put a geometric structure on the Poincare dodecahedral space, find the regular spherical dodecahedron with dihedral angles of 120°, and identify them in the given pattern. Note that the universal cover of the Poincare dodecahedral space is S3, with a tiling by 120 dodecahedra. (Poincare first discussed this example in (Poin], in a very nongeometric form.) It is a strange fact that all spherical manifolds also have a stronger structure, based on X = S3 and G = S3 X SI/Z/2. 2. EUCLIDEAN GEOMETRY. X = E 3 , G = R3 X SO(3), the group of Euclidean isometries. There are only 10 nonhomeomorphic 3-dimensional Euclidean manifolds. Each one is finitely covered by the 3-torus. 3. HYPERBOLIC GEOMETRY (ef. the article by Milnor in this volume). One good picture of hyperbolic space is the Poincare upper half space, X {x, y, z Iz > 0). The group G is the group PGL2 (C) of nonsingular 2 X 2 complex matrices of determinant 1, up to multiplication by ±J scalars. The action of G on X may be defined as fractional linear transformation using quaternionic

=

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notation, q = x

+ yi + zj. The formula is q

[~tl( -->

aq

+ b) ( cq + d) -1 .

One easily-described example of a hyperbolic manifold is the Seifert-Weber dodecahedral space, obtained by identifying opposite faces of a dodecahedron by 3/10 right-handed rotations. One checks that edges of the dodecahedron are identified in quintuples. To form a geometric model, use the regular hyperbolic dodecahedron whose dihedral angles are 72°. For the next four cases, the identity component of Gx is SO(2). 4. X = S2 X El = G consists of (spherical isometries) X (isometries of the Euclidean line). Note that an isometry which reverses the orientation of both factor preserves the total orientation. There are only two nonhomeomorphic examples of compact manifolds with this geometry; can the reader find them? 5. X = H2 X El, G consists of (isometries of H2) X (isometries of El). Every manifold modelled on this geometry is finitely covered by the product of a surface and a circle. Can the reader find an example which is not itself a product? 6. The space X is the universal covering space set of unit length vectors in the hyperbolic plane, T 1(H 2 ), and the group Gis R X (the universal covering of the group of isometries of H2). The isometries of H2 act by their derivatives, and R acts as simultaneous rotations of all vectors, keeping their based points fixed. Note that even orientation-reversing isometries of H2 preserve the orientation of T 1(H 2 ). The space of unit tangent vectors to any hyperbolic surface is an example of a manifold with this geometry. Another example is the quotient of Tl(H2) by, say, the group of orientation-preserving automorphisms of the regular (2, 3, 7) tiling.

FIGURE 7. A tiling of the hyperbolic plane by congruent triangles with angles

7T /2, 7T /3, 7T

/7.

Upper half-space projection.

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7. X is a twisted product of E I with E 2, and G is an extension of the group of isometries E 2 by R, R -. G -. isometries of E 2 •

To see a picture of X, consider R3 with a certain 2-plane field 'T. Along the z-axis, T is horizontal (orthogonal to the axis). Along any ray emanating from the z-axis and orthogonal to it, 'T always contains the tangent to the ray, and its slope in the direction orthogonal to the ray increases linearly, so this slope always equals one half the distance from the z-axis. In coordinates, if e I' e 2 and e 3 are the three coordinate vector fields in R3 , 'T is spanned by el

-

(1/2)y e 3'

e2

+ (l/2)x e 3 •

Any isometry of the x-y plane lifts to an affine map of R3 which preserves 'T and also preserves distances along lines parallel to the z-axis. In fact, there is a whole I-parameter family of such lifts. (These affine maps may be thought of as obtained by an isometry of E 3 , followed by some shearing map to adjust T.) These affine transformations of R3 constitute G. Note that an orientationreversing isometry of E2 lifts only to maps which reverse the z-direction. Any oriented circle bundle over a 2-torus which is not the 3-torus has this kind of geometric structure. The simplest circle bundle over the 2-torus is the one with Euler class 1, and it is obtained as the quotient of X by the discrete, nilpotent subgroup r of G generated by

(x, y, z) -. (x + 1, y, z + y /2) and (x, y, z) -. (x, y + 1, z - x /2). Notice that the commutator of these two generators is the vertical translation

(x, y, z)

->

(x, y, z + 1).

Every other example of a manifold with this geometry is finitely covered by a manifold homeomorphic to x/r. An alternate description of this geometry is to define X as the Heisenberg group,

and G as the semidirect product of H by Sl acting as a group of automorphisms of H which rotate the x-y plane. In this form, the group r can be taken as the subgroup of H where x, y and z are integers. 8. Finally, there is one example for which the identity component of Gx is trivial. In this case, X is very naturally a Lie group, the solvable Lie group R2

->

X

->

R

where R acts on R2 (by conjugation) with the formula (x, y) -. (e1x, e-1y).

The group G is an extension of X by (Z2? acting as a group of automorphisms, whose 3 nontrivial elements are the 180 0 rotations

(x, y, t)

22

->

(-x, -y, t), (x, y, t)

->

(y, x, -t) and

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Any torus bundle over SI whose monodromy is a linear map with distinct real eigenvalues has a geometric structure of this form. S. The varying geometry of K1einian groups. The Mostow rigidity theorem applies only to hyperbolic 3-manifolds with finite volume. which are interiors of compact 3-manifolds whose boundary consists of tori. Hyperbolic structures for the interior of a manifold with more complicated boundary are not rigid. and there is a wonderful deformation theory for them. The proofs by induction of Theorems 2.3 and 2.5 involve a study of the varying geometry of such manifolds. The theory of this varying geometry is tricky and rich. and a number of intriguing and significant questions remain. Here we will outline a few of the considerations. In §4 we mentioned the Poincare upper half-space picture for H3 If one adjoins a single point at 00 in R3 , then the action of the group PGL 2 (C) extends to the closure of upper half-space, which is a ball. The boundary of the ball is the Riemann sphere

c = c U {oo}

= Cpl,

and the action of PGL 2 (C) is the usual action as Moebius transformations or complex projective transformations [~~laz+b

Z-,locz+d· A Kleinian group f is a discrete subgroup of PGL 2(C), that is, a subgroup for which each element has a neighborhood in the group containing only that point. If f has no elements of finite order, then it acts freely on H 3 , and H 3 /f is a complete hyperbolic manifold. Henceforth we will assume this to be the case. Each orbit of f acting on C has accumulation points (provided f is not finite). The set of all accumulation points of orbits is called the limit set Lr of f, while its complement is called the domain of discontinuity, Dr. The Kleinian group f acts nicely on Dr (properly discontinuously) so that its quotient Dr/f is a surface, which inherits a conformal structure (= complex structure) from Dr. In fact, Dr/f is the boundary of the three-manifold (H' U Dr)/f. Sometimes this "Kleinian" 3-manifold is compact, and sometimes it isn't. The basic deformation theorem, developed by Ahlfors, Bers, Mostow, Sulli-van and others, says that a certain class of relatively mild deformations of f, the quasi-conformal deformations of f, are controlled precisely by conformal structures on Dr/f. Except in degenerate cases (when f is abelian) the conformal structure on Dr/f is represented by a unique hyperbolic metric, by the classical uniformization theorem.

5.1. AHLFORS FINITE AREA THEOREM. The total hyperbolic area of Dr/f is finite. Thus one has a correspondence between the space of quasiconformal deformations of f and the space of finite-area hyperbolic metrics on Dr/f, which is called the Teichmiiller space of Dr/f.

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FIGURE

24

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8. Six limit curves. The last curve is omnipresent.

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Here are some computer sketches of examples in which the limit set is a curve. Poincare was taken by the nondifferentiable nature of the curves which even have Hausdorff dimension> 1, as described in Sullivan's article [Sui I]. The first limit set is a circle. This means the group leaves invariant a hyperbolic plane inside H3; in Poincare upper half-space, hyperbolic planes appear as Euclidean hemispheres resting on C. The other groups are obtained by bending the original group. All the curves except the last are Jordan curves; the last curve actually fills C, but to make the computer actually fill C would require an amount of computer money (and foolishness) approximating the "defense" budget. A group whose limit set is a circle is called the Fuchsian group (because of Poincare's modesty), and groups whose limit sets are Jordan curves are called quasi-Fuchsian groups; they are obtained by quasi-conformal deformations of Fuchsian groups. The final group in our sequence of pictures cannot be obtained by a quasi-conformal deformation of a Fuchsian group, since the topology of the limit set has changed. How can we explain limiting phenomena such as this? First we need to know something about the limiting geometry of hyperbolic surfaces, as the hyperbolic structure goes to infinity in Teichmtiller space. One way that hyperbolic structures can go to infinity is that a curve, or system of curves, can be pinched:

FIGURE 9. A surface with a pinched waist.

The hyperbolic metric develops a long, skinny waist to accomplish this. To describe more generally how surfaces can go to infinity in Teichmuller space one needs a generalization of the notion of a simple closed curve. A geodesic lamination A of a hyperbolic surface S is a closed subset of S which is a disjoint union of complete geodesics on S (called the leaves of A). A simple closed geodesic is one example of a geodesic lamination. To get other examples, consider a sequence of longer and longer simple closed geodesics. There is always a subsequence so that the pictures "converge", usually to an uncountable set of geodesics. A typical local cross-section of a geodesic lamination is a Cantor set, but other behavior can also occur. The 2-dimensional Lebesgue measure of a geodesic lamination is always O. A transverse invariant measure, fl, for a geodesic lamination can be thought of as a rule which assigns to each transverse arc IX to A a measure that is supported on A n IX and invariant under maps from one arc IX to another arc f3 which take each point of intersection of IX with a leaf of A to a point of intersection of f3 with the same leaf. One can think of fl as a weighted counting of the leaves of A, or as a measure of the amount of exertion required to cross a certain set of leaves. A lamination equipped with a transverse invariant measure of full support is a measured lamination. We also include here the trivial lamination consisting of no leaves and having 0 measure.

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There is a good topology for the set of measured laminations, making it a Hausdorff space. In this topology, a bunch of leaves can continuously get smaller and smaller measure until they vanish. Let 6JlLe(S) denote the space of measured laminations on S, and 6JlLeo(S) denote the space of compactly supported measured laminations (in the case S is noncompact). Despite the apparently erratic behavior of laminations, we have 5.2. THEOREM [111 1]. The space 6JlLeo(S) is homeomorphic to Euclidean space of the same dimension as the TeichmiUler space for S.

There are also projectivized versions of the lamination spaces. That is, any transverse invariant measure may be multiplied by a positive constant to give a new transverse measure. We define the projective lamination spaces

Fe(S) = (6JlLe(s) - OJ/multiplication by scalars and

Feo(S) = (6JlLeo(s) - OJ/multiplication by scalars. 5.3. THEOREM. There is a natural topology on 5(S) U Feo(S) which makes it a ball, where 5(S) is the Teichmuller space for S.

Intuitively, the interpretation is that a sequence of hyperbolic structures on S can go ,to infinity by "pinching" a certain geodesic lamination A; then it converges to A. As a lamination is pinched toward 0, lengths of paths crossing it are forced toward infinity. The ratios of these lengths determine the transverse invariant measure. A good exposition of this theory may be found in the book by Fathi, Laudenbach, Poenaru et al [F, 1" P], although this work deals with the closely related theory of measured foliations. Geodesic laminations enter into the theory of hyperbolic 3-manifold in a number of guises, but we will describe only one result. Two compactly supported laminations Al and A2 fill up S if S - Al U A2 consists of chunks which, when lifted to S, are bounded if S is compact, or possibly contained in a horoball neighborhood of a cusp. 5.4. DOUBLE LIMIT THEOREM [111 3]. Let Al and A2 be a pair of projective laminations which fill up S. Then if (gl) - Al and (glJ are sequences in 5(S) tending toward Al and A2 , the sequence of quasi-Fuchsian groups whose two components of DrIf have conformal structures given by gl and g~ has a subsequence which converges algebraically to some Kleinian group f isomorphic to "I(S), This was proven first in the case S is a punctured torus by J0rgensen, [Jar]. This theorem implies the existence of many geometrical complicated Kleinian groups isomorphic to "I(S), since from the theory developed in [111 1] the laminations Al and A2 can be reconstructed from r although the information of the projective class of measures is lost. A geodesic lamination is arational if it intersects every simple closed geodesic. When Al and A2 are arational (which is the likely case), then the limit set of r is the entire 2-sphere.

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There is one construction for 3-manifolds which relates very directly to the theory of surfaces, and which gave me considerable difficulty for a time in the proof of Theorem 2.5. In fact, when I first heard the suggestion that Theorem 2.5 was true (by one mathematician giving a distorted quote of another), I immediately came up with this class of 3-manifolds as "obvious" counterexamples. This construction is the mapping torus construction, which produces a 3-manifold M. depending on a diffeomorphism : S .... S of a surface S. One simply forms the product S X I and identifies each point (x, I) to ((x), 0). It is easy to construct quite complicated diffeomorphisms q> by taking compositions of simple diffeomorphisms, called Dehn twists, which are the identity outside the neighborhood of a simple closed curve on S. The diffeomorphism of S gives rise to a natural transformation of 'J(S), which extends continuously to the ball 3(S) = 'J(S) u '8'eo(S). By the Brouwer fixed point theorem, has at least one fixed point in 3(S). 5.5 THEOREM. One of the following 3 alternatives holds: (a) There is a fixed point in 'J(S), and is isotopic to a diffeomorphism of finite order. (b) There is a finite system of disjoint simple curves invariant (up to isotopy) by .

(c) There are precisely two points in peo(S) fixed by . These are arational laminations which together fill up S.

The deduction in part (a) that if fixes a point in Teichmillier space, is isotopic to a diffeomorphism of finite order is the same as the deduction of Corollary 3.2 from the Mostow rigidity Theorem 3.1. Theorem 5.5 is part of the classification of conjugacy classes of diffeomorphisms of surfaces up to isotopy [Th 4] or [F, L, P], analogous to Jordan form. In case (b), one can cut the surface along the system of curves and analyze the diffeomorphism into diffeomorphisms of simpler surfaces. In case (c), there is a very nice, canonical representative of the isotopy class of q> (up to conjugacy), called a pseudo-Anosov homeomorphism. 5.6. THEOREM [Th 3]. The mapping torus M. has a hyperbolic structure if and only if satisfies condition (c) ( is isotopic to a pseudo-Anosov homeomorphism).

This was proven first for the case S is a punctured torus by Jorgensen, [Jar]. This is a special case of Theorem 2.5 or 2.3. An exposition of this can also be found in [Su 2]. The proof in [Th 3] is by applying the double limit Theorem 5.4 to the two laminations in peo(S) fixed by . One obtains an action of "\(S) on H 3 , which by an extension of the Mostow rigidity theorem from [Th 1] or [Su 3] can be shown to be conjugate to the action obtained by composing with the automorphism . The conjugating isometry, when adjoined to the isometries coming from ,,\(S), gives a discrete, faithful action of ,,\(M.). The quotient manifold is homeomorphic to M. by 3-manifold theory. This was first proven by Stallings [SIa].

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Consider now a copy of the surface 8 inside the 3-manifold M.p. The universal covering space 8 sits inside the universal covering space of M.p, which is H3. The closure of 8 must contain all of C, because as we have indicated the limit set of '1TI(8) is all of C. (This is also easy to deduce directly from the existence of a hyperbolic structure on M.p.) If we consider any hyperbolic metric on 8, this gives a diffeomorphism of 8 to H2. Does the map of H2 to H3 so defined extend continuously to a map of the 2-disk to the 3-ball? This question is more subtle than it first appears, but it was answered affirmatively by J. Cannon and me: 5.7. THEOREM [Can-Th]. The circle at infinity in the universal cover of a fiber of a mapping torus maps continuously to the sphere at infinity in H 3 , to give a sphere-filling curve.

The topology of the map of 8 1 to 8 2 can be described exactly, as follows. Let Al and A2 be the two geodesic laminations invariant by . Put two copies of 8 on a 2-sphere, one filling the northern hemisphere and one filling the southern hemisphere, with corresponding points on the circles at infinity glued together. Put a copy of Al lifted to 8 on the northern hemisphere, and A2lifted to 8 on the southern hemisphere.

10. A pattern of identifications of a circle, here represented as the equator, whose quotient space is topologically a sphere. This defines, topologically, a sphere-filling curve.

FIGURE

Now, form the identification space of 8 2 obtained by identifying the closure of each leaf of a lamination to a point and the closure of each component of the complement of the laminations to a point. It can be deduced readily from a theorem of R. L. Moore that the identification space is homeomorphic to the 2-sphere. Since each equivalence class meets the equator, the image of the equator is the entire 2-sphere. This is the topological model we promised. There are similar models for the possible behaviour of the boundary of 8 in all the cases covered by the double limit theorem. It is not hard to show that these are the correct topological models provided the maps of 8 1 are continuous, but continuity is not known in the general case.

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We have already given one picture of a space-filling curve, which, indeed, came from this construction. The surface in question was a punctured torus in the figure eight knot complement.

FIGURE II. The figure eight knot is spanned by a surface (topologically, a punctured torus) which can be swept around through all of S3-(the figure eight knot) and brought back to its starting position!

This knot complement is homeomorphic to the mapping torus of the diffeomorphism of the punctured torus determined by the linear map [? II of the torus. Here is another example. Note the definite sense of the spirals. This reflects the fact that, unlike in the case of the figure eight knot complement, M is not diffeomorphic to its mirror image. In this case, is the diffeomorphism of the punctured torus coming from the linear map [f i 1of the torus.

FIGURE 12. The sphere-filling curve determined by the punctured torus bundle over the circle

with monodromy

[it].

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The double limit theorem is a powerful result for analyzing surface groups, which are among the most flexible of Kleinian groups. There are several other

significant theorems which back up the double limit theorem by deriving information about the geometry of the quotient manifolds in the limits which the double limit theorem produces (see [Th 1] and [Th 2]). There are still many basic conjectures which have not been proven in general, however. Information

about the infinitely complicated manifolds obtained by the double limit theorem has relevance because it relates to the geometry of closed manifolds,

especially to the mapping tori M., but also to much more general manifolds. We will now give a sample of a theorem which complements the double limit theorem by analyzing a certain class of Kleinian groups which are more complicated and more rigid than surface groups.

5.S. THEOREM [Th 2]. Let r be a Kleinian group such that (a) the Kleinian manifold M = (H 3 U Dr)/r is compact, (b) the components of Dr are simply connected and (c) the closures of any two components of Dr are disjoint. Then the space of discrete, faithful representations of r in PSL 2 C, up to conjugacy, is compact.

Condition (b) is equivalent to the topological condition that aM is incompressible, and condition (c) is equivalent to the topological condition that Mis acylindrical: there are no essential cylinders in M. The space of conjugacy classes of discrete faithful representations of r is called the algebraic deformation space. For a precise definition of this and other topologies or sets of Kleinian groups, see [Th 2]. Since the quasi-conformal deformation space is noncompact, Theorem 5.8 produces many limits of sequences of quasi-conformal deformations of r, which are not themselves quasi-conformal deformations. Just as for the double

limits of surface groups produced by Theorem 5.4, the limit set of most of the limiting groups is the entire sphere C, and there is an "ending lamination" such that the associated components of associated to each component of Dr collapse in the limit.

aM

The limiting groups produced by Theorem 5.S, unlike the ones produced by 5.4, do not occur as subgroups of groups with compact quotients. Nonetheless, 5.8 is very important in the proof of Theorem 2.5 because it implies a good

deal about what can happen, before passing to the limit. In the inductive proof of Theorem 2.5, this enables one to show that manifolds such as M can be

deformed until they fit together to give closed hyperbolic manifolds. Kleinian groups are very beautiful and rich, with an amazing variety of

productive ways to think about them. (Compare the articles of Bers [Bers] and Sullivan [Sui 1] for two different points of view. These samples are not exhaustive.) We have seen a great deal of progress in understanding them, but we are still in the midst of a number of significant and intriguing problems.

Perhaps by the year 2000 our understanding of 3-manifolds and Kleinian groups will be solid, and the phenomena we now expect will be proven.

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6. Some open questions. Here are a few questions and projects concerning 3-manifolds and Kleinian groups which I find fascinating. 1. Do all 3-manifolds have decompositions into geometric pieces? 2. Is every finite group action on a 3-manifold equivalent to an action respecting the geometry? In particular, let : M ~ M be a diffeomorphism of finite order of a hyperbolic manifold. Is there an equivarian t isotopy of cp to an isometry? In particular, is the fixed point set of cp isotopic to a union of geodesics?3 3. Does every 3-dimensional orbifold which contains no bad 2-dimensional suborbifolds admit a geometric decomposition? (For terminology, see [Th 1]. This question contains 2.)3 4. Develop a global theory of hyperbolic Dehn surgery. Give specific general upper bounds for nonhyperbolic Dehn surgeries. Describe the limiting geometry which occurs when hyperbolic Dehn surgery breaks down. (See [Th 1]. From a number of examples, we know that the behavior in the limit can be very beautiful, but there is no general theory. This is a possible approach to solving I and 3.) 5. Are all Kleinian groups geometrically tame? (See [Th 1] for a definition, which should be extended appropriately to the general case.) 6. Is every Kleinian group a limit of geometrically finite groups? (In many cases, this implies 5.) 7. Develop a theory of Schottky groups and their limits analogous to the theory of quasi-fuchsian groups and their limits developed in [Th 1]. 8. Analyze limits of quasi-fuchsian groups with accidental parabolics. (6, 7 and 8 would combine to prove 5 in a very satisfying way.) 9. Is H3/r, where r is finitely generated, always homeomorphic to the interior of a compact manifold? (This was proven in many cases for geometrically tame groups in [Th 1].) 10. (AHLFORS MEASURE 0 PROBLEM). Does the limit set of a finitely-generated Kleinian group always have full measure or 0 measure, and in the former case does r act ergodically? (This was proven for many cases of geometrically tame groups in [Th 1].) II. Classify geometrically tame representations of a given group. Are they parametrized by their ending laminations and their parabolics, together with the conformal structure on the domain of discontinuity? 12. Describe the quasi-isometry type of an arbitrary Kleinian group. In other words, give a formula for a Riemannian manifold that has a diffeomorphism to H 3/r such that the metrics have a bounded ratio. (A good deal of information about the quasi-isometry types is already known, but it is not yet complete.) 13. If the limit set of a finitely-generated Kleinian group has Hausdorff dimension less than 2, is it geometrically finite? (This would probably be a consequence of 5.) 14. Suppose r has the property that (H 3U Dr)/r is compact. Then is it true that the limit set of any other Kleinian group f' isomorphic to f is the JAdded ill proof. This is now proven, provided, for (3), the complement of the singular locus is

irreducible.

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homeomorphic image of the limit set of

r,

by a homeomorphism taking the

fixed point of an element y to the fixed points of the corresponding element y'?

(Theorems 5.7 is a special case of this.) There are examples to show that there is no continuous map

Lr X {algebraic deformation space of r} .... S2 which parametrizes the limit sets. Perhaps, though, there is a parametrization which is continuous separately in the two factors.

15. Can finitely-generated subgroups of a finitely-generated Kleinian group be residually separated from the group? In other words, given a subgroup Her and y E r - H, is there a finite quotient of r in which the image of y is not in the image of H? Peter Scott proved this property for surface groups. It is useful for a number of topological arguments, even for special subgroups H.

16. Does every aspherical 3-manifold, or every hyperbolic 3-manifold, have a finite-sheeted cover which is Haken? This is related to (15). By applying Mostow's theorem, and (2.5), it is easy to see that a homotopically atoroidal manifold with a finite-sheeted cover which is Haken is homotopy equivalent to

a hyperbolic manifold. Unfortunately, there seems to be little prospect of finding such finite-sheeted coverings without first knowing the manifold is hyperbolic. 17. Does every aspherical 3-manifold have a finite-sheeted cover with positive first Betti number? This is stronger than 16.

18. Does every hyperbolic 3-manifold have a finite-sheeted cover which fibers over the circle? This dubious-sounding question seems to have a definite chance for a positive answer.

19. Find topological and geometric properties of quotient spaces of arithmetic subgroups of P SL 2 C. These manifolds often seem to have special beauty. 20. Develop a computer program to calculate the canonical form for a general diffeomorphism of a surface, and to calculate the action of the group of diffeomorphisms on peo(S). Use this to implement an algorithm for the word problem and the conjugacy problem in the group of isotopy classes of diffeomorphisms of a surface. 21. Develop a computer program to calculate hyperbolic structures on 3-manifolds. 22. Tabulate the volumes and the Chern-Simons invariants and other simple information for a bunch of 3-manifolds: for instance, the knots in the knot tables. Try to develop a practical feel for the well ordering. 23. Show that volumes of hyperbolic 3-manifolds are not all rationally related. Cf. [Mil 2]. 24. Show that most 3-manifolds with Heegard diagrams of a given genus have hyperbolic structures-in analogy to the hyperbolic Dehn surgery theorem. This would be the next step after 7. ACKNOWLEDGMENT.

I would like to thank George Francis for the illustra-

tions.

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BIBLIOGRAPHY

[BersJ L. Bers, Finite dimensional TeichmiUler spaces and generalizations, Bull. Amer. Math. Soc. (N.S.)5(1981),131-172. [Can, TItJ J. Canno~ and W. Thurston, Sphere-filling mrves and limit sets of Kleinian groups (to appear). [C, J, SJ M. Culler, w. Jaco and H. Rubinstein, Incompressible surfaces in once-punctured torus bundles (to appear). [F, H] W. Floyd and A. Hatcher, Incompressible surfaces in 2-bridge link complements (to appear). [F, L, P] A. Fathi, F. Laudenbach, V. Poenaru et al., Travaus de Thurston sur les surfaces, Asterisque 66·67, Societe Mathematique de France, 1979. [Hak] W. Haken, Theorieder Normal Flachen, Acta Math. lOS (1961), 5. [HatJ A. Hatcher, Incompressible surfaces in once-punctured torus bundles (to appear). [H, T] A. Hatcher and W. Thurston, Incompressible surfaces in 2-bridge knot complements, Invent. Math. (to appear). (Ja, ShJ W. H. Jaco and P. B. Shalen, SeIfert fibered spaces in 3-mamfolds, Mem. Amer. Math. Soc. No.2 (1979). [JohJ K. Johannson, Homotopy equivalence of 3-mamfolds with boundaries, Lecture Notes in Math., no. 7761, Springer-Verlag, Berlin, Heidelberg, New York, 1979. [M, YJ W. Meeks and S-T Yau, Topology of three dimensional mamfolds and the embedding problems in minimal surface theOJY, Ann. of Math. 112 (1980),441-484. [Mey] R. Meyerhoff, The Chern-Simons ilWariant for hyperbolic 3-mamfolds, thesis, Princeton University, 1981. [Mil IJ J. "Milnor, A unique factorization theorem for 3-mamfolds, Amer. J. Math. 84 (1962), 1-7. [Mil 2J ___ , Hyperbolic geometry: the first 150 years, proceedings. [Mos 1] G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. Math. Studies, No. 78, Princeton Univ. Press, Princeton, N.J., 1973. [Mos 2J G. D. Mostow, Inst. Hautes Etudes Sci. Publ. Math. [PoinJ H. Poincare, Cinquieme complement a l'analysis situs, Rend. Circ. Mat. Palermo 18 (1904),45-110, or Oeuvres, t. VI, pp. 435-498. [PrasJ G. Prasad, Strong rigidity of Q-rank I lattices, Invent. Math., 5-6. [RilJ R. Riley, Discrete parabolic representations of link groups, Mathematika 22 (1975), 141-150. [Seif] H. Seifert, Topologie drei dimensiolUlles gefasertei Raume, Acta Math. 60 (1933), 147-288. [SmiJ Proceedings of the Smith Conjecture Conference at Columbia University, (to appear). [StaJ J. Stallings, On fibering certain. 3-mamfolds, Prentice-Hall, Engelwood Cliffs, N.J., 1962, pp. 95-100. (SuI 1] D. Sullivan, Discrete conformal groups and measurable dYlUlmics, these proceedings. [Sul21 _ _ • [Th IJ W. Thurston, The geometry and topology of 3-mamfolds, preprint, Princeton Univ. Press

(to appear). [Th 2] Hyperbolic structures on 3-mamfolds, I: deformations of aLylindrical mamfolds, preprint. Hyperbolic structures on 3-mamfolds, II: surface groups and 3-mamfolds which [Th 3] fiber over the circle. [Th 4J ___ , On the geometry and dynamics of dlffeomorphisms of surfaces, I, preprint. [Wall F. Waldhausen, On irreducible 3-mamfolds which are sufficiently large, Ann. of Math. (2) 87 (1968),56-88 DEPARTMENT OF MATHEMATICS, UNIVERSITY OF COLORADO, BOULDER, COLORADO

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Annals of Mathematics, 124 (1986), 203-246

Hyperbolic structures on 3-manifolds I: Deformation of acylindrical manifolds By WILLIAM P. THURSTON

O. Introduction This is the first in a series of papers dealing with the conjecture that '.11 compact 3-manifolds admit canonical decompositions into geometric pieces. This conjecture will be discussed in detail in Part IV. Here is an easily stated special case, in which no decomposition is necessary: CONJECTURE 0.1. Indecomposable implies geometric. Let M 3 be a closed, prime, atoroidal 3-manifold. Then M3 admits a locally homogeneous Riemannian metric, i.e., a Riemannian metric such that for any two points p and q there is an isometry from a neighborhood of p to a neighborhood of q carrying p to q. The main result of this series of papers will be to prove Conjecture 0.1, Indecomposable implies geometric, and generalizations for a large class of manifolds. THEOREM 0.2. Atoroidal Haken is hyperbolic. Let M3 be a closed, atoroidal Haken manifold. Then M 3 admits a hyperbolic structure, i.e., a Riemannian metric with all sectional curvatures equal to - 1. A hyperbolic structure is, of course, locally homogeneous; it is locally isometric to hyperbolic 3-space H3. Conjecture 0.1, Indecomposable implies geometric is far from being proven in general, but it seems well-supported by a great many examples. A special case is:

CONJECTURE 0.3 (Poincare). Every simply-connected closed three manifold is homeomorphic to the 3-sphere. This would easily follow from 0.1, Indecomposable implies geometric, because it is easily seen that the only 3--i ~

Existence of hyperbolic structures on 3-manilolds which do not fiber

~-----,

II Orbilolds or V-manifolds

~

>-i

- - - - - ---1

i-i

FIGURE 0.4. F10wchart of ideas, This is a flowchart of some of the ideas which enter in the proof of Theorem 0,2, Atoroidal Haken is hyperbolic,

HYPERBOLIC STRUCTURES ON 3-MANIFOLDS, I

207

hyperbolic structures. We will define and briefly discuss in turn the algebraic topology, the geometric topology, and the quasi-isometric topology. When H(M) is thought of in terms of representations of the fundamental group of M, an obvious topology comes to mind, namely the topology of convergence of representations on finite generating sets. This can be alternately phrased as the compact-open topology on homomorphisms between the two groups. The set H(M) inherits a topology as a quotient space under the action of Isom(Hn) by conjugacy on the space of representations. This topology is the algebraic topology. H(M) with this topology will be denoted AH(M). In the case M is a closed 2-manifold, AH( M) is commonly called the Teichmuller space ofM. PROPOSITION 1.1. AH( non-elementary) is Hausdorff. If '1T 1( M) does not contain an abelian subgroup with finite index, then AH( M) is Hausdorff.

A discrete group of isometries of hyperbolic space which admits an abelian subgroup with finite index is called an elementary group. Proof Suppose that '1T 1(M) does not have an abelian subgroup of finite index and that N1 and N2 are elements of AH(M) such that every neighborhood of N1 intersects every neighborhood of N2. We will show that N1 = N2. We translate to the context of representations of groups. Suppose that '1T 1( M) does not contain an abelian subgroup of finite index. Represent Ni as Hn / Pi( '1TM). Let g1 ... gk be generators for the fundamental group of M. For any representation 'T, the translation distance for the i-th generator is a function T..(g;)(x) = d(x, 'T(g;)(x)) on hyperbolic space. The translation distance of gi goes to infinity with distance from the axis of 'T( g;) if 'Tgi is hyperbolic. Otherwise, 'Tgi has a unique fixed point at infinity, and T..( g i) tends to infinity away from this fixed point. It cannot happen that all generators for a discrete faithful representation 'T have a common fixed point at infinity, for otherwise the entire group would be a discrete subgroup of the group of similarities of Euclidean space, which automatically has an abelian subgroup of finite index. (This is a standard part of the theory of the thick-thin decomposition, or Margulis decomposition, of a hyperbolic manifold.) Therefore, the total translation distance

is a proper function, that is, it goes to infinity near the sphere at infinity. Also, d .. is strictly convex, so that d~ 1«0, KD is a convex ball in Hn for any sufficiently large K.

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Suppose that a sequence of representations cf>j converges to PI' while a conjugate sequence 1/Ii converges to P2. Then the sets d;i l ((O, K)) must converge to d;1 1((0, K)) in the Hausdorff topology, and similarly for 1/Ii. The conjugating isometries must carry these sets for the cf>i to the corresponding sets for the 1/1 i. This implies that the conjugating isometries remain in a compact subset of the group of isometries of hyperbolic space; so they have a convergent subsequence, whose limit conjugates PI to P2. The case that M is a closed surface has been analyzed classically. Fricke proved that for a closed surface of negative Euler characteristic, AH( M), which may be identified with the Teichmilller space T(M) for M, is homeomorphic to Euclidean space of dimension 31 X( M) I. When M is a surface with non-empty boundary, AH( M) is not connected. The explanation is that a homotopy equivalence between two surfaces with boundary is not generally homotopic to a homeomorphism. The components of AH( M) are in one-to-one correspondence with homotopy equivalences of M to surfaces, up to homeomorphisms. Each component of AH( M) is diffeomorphic to the product of Euclidean space with a number of half-lines. A three-manifold (M, aM) is acylindrical if aM is incompressible and if every map of the Uimensional cylinder C = Sl X I,

f:

(C, ac)

-+

(M, aM)

which takes the components of ac to non-trivial homotopy classes in aM is homotopic into M. The main result of this paper is:

a

THEOREM 1.2. AH(acylindrical) is compact. If M is any compact acylindrical 3-manifold with boundary, then AH(M) is compact.

The case when aM is empty follows from Mostow's theorem, [Mos] which asserts that AH(M) consists of at most one point. The theorem is false without the hypothesis that M is acylindrical; a simple example is (surface X I). Another appealing and often useful topology for H(M) is the geometric topology, denoted GH(M). To define a neighborhood basis for GH(M), consider (N, f) any element of H( M), where f is a homotopy equivalence from M to the hyperbolic manifold N. A (small) neighborhood of (N, f) in the geometric topology is defined by the choice of a (large) compact set KeN and a (small) positive real number f. The neighborhood Nbhd K,.( N) consists of those elements (N', f') of H( M) such that there exists a diffeomorphism cf> of K to a subset K' c N' whose derivative is within f of being an isometry at every point of K, and where cf> has the correct homotopy class. Explicitly, (f') -1 0 i' 0 cf> must be homotopic to f- l 0 i, where i: KeN and i': K' eN' are the inclusions and f- 1 and (f') - 1 are homotopy inverses of f and f'.

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To measure the deviation of a linear map L from being an isometry, one can use the quantity sup Iog ( v*o

IIL(V}II )

IIVII

.

This geometric topology is not to be confused with the geometric topology which can be defined on other sets. The most useful example is the set of based hyperbolic manifolds, with no specification of homotopy type or homotopy classes of maps. The definition of the geometric topology on that set differs in that the map cf> must preserve base point, and its homotopy class is not restricted. A third topology on the set H(M) is the quasi-isometric topology, denoted QH(M). A small quasi-isometric neighborhood of an element f: M ~ N of H(M) consists of those f':' M ~ N' E H(M) for which there exists a global diffeomorphism g: N ~ N' with go f =::: f', whose derivative is uniformly close to being an isometry. In general, a diffeomorphism g between two Riemannian manifolds P and Q which has the property that its derivative is within a uniform distance of an isometry is called a quasi-isometry. More particularly, when the derivative of g satisfies log

(

IIdg(V}1I ) II VII

~

K, for all tangent vectors V to P,

g is called a K-quasi-isometry. When such a quasi-isometry exists, then P and Q are quasi-isometric. These three topologies, the algebraic, geometric, and quasi-isometric, are successively finer, so that the maps

QH(M} ~ GH(M} ~ AH(M} induced from the identity on H( M) are continuous. This is an easy exercise from the definitions. Since AH( M) is Hausdorff, it follows that the other two are also Hausdorff. The inverse maps, in general, are discontinuous. For example, it is easy to see that in the geometric topology any non-compact hyperbolic surface of finite area is a limit of hyperbolic surfaces with infinite area. In the quasi-isometric topology, area is clearly a continuous function (where the values of the area function are endowed with the topology making 00 an isolated point). The demonstration that AH( M) ~ GH( M) is not continuous is much trickier, and requires at least three dimensions. There exist examples for which

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M is the product of a surface of genus 2 with an interval. See [Th-K] for one discussion of such an example. The sphere at infinity divides into two essentially different parts under the action of any discrete group f of hyperbolic isometries: the limit set L r , which is closed, and its complement, the region of discontinuity Dr, which is open. Dr is the unique maximal open set where f acts properly discontinuously, and L r is the unique non-empty minimal closed subset invariant for f (except in certain cases when f is elementary, such as f = Z). Since facts conformally on Dr' the quotient space Dr/f inherits a conformal structure. Let us specialize to the case of dimension n = 3. The TeichmUller space of (Dr' f) can be defined as the space of all conformal structures on Dr invariant by f which are quasiconformal to the initial conformal structure, up to conformal equivalences between structures which are equivariant with respect to f. Except in the case of elementary groups, each such conformal structure is represented by a unique f-invariant hyperbolic metric on Dr, its Poincare metric. The Teichmilller space T(Dr' f) is sometimes the same as the Teichmilller space of the quotient surface Dr/f, but not always. Consider, for simplicity, the case that f has no elliptic or parabolic elements. Then the AhHors finite area theorem says that the quotient surface consists of a finite number of components, each compact. Any conformal structure on the quotient surface gives rise to an equivariant conformal structure on Dr, and all equivariant conformal structures are quasiconformally related. This says that an element of the Teichmilller space of the quotient surface defines an element of the Teichmilller space of (Dr, f). When the components of Dr are simply-connected, then any equivariant homeomorphism of Dr is equivariantIy isotopic to the identity, and the two Teichmilller spaces are the same. In general, T( Dr, f) is the quotient of T( Dr/f) by the group of isotopy classes of quasiconformal homeomorphisms of Dr which commute with the action of f. Any quasi-isometry between two complete hyperbolic n-manifolds lifts to a quasi-isometry from Hn to Hn. Any such quasi-isometry extends continuously to the sphere at infinity, where it induces a quasiconformal map. (Indeed, this fact was an important point in Mostow's proof that hyperbolic structures are rigid.) Therefore, if N is any hyperbolic 3-manifold and N' is any other hyperbolic 3-manifold quasi-isometric with N, the conformal structure on D"'l(N') defines an element

We shall be concerned with the case that each component of

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connected, so that the conformal invariant is more directly thought of as an element of the Teichmilller space of the quotient surface. Here is a fundamental result, which has evolved through the work of a number of people, beginning with Ahlfors and Bers, including Mostow, Maskit and others, and has been put in a clean form through the work of Sullivan. 1.3. Quasiconformal deformation theorem. Let M be any complete hyperbolic 3-manifold with finitely generated fundamental group. The component QHo(M) ofQH(M) which contains M consists ofallf: M - N such that f is homotopic to a quasi-isometry cp: M - N. The map THEOREM

conf: QHo{M) -

r(

D"'l(M) ,

'lT1{M))

is a homeomorphism. In particular, the quasi-isometric deformation space QHo always consist of a single point, or it is noncompact. An important special case occurs when M is compact or has finite volume. Then Dr is empty, so that QHo(M) is a single point; this case is the Mostow rigidity theorem. Since QHo is often not compact, Theorem 1.2, AH(acylindrical) is compact, may be thought of as an existence theorem for many sequences of elements of H(M) which converge in AH(M) but not in QH(M). A complete description of the three spaces AH(M), GH(M), and QH(M) is certainly not rigorously known, but here is a conjectural image, of which certain features can be rigorously proved. Let us stick to the case that M is a compact, acylindrical manifold. Then H( M) is a hard-boiled egg. The egg complete with shell is AH(M); it appears to be homeomorphic to a closed unit ball. GH(M) is obtained by thoroughly cracking the egg shell on a convenient hard surface. Apparently no material is physically separated from the egg, but many cracks are developed-cracks are dense in the boundary-and at the same time, the material of the egg just inside the shell is weakened, so that neighborhood systems of points on the boundary become thinner. Finally, QH(M) has uncountably many components, which are obtained by peeling off the shell and scattering the pieces all over. Each component is homeomorphic to some Teichmilller space-it is parametrized by Euclidean space of some even dimension. "Most" of the components have dimension zero, for they describe groups whose limit set is all of S!. 2. Ideal triangulations The plan for the proof of Theorem 1.2, AH(acylindrical) is compact is to study maps of a triangulation of a compact acylindrical manifold M to hyper-

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bolic manifolds homotopy equivalent to M. In a sequence of such maps, we can study how the shapes of simplices might degenerate. Hyperbolic simplices can degenerate in only very special ways; qualitative information about the geometry of the three-manifolds can be deduced from these shapes and how they are assembled. It is desirable to impose conditions on the maps we study so that properties of the maps indicate properties of the target manifold, and are not so dependent on the arbitrary choices we make in choosing a particular map in a homotopy class. From the beginning, we abandon any thoughts of choosing the map to be an embedding, and concentrate on more attainable conditions. If 'T is a triangulation of M, and if f: M ~ N is any map in the homotopy class, then f is homotopic reI the vertices of 'T to a piecewise straight map s, that is, having the property that when s is lifted to a map s between universal covers, the image of each simplex is the convex hull of the image of its vertices. It is not important exactly how each simplex maps to its image, but one way to choose an explicit map is by way of the Lorenz model for hyperbolic space. The lift lof f to a map between universal covers sends the vertices of any simplex to pOints which in the Lorenz model E 3. 1 lie on H3 is regarded as a sheet of a hyperboloid. The map on vertices extends to a unique map S of if to E 3.1 which is linear on each simplex; when this is projected back to the hyperboloid, it gives a map s to H3 = N. Since s is canonically defined, it is equivariant; so it projects to a map to N. S is equivariantly homotopic to 1 reI vertices, by a linear homotopy. The homotopy can be projected back to H3 and thence to N, giving a homotopy of f to s reI vertices. It is clear that any other piecewise-straight map is homotopic to s through maps such that each simplex has the same image. This property also holds for the lifts of the maps to the universal covers. The use of piecewise straight maps removes much of the arbitrary choice involved in choosing a map in the homotopy class-the choice which remains is the choice of positions of the vertices, except for the irrelevant choice of parametrization of images. To be precise, the choice of positions of vertices is really made not in N, but in its universal cover N = H3. Any equivariant map of the vertices of the triangulation T of if determines a piecewise straight map, up to trivial homotopy. If the images of the vertices are merely chosen in N, the homotopy classes of edges must still be determined. To minimize choices, we will use a limiting case of piecewise-straight maps where we choose positions of the vertices not in H3 = N, but on the sphere at infinity S!. By doing this, we immediately lose the map from M to N. We will have at most a map from M minus the vertices of the triangulation to N. What we gain is much simpler variations in the shapes of images of simplices.

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An ideal k-simplex in Hn is the convex hull of a set of k + 1 distinct points on S~-l, subject to the condition that this convex hull is k-dimensional. If the convex hull is lower dimensional, then it is a flattened ideal simplex. An ideal simplex is homeomorphic to a simplex minus its vertices. To choose an explicit parametrization, one can choose a point in En,l to represent each vertex. These representatives lie on the light cone; the choice can be anywhere on the appropriate ray. Once these choices are made, map the simplex to En, 1 by linear extension, and project the simplex minus its vertices to Hn. This determines the parametrization up to precomposition with projective diffeomorphisms of the simplex to itself. Similarly, there is a map of a simplex minus its vertices to a flattened ideal simplex which is well-determined up to precomposition with projective diffeomorphisms. An ideal simplicial map of T to N is defined to be an equivariant map from the universal cover T to the completion H3 U of N such that each vertex of T goes to and the image of each simplex is the convex hull of the image of its vertices. Any ideal simplicial map r determines a special subcomplex at infinity 1(r) of T, consisting of each simplex whose image is a single point at infinity. The image of 1( r) in T is denoted L( r). We shall also refer to L as the subcomplex at infinity. There is a map D( r) defined downstairs, but its domain is only T - L. Each simplex of T minus its intersection with L maps either to an ideal simplex, a flattened ideal simplex, or in the case its intersection with L contains more than vertices, it maps degenerately to a lower-dimensional simplex. The map D( r) is a limit of ordinary piecewise-straight maps restricted to T - L.

S;,

S;

2.1. Ideal simplicial pants. An ideal simplicial map for a pair of pants can be obtained by spinning a triangulation. The initial triangulation has 8 triangles, but only two of them remain nondegenerate after spinning.

FIGURE

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Figures 2.1, Ideal simplicial pants, and 2.2, 2-Dimensional ideal simplicial maps, illustrate some examples in 2 dimensions. These examples have been obtained from piecewise straight maps of T by spinning, that is, homotoping each vertex repeatedly around a simple closed curve on the target surface and passing to the limit. In Example 1 on a pair of pants, only 2 out of 8 two-simplices are mapped non-degenerately, and in Example 2 on a surface of genus 2, only 4 out of 6 are non-degenerate. These examples were chosen to be particularly nice, so that the nondegenerate simplices are mapped disjointly and with positive orientation. In the general case for a surface of genus 2, each ideal triangle would be likely to be dense on the surface, although most of it would be extremely thin.

Poincare disk model:

EX,II11plt' 2.

!pin a.roUltd

FIGURE 2.2. 2-Dimensional ideal simplicial maps. Examples of ideal simplicial maps for surfaces. These can be constructed starting with an ordinary piecewise straight map, then homotoping the vertices through homotopy classes tending to infinity.

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For the proof of Theorem 1.2, AH(acylindrical) is compact, we will study the limiting geometry of the map r or D( r), and try to reconstruct a limiting manifold from this geometry. Before trying to pass to limits, however, we should ask ourselves whether we can even reconstruct N from the local geometry of D( r), that is, from the shape data-the shapes of nondegenerate images of simplices minus t-and the gluing data-the information describing how the nondegenerate images of simplices fit together near any common face in N. In Example 2 of Figure 2.2, 2-Dimensional ideal simplicial maps, this information does not suffice to determine the hyperbolic structure. In this case M - t is not connected, and it is impossible to reconstruct how the two halves are joined together. We shall see that the situation is much better in dimension 3. A group G is an amalgam if it can be described as either (1) a free product G = A *c B, or (2) an HNN amalgam G = A * c' The group C is the amalgamating group. A representation as an amalgam is trivial if it is of type (1) and C includes into A or B as the entire group. PROPOSITION 2.3. Abelian means small in acylindrical manifolds. The fundamental group of a compact aspherical atoroidal acylindrical 3-manifold with incompressible boundary cannot be represented as a nontrivial amalgam such that the amalgamating group is abelian or contains an abelian subgroup with finite index.

Proof For any representation of a 3-manifold group as a nontrivial amalgam, there is an incompressible surface whose fundamental group is some subgroup of the amalgamating group, by a standard 3-manifold argument. (Construct a classifying space by forming the product of a classifying space Xc for C with an interval, and gluing its ends to the classifying space for A or spaces for A and B. Map the three-manifold to the classifying space, and make the map transverse to Xc X {.5}. The preimage of Xc X {.5} is a surface. Apply the loop theorem repeatedly to construct homotopies of the map which make this surface incompressible. ) If the amalgamating group is abelian or contains an abelian subgroup of finite index, then the surface could only be a sphere, projective plane, torus, Klein bottle, or disk, annulus or Moebius band. The hypotheses rule out any such surface which cuts the 3-manifold non-trivially. PROPOSITION 2.4. Iota bounded by abelian. If S is any component of the boundary of the regular neighborhood of the subcomplex at infinity t for any ideal simplicial homotopy equivalence of a compact acylindrical 3-manifold M

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to a hyperbolic 3-manifold N, then the image of '7T 1(S) in '7T 1(N) contains an abelian subgroup with finite index. Proof Each component of r maps to a single point on S! . Therefore, the fundamental group of any component of the boundary of a regular neighborhood of t acts on N = H3 with a fixed point at infinity. Any discrete group with this property contains an abelian subgroup with finite index. COROLLARY 2.5. The boundary of iota is large . The fundamental group of some component of the complement of t maps surjectively to the fundamental group ofM.

Proof For each component S of the boundary of the regular neighborhood of t , there is a representation of '7T1(M) as either an amalgamated free product or an HNN amalgam, where the amalgamating group is the image of '7TlS) in '7T l M ). By 2.4, Iota bounded by abelian, the amalgamating subgroup is virtually abelian. Thus, by 2.3, Abelian m eans small in acylindrical manifolds, the amalgam is trivial. This implies that S must separate, and one of the two pieces must have the same image for its fundamental group as S. Throwaway all the pieces of this type; what remains is a single component the image of whose fundamental group is '7TrCM). Examples (rather artificial) can be easily constructed where the complement of t is not connected. For example, choose T to be a triangulation of M which admits an embedding of a sphere in its 2-skeleton, and so that the ball bounded by the sphere contains at least 1 vertex. Choose the ideal simplicial map so that the vertices of each upstairs sphere (in M) map to a single vertex on S!, while the vertices in the upstairs balls map to points distinct from the image of its boundary and distinct from each other. Somewhat more interesting examples may be constructed using a solid torus whose fundamental group injects in '7T 1(M). In this case, the vertices on the bounding torus must be mapped to one of the fixed points for the action of its fundamental group on S! . We can now characterize exactly which subcomplexes t of a triangulation T can be the subcomplex at infinity for an ideal simplicial homotopy equivalence. Certainly t must contain all the vertices of T, and it must satisfy the condition that if the boundary of any simplex of dimension 2 or more is in t, then the simplex is in t. (A simplex of dimension greater than 1 has a connected boundary, which implies that if its boundary is in t, it maps upstairs to a single point.) If t also satisfies the necessary condition that the image of the fundamental group of each component of t in '7T 1(M) is virtually abelian, then an ideal simplicial map is easily constructed by mapping each component of r to some point fixed by the action of its stabilizer in '7T i M) on S!.

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FIGURE 2.6. Image may be dense. The image of a simplex may be dense under an ideal simplicial map.

The final ideal simplicial map r can easily have the property that each image simplex by D( r) is dense in N. However, most of each image simplex consists of long, thin spikes; because the spikes have small volume, we shall see that they have little real consequence.

3. Shapes and limits We will now analyze in detail how the shapes of ideal simplices determine a hyperbolic manifold N, and what kind of limiting behavior they can have. From this analysis, we will derive a quick proof of Theorem 1.2, AH( acylindrical) is compact, in the case that the manifold is compact; of course, the theorem is a consequence of the Mostow rigidity theorem in this case, but the simple proof helps illustrate the idea. The remaining ingredient needed for the general case of 1.2, AH(acylindrical) is compact, is some way to control the boundary of M. We shall do this using an analysis of pleated surfaces, to be introduced in the next section. For simplicity, we will stick to analyzing the case that M is oriented. The result for an unorientable manifold will be a simple corollary of the oriented case. All ideal I-simplices in Hn are congruent. In fact, they are overly congruent: any ideal I-simplex (line) has many isometries taking it to any other ideal I-simplex. This is the reason that the local geometry of an ideal simplicial map does not determine the target manifold in dimension 2. In the case of an ideal

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I-simplex in H 3 , there is a two dimensional or one complex dimensional group of isometries stabilizing it. All ideal triangles in Hn are congruent, since the group of isometries of Hn acts transitively on triples of points. In the case n = 3, there is a unique orientation-preserving isometry sending any ordered triple of points to any other. Ideal tetrahedra are not all congruent. There is a one dimensional or one complex dimensional family of different shapes. The shape of an ordered simplex in oriented H 3 has a convenient parametrization obtained as follows: Transfonn by the unique orientation-preserving isometry which sends the first three vertices to the points 0, 1 and 00 in the upper half-space model. Its congruence class is detennined by the position of the last vertex, which is at some point z E C {O, 1, oo} in the thrice-punctured sphere. If z is on the real line, then the ideal simplex is a flattened simplex. If the imaginary part of z is positive, then the map of the simplex preserves orientation, while if it is negative it reverses orientation. The invariant z is the same as the cross-ratio of the four vertices of the tetrahedron. There are 24 possible orderings of the vertices of a tetrahedron, 12 of them compatible with a given orientation. How can the shape of an oriented but unordered simplex be described? The description is simplified by the fact that every ideal tetrahedron has a certain beautiful symmetry, described by an action of Z2 X Z2. In fact, consider the common perpendicular to any opposite pair of edges of an ideal simplex, that is, edges with no common ideal vertex. Rotation of 'IT about such a perpendicular preserves the two edges in question, which implies it preserves their set of endpoints. Therefore, it acts as an isometry of the simplex to itself. There are three pairs of opposite edges; so this construction gives rise to three nontrivial isometries; together with the identity, they fonn the group Z2 X Z2. The three axes intersect in a common point, where they are mutually perpendicular. This group of symmetries of the ideal tetrahedron is a normal subgroup of the alternating group for the four vertices, and it preserves the invariant z. The quotient group is Z3' which may be identified with the group of even pennutations of the three pairs of opposite edges. This means that the invariant z is associated with a pair of opposite edges. Here is a more direct way to describe the association of an edge invariant z( e) to an edge e of a congruence class of oriented ideal simplices. If e is (temporarily) given an orientation, then in terms of the orientation of the standard tetrahedron, the two triangles which have e as an edge can be distinguished as the clockwise face when viewed from above, and the counterclockwise face. There is a unique orientation-preserving isometry cp of H 3 taking the image of the clockwise face to the image of the counterclockwise face, while

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preserving the two endpoints of e. The edge invariant is the complex number whose modulus is the translation distance of cp, and whose argument is the angle of rotation of cpo Note that cp will rotate clockwise or counterclockwise about e, depending on whether the simplex is mapped to preserve or reverse orientation. If the opposite orientation of e is chosen, then cp must be replaced by cp -1, but the complex number z( e) remains the same. The edge invariant is consistent with the parametrization of an ordered simplex: If an ideal tetrahedron is arranged so that the first three vertices are at 0, 1 and 00 with the fourth vertex at w, then the w = z(Ooo) = z(lw). The transformation taking the clockwise face for 0,00 to the counterclockwise face acts as multiplication by w on the complex plane. More generally, if one of the vertices is put at 00 then the invariant for the edge from any of the finite points to 00 is the ratio of the two adjacent sides of the Euclidean triangle spanned by the three finite vertices; here, a side of a triangle is interpreted as a vector, which is identified with a complex number. It is easy to deduce the formula relating the various edge invariants to each other. They are illustrated in Figure 3.1, Edge invariants.

o

FIGURE 3.1. Edge invariants. The shape of an oriented ideal simplex is detennined by any of its edge invariants. The invariants for various edges are related as shown.

It is easy to understand the ways in which a single simplex can degenerate. The illegal values for the edge invariant in Care 0, I, and 00. Note that if one of the edge invariants tends toward anyone of the three illegal values, the other two edge invariants tend toward the other two illegal values. When this happens, the ideal simplex stretches apart into two pieces that look nearly like ideal triangles, connected by a long thin nearly l-dimensional part (Figure 3.2, Stretched simplex). Now let M be any compact three-manifold, and consider a sequence of elements {Nk} E AH( M), parametrized by ideal simplicial maps fk using

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FIGURE 3.2. Stretched simplex. As an ideal simplex changes, the only kind of degeneration which can happen to its shape is that it stretches out into two parts which look like ideal triangles, joined to each other by a long stringy part. The edge invariants for the two edges which remain on one side converge to 1. The other edge invariants converge to 0 or infinity, depending on the orientation of the simplex.

domain triangulation T. Let Pk: '1T l M) ~ Isom H 3 denote the holonomy for N k . We will pass to subsequences as necessary to try to get convergence of as much of the data as possible; each subsequence will be renamed like the entire sequence. We first pass to a subsequence so that the subcomplex l at infinity is constant. Next, we pass to a subsequence so that the edge invariant of each image ideal simplex or flattened ideal simplex converges in C. Let D c M be the union of all simplices with at least one edge in t. If the limiting edge invariants are all in C - {O, I, 00 }, then clearly the sequence of developing maps for M - D converges; so the sequence of representations restricted to the image of '1T i M - D) in '1T l M) converges, possibly after conjugating. This gives us a limit in AH(M), provided we show that the degenerate simplices do not fill up too much of M. In general, some of the edge invariants may very well go to illegal values. We will define a "submanifold" GeM so that the sequence of representations for the fundamental group of each component of G automatically converges, possibly after taking a subsequence and conjugating. For this, it is sufficient that fk restricted to G is homotopic to a map which stretches the metric of G by only a bounded amount, since the set of isometries which move a given point by a bounded amount is compact. G is obtained from M by removing a neighborhood of a "bad" 2-complex B. Now B is defined to consist of all l-cells (not because they are bad, only because it Simplifies the picture to discard them), all of D, together with a twisted square embedded in any 3-simplex whose edge invariants are converging to illegal values. The twisted square is embedded in such a way that its four or 00. It edges are the four edges with edge invariants tending toward separates the two edges whose edge invariants are tending toward 1 (Figure 3.3, Twisted square).

°

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FIGURE 3.3. Twisted square. A twisted square like this is adjoined to the 2-complex B in each 3-simplex (of or) whose edge invariants tend toward the three illegal values 0, 1, and 00. The edges of the square are on the four "stretched" edges with invariants tending to and 00, and the square separates the two edges with invariants tending to 1.

°

Denote by ~ the set of all non-degenerate 3-simplices with edge invariants tending to illegal values. The intersection of G with any 3-simplex a is determined by a n L and by whether or not a E ~ . There are four cases, illustrated in Figure 3.4, G intersect simplices.

3.4. G intersect simplices. These are the four ways in which G can intersect a 3-simplex up to symmetries. There are either 0, 2, or 4 degenerate simplices on the boundary of IX, depending on the intersection of t with IX. (Note that if IX n t contains as much as two opposite edges or a 2-simplex, then all the 2-simplices of JIX are degenerate.) FIGURE IX,

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It is clear from the definition that we can conjugate the sequence of representations restricted to the fundamental group of a component of G until they lie in a compact set. Let K denote the frontier of G, that is, K;:

aG - aM.

The strategy for proving Theorem 1.2, AH(acylindrical) is compact, will be to show that '1T l( G) is large in '1T l( M); we will do that by showing that K is in a certain sense "small" in M. The analysis is related to a certain quotient map p: K ~ y, where y is a graph or l-complex. The quotient graph will be determined by defining an equivalence relation or decomposition of K. The first step is to construct a subsurface with boundary KO C K which consists of the intersection of a regular neighborhood of l with K, together with a diagonal band on each side of the twisted squares of E, as illustrated in Figure

FIGURE 3.5. Local decomposition of K. A quotient graph y of the internal boundary K of the good submanifold G is constructed by defining an equivalence relation on its intersection with each 3-simplex.

3.5, Local decomposition of K. Each diagonal band "indicates" vertices of ideal simplices which are converging together, from the point of view of its component of G n the simplex. Each component of Ko is to be collapsed to a point to form the vertices of y. The second step is to form a foliation F of K - KO' In fact, any component of the intersection of K - KO with a 3-simplex is a rectangle, two of whose edges are on the boundary of the simplex and near two edges of the simplex. Let F be the trivial foliation of each such rectangle transverse to these two edges. The leaf space of F within the rectangle is parametrized by the unit interval [0, I].

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Arrange so that the foliations in neighboring rectangles are joined so that this parametrization is either preserved or reversed. The leaves of F are compact, so that each leaf is a circle if M is closed, and either a circle or an interval if M has boundary. The components of KO together with the leaves of F define an equivalence relation, whose quotient is the graph y. We will need the following, due to Chuckrow: PROPOSITION 3.6. Non-elementary limit is discrete. Let r be any group, and suppose that (JK: r ~ Isom(Hn) is a sequence of discrete faithful representations which converges to a representation (Joo' Then either (Joo(f) is discrete and (Joo is faithful , or r is virtually abelian, i.e. , it contains an abelian subgroup with finite index.

Proof For any K > 0 there is an E > 0 such that any discrete group of isometries of Hn which is generated by elements Yl" .. , Y/ having the property that for some point x o, d(x o, Yl(X O» < E and for all i > 1, d(x o, y/x o < K has an abelian subgroup with finite index. (Cf. [ThI, pp. 5.51-5.56] for one discussion). To prove 3.6, Non-elementary limit is discrete, let K be greater than the distance that (Joo of the generators of r moves some Xo' If there is no element of r such that d«(JooYx o, x o) < E, then (Joo r is discrete. Otherwise r contains an abelian subgroup with finite index. What are the possibilities for a virtually abelian subgroup of the fundamental group of an orientable hyperbolic 3-manifold? A subgroup which is virtually abelian is in fact abelian.

»

PROPOSITION 3.7. Y locally abelian. For each Ckell or I-cell Y, 7T l( p - l( f3» is abelian.

f3 of the graph

Proof If f3 is an edge of Y, then p - l( {3) is a rectangle or a cylinder and in either case has an abelian fWldamental group. Now consider a O-cell f3 corresponding to a component K o of K. Since K is in the good set, the sequence of representations Pk converges on 7T l( K0) to a limiting representation Poo' The limiting representation fixes a point on S! . This point is a limit of images of vertices under the ideal simplicial maps. If Poo 17T i Ko) is not discrete and faithful, it is abelian by the preceding proposition. On the other hand, any discrete group of orientation preserving isometries of H 3 which fixes a point at infinity is abelian. It is likely that for some of the cells {3 of Y, the image of 7T1(p - l({3» in 7T l( M) is trivial. Let Yo be the union of cells whose image group is not trivial.

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3.8. Yo has abelian pieces. For each component c of Yo, the 'IT l M) is abelian.

image of 'IT l( p - l( c)) in

Proof This will follow from Proposition 3.7, y locally abelian, together with the following lemma: LEMMA

3.9. Commutation transitive.

(i). If x, y and z are any three orientation preserving isometries of infinite order in H 3 such that x commutes with y and y commutes with z, then also x commutes with z. (ii). If x and yare two orientation preserving isometries of infinite order in H 3 such that y - lxy commutes with x, and if x and y generate a discrete group, then y commutes with x.

Proof Orientation preserving isometries of infinite order commute if and only if they have the same axis or the same parabolic fixed point at 00. Part (i) immediately follows. Under the hypothesis of part (ii), if x is hyperbolic, then y must also be hyperbolic with the same axis; so they commute. If x is parabolic, then y must fix the parabolic fixed point of x. The hypothesis of discreteness implies that y cannot be hyperbolic; therefore it is parabolic, with the same fixed point. The image of 'IT l( p - l( in 'IT l( M) is built up by amalgamated free products and HNN amalgams from abelian subgroups. By the lemma, the entire group is abelian.

c»

4. The deformation space for closed manifolds is compact We will now specialize and complete the proof of Theorem 1.2,

AH( acylindrical) is compact in the case that M is a closed manifold or a manifold whose boundary consists of tori. This is equivalent to the condition that Nk have finite volume. Even though this case follows from Mostow's rigidity theorem, it is worth presenting in order to motivate the more technical version needed to handle the case when M has boundary. Here is a quick synopsis of the proof. The geometry of the maps from M to Nk (pinned down via ideal simplicial maps) remains bounded unless something is stretched a great deal. In terms of the simplices, M decomposes into pieces where the geometry of the map remains bounded, joined by other parts of M where the geometry is going to infinity. The interface is a surface which maps with a small area, and which turns out to be made up of pieces whose image fundamental group in M is abelian. By topological considerations, such surfaces cannot separate an atoroidal manifold in an essential way. Thus, one of the

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pieces where the geometry of the map remains bounded carries all of the fundamental group of M. Since M consists of tori, we can arrange that C t, by mapping all simplices of if to the point at infinity corresponding to the given cusp. This implies that K will be a closed surface. We can make an isotopy of the surface K in M so that its image in Nk by the ideal simplicial map has small area, tending to 0 as k ~ 00. This can be accomplished by isotoping the intersection of K with any 2-simplex so that it is near the boundary of the 2-simplex, and then extending individually to each 3-simplex. Recall that components of K - KO intersected with a 3-simplex are foliated rectangles. We arrange that these rectangeles be mapped to long thin strips in N, where the leaves of the foliation give the short direction. In fact, we make the lengths of the rectangles go to infinity with k, while the widths go to O. For any edge e of Y - Yo' the cylinder p-\e) C K is made up of a fixed number of thin strips, glued together edge to edge. These edges are the intersections of the cylinder with 2-simplices of T. Since the fundamental group of the cylinder has a trivial representation in this case, fk restricted to the cylinder can be lifted to H3. Since the cylinder lies in the good set, the amount of shearing of one of these strips in the cylinder with its neighbors is bounded as k ~ 00. Since the strips are becoming infinitely long, there is a short cross sectional circle. Such a circle bounds a disk of small area. Construct a manifold C 1 by attaching 2-handles to C along the core curve of each of these trivial cylinders. The inclusion C C M extends to a homotopically unique map gl: C 1 ~ M, since the core curves are trivial in 7T 1(M). The composed map fk 0 g 1 into Nk is homotopic to a map such that the image of K1 = aC 1 has small area. Note that the image of the fundamental group of any component of K1 in 7T1(M) is abelian; in virtue of 3.8, Yo has abelian pieces.

a a

aM

If C is any piecewise-differentiable singular 3-chain in a Riemannian manifold P, let degreec : P ~ R be the step function which gives the degree of C at a point in P. It is defined almost everywhere. The mass of C is defined as mass( C) = f p Idegreeci dV. The mass of a 2-chain is defined similarly, if we use the degree of C on each tangent 2-plane in P and integrate with respect to 2-dimensional Lebesgue measure. For a generic 2-chain, the mass is just the area of its image. PROPOSITION

4.1. Snwll abelian 2-cycle bounds snwll 3-chain. Let P be

any hyperbolic 3-nwnifold with abelian fundamental group. Then every 2-cycle Z in P bounds a 3-chain C in P, possibly non-compact but locally finite, satisfying mass( C) ~ mass( Z). The function degree c is uniquely determined by Z.

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Proof There are two cases-the fundamental group of P can be either parabolic or hyperbolic. Consider first the hyperbolic case. Let (r, x, 0) be cylindrical coordinates about a line in H 3 , so that r is the distance from the line, and x is the height of the projection to the line. The hyperbolic metric is

ds 2

=

dr 2

+ cosh2rdx 2 + sinh2rd0 2 .

0:

The 2-form = cosh r sinh rdxdO, which is the area form for equidistant surfaces about the line, has norm 1. Its exterior derivative do: = (cosh2r

+ sinh2r) drdxdO

is greater than the volume form

dV

=

cosh r sinh rdrdxdO.

0:

The form descends to P. If Z is any 2-cycle, Z is the boundary of a compactly supported 3-chain whose degree function is uniquely determined. It suffices to prove the inequality for the positive and negative parts of degreee , after perturbing C to make C transverse to itself. Since we can change the sign of the negative part, we need consider only the case degreee ~ O. We have

a

mass(C) =

1c dV < 1c do: 10: S z =

mass(Z).

The case that P is parabolic is similar, except that H 2( P) may not be zero; however, each 2-cycle bounds a unique chain of finite volume. For this case, take the area form for a family of horospheres. Then do: = 2 dV, and the proof as goes through. For each component C of K 1, the group gl*1T 1(C) in 1Tl(M) is abelian-either 1, Z, or Z2. In the first two cases, choose some 3-manifold Me with aMe = C such that the homomorphism extends to 1Tl(Md. The map of C to M extends over Me in a homotopically unique way. In the third case, define Me = C X [0,1], and glue it on to C at C X O. The map gl on C extends over Me taking C X 1 to aM. Let G 2 be the manifold obtained from G 1 by attaching the Me's, and let g2: (G 2 , aG 2 ) - (M, aM) be the extension of glover the Me's. Clearly each component of G 2 contains a unique component of G and the corresponding components of G 2 and G have fundamental groups with the same image in 1T 1(M). Thus, if G~ is a component of G 2 then, after conjugations, the representations Pk 0 g2: 1TtCG)~ - Isom(H 3 ) converge.

0:

PROPOSITION

4.2. Degree G is 1. The degree Ofg2: (G 2 , aG 2) - (M, aM)

is 1.

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Proof The degree can be computed by composing with the map fk: (M , aM) ~ (N, cusps). The map fk , as an ideal simplicial map, is not continuous on all of M, but by modifying f in a small neighborhood of L, it can be approximated by a continuous map which is a homotopy equivalence. The modification takes place on a portion of M whose image has a small mass. The degree of fk can be defined at almost every point in Nk by adding up the degrees of the 3-simplices which hit it. It follows that the local degree of fk is 1 almost everywhere. The singular chain defined by g 2 is obtained from that defined by fk by a sequence of moves, each of which affects only a small volume: The mass of the image of B (the complement of G) is small; G 1 was obtained by adding 2-handles with small mass; and finally, G 2 was obtained from G 1 by adding 3-chains of small mass. Thus, the total mass of the chain defined by g 1 is close to the volume of N k • Since this mass must be an integral multiple of the volume of N k , and since there is an a priori lower bound to the volume of a hyperbolic manifold, it follows that its degree must be 1. This proposition tells us that G 2 must " capture" most of the fundamental group of M, so that the proof is almost finished. In fact, if (P, aP) ~ (Q, aQ) is a map of degree k between connected n-manifolds, the image of 'IT l( P) in 'IT l( Q) has index at most k: otherwise, the map would lift to a covering space of degree (possibly infinite) greater than k; so the composed map from P to Q would have degree greater than k. The proof in this special case (a M consists of tori) of Theorem 1.2, AH(acylindrical) is compact is finished, except for the possibility that G, and hence G 2 , is not connected. Even if G 2 is not connected, at least one component must map to M with positive degree. The easy case of Theorem 1.2 will be completed by the following : LEMMA 4.3. Finite index convergence implies convergence. Let r be a group which does not have an abelian subgroup of finite index. Let H e r be any subgroup of finite index. Suppose that {p k} is a sequence of discrete faithful representations of f in the group of orientation preserving isometries of H3 such that {Pk} restricted to H converges. Then {Pk} converges on the entire group f .

Proof One may as well assume that H is a normal subgroup of f, since in any case H contains a normal subgroup with finite index. Any non-elementary group, such as H, contains at least three non-commuting elements of infinite order. (In fact, H must contain a free group on 2 generators.) These three hyperbolic elements determine three distinct points, their attracting fixed points, on S!. Any orientation preserving isometry g is

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detennined by its action on these three points; but if g E r, this action is detennined by the action of g by conjugation on H. If { Pk} restricted to H converges, then also the attracting fixed points of its hyperbolic elements converge. This implies that {Pk} converges on all of r.

5. The geometry of pleated surfaces The proof of the easy case of Theorem 1.2, AH(acylindrical) is compact, given in the preceding section is insufficient in the general case for lack of control over the behavior at aM. To gain the needed control, we will use pleated surfaces. A pleated surface in a hyperbolic 3-manifold N is a complete hyperbolic surface S, together with a continuous map f: S ~ N which satisfies: (a) f is isometric, in the sense that every geodesic segment in S is taken to a rectifiable arc in N which has the same length, and (b) for each point XES, there is at least one open geodesic segment Ix through x which is mapped to a geodesic segment in N. Generally, a pleated surface is not smooth, and generally f is not even locally an embedding. These qualities are sacrificed for the sake of properties (a) and (b) above, which turn out to generate a useful theory. This theory has been extensively analyzed in [Thl]. We will review a few basic facts. PROPOSITION 5.1. Folded or flat. If xES is any point in the domain of a pleated surface, then either there is a unique geodesic segment Ix through x which is mapped to a geodesic segment, or x has a neighborhood which is mapped isometrically to a portion of a hyperbolic plane in the image.

Proof Suppose I and m are two distinct line segments through x which are mapped to line segments. Consider a quadrilateral surrounding x, with corners on I and m. Since its sides cannot increase in length, none of the angles fonned by I and m at x can inGrease under f. This forces them to be the same in the range as in the domain; therefore the four sides of the quadrilateral are mapped to geodesics. This in turn forces each geodesic segment from x to a point on the quadrilateral to map to a geodesic segment; so the image is .a portion of a plane. The set of points for which the genn of the segment I" is unique is called the pleating locus I1( f) for the pleated surface f. A geodesic lamination A. on a hyperbolic surface S is a closed set L, together with a foliation of L by geodesics of S. One example is a simple closed geodesic; other examples can be obtained by taking for L a Hausdorff limit of a sequence of longer and longer simple closed geodesics.

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When we are not being fonnal, we will ignore the distinction between A and its support L. Indeed, it is an easy consequence of Proposition 5.3, Lami1Ultions have measure 0, below, that L detennines A. PROPOSITION 5.2. Pleating locus is lamination. The pleating locus for a pleated surface has the structure of a geodesic lamination.

Proof It is obvious that the pleating locus is closed, since its complement is open.

If y E I1( f), and if there is any geodesic segment m with one endpoint at y in a direction not on ly, then we claim there is a portion of a half-plane bounded by ly containing m which is mapped by f to a portion of a half-plane. The proof is half the proof of 5.1, Folded or flat. With these two observations, it follows that for any x E I1( f), the line segment Ix extends to a complete geodesic which remains always in the pleating locus and defines ly for any point on it. These geodesics must be coherent, defining a foliation of I1( f), in virtue of the fact that they cannot cross. PROPOSITION 5.3. Lami1Ultions have measure o. The support L of a geodesic iami1Ultion A on any surface of finite area has measure o.

Proof This is due to Nielsen, who studied geodesic laminations extensively in a somewhat disguised fonn ([Nie]). The method is to show that the complement of L has full measure. Each component of the complement is a hyperbolic surface, with boundary consisting of geodesics, possibly with comers (half-cusps) like the comers of an ideal triangle and cusps. The area of such a region can be easily computed by the Gauss-Bonnet theorem, in tenns of its Euler characteristic and number of cusps of each kind. On the other hand, a line field can be defined in a neighborhood of L which agrees with the tangents to the lamination. The line field can be extended to all of S but with isolated singularities. An index is defined for such singularities, just as for vector fields; for a line field, the index takes half-integral values. The total index is the Euler characteristic of S; when this index is allocated among the regions in the complement of L, one finds that their total area matches the area of S. Later we will describe how an ideal simplicial map on the boundary of M can be made to describe a pleated surface in N k , much as in the example illustrated in Figure 2.2, 2-Dimensional ideal simplicial maps. The proof of l.2, AH( acylindrical) is compact, will depend on knowing that these bounding pleated surfaces remain definitely separated: they cannot collapse like a punctured innertube.

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Mathematically, the infonnation will be conveyed in tenns of the mapping Pf of the pleating lamination I1( f) on the domain surface S into the tangent line bundle P(N) of the target hyperbolic manifold. Even though f itself is not usually an embedding, we will show that under mild assumptions Pf is an embedding, and from this we will deduce that laminations in S embed unifonnly in Pf. A geodesic lamination A on a hyperbolic surface is minimal if the closure of any leaf is all of A, or in other words, if there are no proper sublaminations. A lamina~ion is recurrent if the closure of any half-infinite segment of any leaf contains the leaf. It is immediate that a minimal lamination is recurrent. There is a partial converse: 5.4. Finite minimal decomposition. A geodesic lamination A on a complete hyperbolic surface of finite area has only a finite number of connected components, and it is recurrent if and only if each component is minimal. PROPOSITION

Proof The complement of A is a hyperbolic surface with geodesic boundary, sometimes with half-cusps along its boundary. The number of boundary components of complementary pieces is bounded in tenns of the area of S, so the number of components of A is bounded. By elementary topology, each component of A has at least one minimal set. Suppose that f.L is any minimal set of A. There is some € such that any geodesic of A which comes within € of the support of f.L must tend toward f.L in one direction or the other: simple geodesics within € of a closed boundary component of S - f.L are trapped into a spiral around f.L, and simple geodesics close to a boundary component of S - f.L which has half-cusps are trapped into some half-cusp. Therefore, if the component of A containing f.L is recurrent, it consists only of f.L. A geodesic laminatio~ on a surface may happen to wander around only on a small part of the surface. For any lamination A on S, denote by S( A) the smallest subsurface with geodesic boundary containing A. As a special case, if A is a simple closed curve, define S( A) to be the curve itself. If the components of A are f.LI' f.L2 ' ···' f.Ln' then S(A) = u S(f.Li)' and the S(f.L;) have disjoint interiors. THEOREM 5.5. Laminations cover. Let f: S ~ N be a map of a hyperbolic surface of finite area, not necessarily connected, to a hyperbolic manifold, which takes any cusps of S to cusps of N, and which is injective on the fundamental group.

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Suppose that A is a recurrent geodesic lamination on S such that f takes each leaf of A to a geodesic of N by a local homeomorphism (as with the pleating locus for a pleated surface). Let g: A ~ P(N) be the canonical lifting, and let Ill'·.·' Iln be the components of A. Then either (a) g is an embedding, or (b) the map f restricted to the disjoint union of the surfaces S(IlJ factors up to horrwtopy through a covering map p: US(IlJ ~ R

and U Il i is the pullback by p of a lamination Il on R which does embed in P(N). Case (a) is actually contained in case (b); it is stated separately because it is the typical case. In particular, note that when S is the boundary of an acylindrical manifold, (a) must hold for topological reasons. Our current application will be in the case of a pleated surface, but the surface S is really only present to control the topology of the map of A into N. In many ways, minimal geodesic laminations behave like simple closed geodesics; this theorem is an illustration. In the case that A is a union of simple closed geodesics, this theorem is quite clear.

Proof Consider any two points Xl' X 2 E A that have the same image under g. Then there are geodesics Ii: R ~ S of A through Xi such that go II = go 12 • Consider the diagonal lamination A X A C S X S. The product map II X 12 : R ~ S X S is a leaf of this lamination. Since A X A is compact, there is a neighborhood U C S X S of arbitrarily small diameter which it enters infinitely often. If t and s are any two times when II X 12 enters U, closed loops can be constructed, based at X 1 and at X 2: i = 1,2,

where EO i is a short arc connecting li( t) to IJ s). The loops g 0 0i are based at the point f( x) = f( y) in N, and they are nearly identical. If the diameter of U is less than half the minimum value for the injectivity radius of N in the image of A, then these loops are necessarily homotopic in N. We will now prove that g is a local embedding of A in P( N). Suppose that X and yare close to each other, and let 0 be an arc on S connecting them. Then the loops 0 * 02 and 01 are homotopic when mapped to N; therefore they must

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be homotopic in S. This implies that 11 [0, t 1 is close to 12 [0, t] for its entire length. This works in both the forward and backward directions, varying the choice of the points sand t, for arbitrarily long segments of Ix and Iy • Consequently, the two geodesics and the two points are identical. This proves that g is locally an embedding. Next we will show that g acts as a covering projection to its image. Define an integer-valued function n on A which gives the number of points in A which have the same image as x in P(N). Since A is compact and since x is not identified with any nearby point, n( x) is bounded. The function n must satisfy the semicontinuity condition

n( lim(xJ) :::; lim{ n(xJ) • • because points identified with x i remain well spaced out, and any limit point of the sets identified with Xi is also identified with Xi ' Therefore, the set { x In( x) ~ m} is closed. It is also saturated by A, so this set is a union of components. It follows that n is constant on each !Li' It is now easy to show that g is a covering projection; this is similar to the proof that a continuous one-to-one map between compact spaces is a homeomorphism. What remains to be proved is that the covering projection of !L extends to more of the surface, in fact, to the disjoint union of the S(!Li)' The assertion is trivial in the case of any of the !L i which are closed geodesics; so we may assume that A has no such components. In the remaining lamination, there is a finite set of special leaves of A, those leaves where A accumulates from only one side. These leaves can be identified topolOgically from A alone, because each end of such a leaf is asymptotic to another end of such a leaf; together they form the boundary of a half-cusp of one of the complementary regions. There are no other pairs of leaf-ends of A which are asymptotic. From this, it follows that identifications in the covering projection of A extend to a neighborhood of A in S, so that f factors through a covering projection at least on a neighborhood of A. Consider now any component A of S(!Li) - !Li' One boundary component of A is made of a chain of geodesics of !Li> the forward end of each asymptotic to the backward end of the next, which maps as a k-fold covering to its image, for some k. There are two possibilities for A: either it is an annulus, with another boundary component a boundary component of S(!Li) (or a cusp), or it is a disk. If A is an annulus, there is no problem: the covering projection defined on one of its boundary components automatically extends.

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If A is a disk, then the map factors up to homotopy through a branched covering to some disk, with one branch point at the center. (Keep in mind here that since N is hyperbolic, any element of finite order in 7Tl(N) is trivial.) This means the map from the entire component S(f. t j ) factors through a branched covering. But if the branched covering had actual branch points, the map could not be injective on 7Tl(S), contrary to hypothesis. Often there are topological conditions that rule out the possibility of the map of a surface into a three-manifold factoring through any non-trivial covering, even on a union of subsurfaces. If S is a hyperbolic surface of finite area and if f: S -+ N is a map to a hyperbolic 3-manifold which takes cusps to cusps, then f is incompressible if: (a) 7T 1( f) is injective. The map f is doubly incompressible if in addition: (b) Homotopy classes of maps (1, a1) -+ (S, cusps( S)) relative to cusps map injectively to homotopy classes of maps (1, aI) -+ (N, cusps(N»). (c) For any cylinder c: SI X I -+ N with a factorization of its boundary ac = fo co: a(SI X 1) -+ S through S, if 7T l (C) is injective then either the image of 7T 1(CO) consists of parabolic elements of 7T1(S), or Co extends to a map of S1 X I into S. (d) Each maximal abelian subgroup of 7T l (S) is mapped to a maximal abelian subgroup of 7TlN). We will use a slightly weaker condition, weak double incompressibility on surfaces where (d) is replaced by: (d') Each maximal cyclic subgroup of 7TlS) is mapped to a maximal cyclic subgroup of 7T 1(N). Condition (d) implies in particular that no curve on S is homotopic to a Z2-.. on S! are either endpoints of leaves of p, or cusps. We will show next that >.. maps injectively, by showing that these endpoints map injectively to the sphere at infinity for N. The set of parabolic fixed points for 'IT is) maps injectively to the set of parabolic fixed points for 'IT1(N), by the hypothesis (b) of injectivity of the relative hll1damental groups. A parabolic fixed point cannot be identified with an endpoint of a leaf of p when mapped to N, in virtue of the fact that the injectivity radius for N is bounded below in the image of p. Consider any two leaves II and l2 of p whose images in PN are asymptotic. Since p - P N is a covering map to its image and since p is compact, the map p - p - P N is also a covering map to its image. It lifts to a map p - P N, also a covering map to its image. This final covering must be trivial, on account of condition (c). Comparing it to the map p - PN, we see that there must be a fixed e such that every e-disk in p maps homeomorphically to its image (since p is compact). This says that II and l2 are already asymptotic in p. It follows that the map from >.. to P N is an embedding. In particular, the map A - P N is a local embedding. Define an integer-valued function n on A which says how many points of A have the same image in PN. As before, we have

n(liJ?{xJ) ~ lim{n{xJ). t

t

If a leaf of A limits on p, then n is 1 on the leaf, since n is identically 1 on p. The other possibility is that the leaf limits on cusps of S at both ends; n takes the value 1 in this case because of condition (b). We will now generalize this injectivity result to a uniform injectivity theorem, by considering geometric limits. For this, we will assume that the surfaces f: S - N are pleated surfaces. In fact, such a pleated surface always exists for any lamination Awhich can be mapped geodesically to N, but in order to keep the logic straight, we will assume that the pleated surface is given to us. The main case of the uniform injectivity theorem will be when A is the pleating locus for the surface, but the theorem needs to be stated with greater generality to handle the case that the surface does not actually bend at A. THEOREM 5.7. Unifonn injectivity. Let eo> 0 and A > 0 be given. Among all doubly incompressible pleated surfaces

f: S-N

of area not greater than A and laminations A c S which are mapped geodesi-

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cally by f, the maps g: A -+ P{N}

are unifonnly injective on the Eo-thick part ofS. That is, for every E > 0 there is a 8 > 0 such that for any S, N, A and f as above and for any two points x and yEA whose injectivity radii are greater than EO' if d(x, y) ~ E then d(g(x), g(y» ~ 8. Proof We will make use of the geometric topology on hyperbolic manifolds equipped with a base point and an orthogonal frame at that point. Two manifolds are close in the geometric topology if there is a diffeomorphism which is an approximate isometry from a ball of radius at least R in one manifold to the other manifold, taking base frame to base frame. No condition is given on the fundamental group. Here is a basic fact about the geometric topology: 5.8. Geometric topology is compact. For any El > 0, the space of hyperbolic manifolds of dimension n with base frame at a point whose injectivity radius is at least El is compact in the geometric topology. PROPOSITION

We can also put a geometric topology on pleated surfaces f: S -+ N in hyperbolic manifolds, where base frames are assumed to match up under f. The pleated surface f is close to f': S' -+ N' in the geometric topology if there are approximate isometries on balls of radius R from S to S' and from N to N' which approximately conjugate f to f'· 5.9. Pleated surfaces compact. The space of incompressible pleated surfaces f: S -+ N in hyperbolic 3-manifolds N, with base frame in S having injectivity radius ~ E l , is compact in the geometric topology. PROPOSITION

Note . Without the hypothesis of incompressibility, one would need a hypothesis on the injectivity radius of N. Proof The base point x 0 . of S has two loops through it of length not exceeding some constant C which depends on A and on El , such that the two loops generate a free group of rank 2. The image f( x o) therefore also has a pair of loops through it with the same property; it follows that the injectivity radius of N at f( x o) is greater than some constant E2 > O. The sets of potential surfaces S and potential 3-manifolds N are therefore compact in the geometric topology. Since the maps f are Lipschitz with Lipschitz constant 1, it follows that in any infinite sequence 1;: Sj -+ Nj of pleated surfaces there would be a subsequence where the domain surface, the range 3-manifold, and the maps 1; would converge to an object f: S -+ N. The

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only thing that remains to be checked is that the limit is actually a pleated surface. If XES, and if x is not an accumulation point of the Aj C Sj> then f in some neighborhood of x is a limit of isometries of a uniform-size hyperbolic neighborhood to a hyperbolic plane in N;, so that f is flat in a neighborhood of x. Otherwise, geodesic arcs in S which are mapped to straight arcs by f can be found as limits of arcs on Aj. It remains to show that the arc length of a geodesic arc a C S is preserved by f. We may assume that the Aj converge. It is clear that the length of the image of a cannot be greater than the length of a, since the approximations have this property. On the other hand, any geodesic arc a can be approximated by a geodesic arc whose length is the sum of the lengths of its intersections with the complement of A, since A has measure 0 on S. Each subarc in the complement of A is mapped to a straight line; so certainly its arc length is preserved; therefore, the length of the image is at least as great as the length of a. Suppose we are given a sequence of doubly incompressible pleated surfaces J;: Sj -+ Np together with geodesic laminations Aj mapped geodesically, and points Xi and Yj on Aj satisfying inj(xj)' inj(Yi) ~ fa and d(gj(x;), gj(Yj» -+ o. Theorem 5.7, Uniform injectivity will follow when we show that d(x j , yj) -+ O. By Proposition 5.9 Pleated surfaces compact, there is a subsequence such that the pleated surfaces with base frames chosen at either Xj or Yi converge, where the first vector in the frame is tangent to the lamination. There is a further subsequence so that the laminations Ai converge in the Hausdorff topology. Let f: S -+ N be the limit pleated surface, A the limit lamination, and g: A -+ P(N) the canonical lifting of f restricted to A.

5.10. Limit doubly incompressible weakly. The limit pleated surface N is weakly doubly incompressible.

LEMMA

f: S -+

Proof We have already seen that condition (a) (injectivity) of the definition holds, by Proposition 3.6, Non-elementary limit is discrete. We will prove (b) (injectivity of the relative fundamental group) by shOwing that the maps

y= [(1, aI): (Sj,(SJthin)]

-+

[(1, aI) : (N; , (Ni)thin)] =%

are injective. Let a and {3 be two arcs on Sj with ends in (SJthin representing two elements of Y. Suppose that they are mapped to the same element of %. This means that there are arcs y and 5 in (NJthin such that f( a) • y • f( {3 - 1) • 5 - 1 is null-homotopic in N. If a(O) and {3(0) are in the same component of (Sj)thin' then they can be connected by an arc 5'. Thus f(5'). 5- 1 is a loop in (N;)thin. By condition (d) (maximal abelian groups preserved) for J;, it is possible to modify 5' to make this

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element trivial, so that 8 is homotopic to f( 8'). On the other hand, if a(O) and {3(0) are in different components of (SJthin, then these components must be cusps, by condition (c) (no cylinders except in cusps) for 1;. A similar argument applies to a(l) and {3(1). There are now various cases, all easy. If the a(O) is in the same component as {3(0) and a(l) is in the same component as {3(1), then one has a closed loop in S to represent the" difference" of the relative homotopy classes, which maps to a trivial loop in N. By condition (a) for S, a and {3 must represent the same class in Y. If all endpoints are in cusps, then a and {3 must represent the same class in Y, by (b). Finally, if a(O) and {3(0) are in cusps and a(l) and {3(1) are in the same component of (S;)thin (possibly after relabeling to reverse endpoints), then the arc a * 8' * {3 - 1 maps to a trivial element of [(I, 1), (N;) thin], SO that [ a] = [{3] in Y. Next we check condition (c). Suppose that we have an incompressible cylinder g: SIX I ~ N in the limit manifold, with a factorization go of its boundary through f. Since arbitrarily large compact subsets of N are approximately isometric to subsets of N; for large i, we obtain similar cylinders C; in N;, and by a small homotopy we can make their boundaries factor though 1;. By condition (c) for 1;, either the boundary components go to parabolic elements of Si' in which case the same property holds in the limit, or there is a cylinder on Si with the same boundary. Consider the covering spaces of Si whose fundamental group agrees with that of the cylinder. If these covering spaces have core geodesics with length going to 0, then the boundary components of C are parabolic, and (c) is satisfied. Otherwise, the core geodesics have length bounded away from zero; two curves of bounded length representing the generator of the fundamental group of such a cylinder are connected by a homotopy of bounded diameter. Therefore, there is a homotopy between the limiting curves; so (c) is satisfied in the limit. As for condition (d'), suppose that a is a non-trivial element of '1T 1(S), and that some power k of {3 E '1Tl(N) gives f*(a). We represent this geometrically by loops a c Sand beN, together with a cylinder C giving the homotopy from b k to f( a). This configuration can be pushed back to the approximations Sp Ni and 1;. It follows, by condition (d') for 1;, that there is a loop Ci on Sj such that c jk =:: ai' where a j is the approximation to a on Si' By considering the cylindrical covering of the hyperbolic surface Si which corresponds to c i' we see that c i can be taken to have a length less than that of a i and to lie in a neighborhood of a i which has a diameter bounded in terms of k and the length of a i • Passing to the limit, we see that a is also a k-th power. This verifies (d'). To complete the proof of Theorem 5.7, Unifonn injectivity, we see by the lemma that we can apply Theorem 5.6, Laminations inject, to our limit of a sequence of pleated surfaces in which Xi and Yi are mapping to points in PNi

a

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which are closer and closer. We conclude that the limit points x = limjx i and Y = limjYi must be equal, since their images in PN are equal. Therefore, d(xj, Yi) -. O. This completes the proof of 5.7, Uniform injectivity.

Remark. For this proof to work, we needed both the notion of double incompressibility and of weak double incompressibility. In fact, the geometric limit of a sequence of doubly incompressible surfaces is often not doubly incompressible: what happens is that a Z-cusp may enlarge to a Z2 cusp, and a limit surface can well be only weakly doubly incompressible. Furthermore, there are hyperbolic 3-manifolds which admit a sequence of weakly doubly incompressible surfaces, all homotopic, such that laminations carried by these surfaces are rwt uniformly injective. The proof breaks down because Y -. % is not injective under this weak hypothesis. 6. Proof that AH(M) is compact Now that we are equipped with some information about the geometry of pleated surfaces, we are prepared to return to the proof of the main Theorem, 1.2, AH(acylindrical) is compact, in the general case. Suppose that we are given an acylindrical manifold M, together with a sequence of elements of AH( M), that is, homotopy equivalences h k: M - Nk to hyperbolic manifolds. We will extract a subsequence which converges. The first step is to replace hk by an ideal simplicial map fk such that for each component of aM, fk describes a pleated surface in a certain sense. Since fk cannot be defined on all of S, it is not possible for fk to be a pleated surface literally. Instead, we will construct pleated surfaces Pk: Sk - Nk together with a factorization of iklaM as a composition of an ideal simplicial map to Sk with the pleated surface,

fkl aM

Pk 0 i k. Triangulate M with a triangulation 'T which has exactly one vertex on each boundary component, and so that the edges that lie on aM represent distinct non-trivial homotopy classes. For each k and for each boundary component Bi' there is at least one edge of 'T on the component which goes to a hyperbolic element under h k • Passing to a subsequence if necessary, we may assume ~hat this edge e j is the same for all k. We will construct the pleated surface P k by spinning, much as in Figure 2.2, 2-Dimensional ideal simplicial maps. Begin with a piecewise-straight map of 'T to Nk such that the vertex of 'T which is on a boundary component of M maps to a point on the geodesic homotopic to the loop formed by e j • A metric is induced on B j : the distance between two points is defined to be the infimum of

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the length of images in Nk of paths joining the two points. This metric is a hyperbolic metric on B j except for one cone point, where the total angle is greater than 2'17, or in other words, where negative curvature is concentrated. Now homotope through piecewise-straight maps by pushing the vertex around the closed geodesic of e j • A family of metrics is obtained in this way, all hyperbolic with one cone point. The cone angle is bounded above (in light of the Gauss-Bonnet theorem), and in the limit it tends toward 2'17. (Most angles of triangles tend toward 0, except in the two triangles which have e j as an edge; an angle of each of these tends toward '17.) All the triangles except the two degenerating triangles which border e j converge in shape to ideal triangles, since fixed points of non-commuting hyperbolic elements in a discrete group are distinct. The gluing maps between ideal triangles also converge along edges which do not border one of the two degenerating triangles. In the limit, as the degenerating triangles get long and thin, two of their edges are glued together. This yields an assemblage of ideal triangles glued edge to edge homeomorphic to B j - e j • The resulting hyperbolic surface is mapped isometrically to Nk • The hyperbolic surface so constructed is not complete, however; in the metric completion, two boundary curves are added, whose length is the length of the closed geodesic of Nk homotopic to e j • Under the map to N k, the two boundary components are glued together. Performing the same operation in the domain, we get a hyperbolic surface homeomorphic to B j mapped as a pleated surface Pk into N k • We can choose a homeomorphism from Bi to the hyperbolic surface so that the composition into Nk is isotopic to hk' enabling us to think of the hyperbolic surface as a hyperbolic structure on B i • This construction gives the promised factorization of iklaM = Pk 0 i k as an ideal simplicial map followed by a pleated surface. We have a fixed lamination A of aM, such that the pleating locus is always a sublamination of A; the hyperbolic structure on aM varies with k. We return to the situation as described in Section 3, Shapes and limits, with a good sub manifold G and its frontier K which encode the way in which 3-simplices of 'T are degenerating in the sequence of ideal simplicial maps i k • We also have the projection p: K - Y describing the pattern of collapsing of K. Recall that Yo is the union of cells whose preimages have non-trivial fundamental groups with non-trivial images in '17 I (M). We will proceed in a similar way to Section 4, The deformation space for closed manifolds is compact, although for the final argument we will analyze the area of the hyperbolic structures on aM rather than the volume of Nk • For each edge of y - Yo whose preimage in K is a cylinder, we add a 2-handle to obtain a new manifold G I . Define KI c aG I , obtained from K U G I

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by adding the two bounding disks of each of these 2-handles, and also deleting a neighborhood of the core transverse arc for each component of p -ICy - Yo) which is a thin strip. If you prefer, you can picture adding 2-handles and semi-2-handles respectively to the homotopically trivial thin cylinders and thin strips. The two boundary components of a thin strip of " are very close to edges of 2-simplices (after isotopies to make the image area small), so that they map by i k to paths which are near edges of ideal triangles of aM - A, hence close to leaves of A. Therefore there are points on A near the two sides of the strip which are mapped to nearby points in PNk • These points can be taken to be in the thick part of the hyperbolic structure on aM, since thick parts of ideal triangles are automatically contained in thick parts of the hyperbolic surface. (Remember that since " is on the boundary of the good submanifold, the displacements of triangles which this thin strip intersects converge to bounded values; so we can choose nearby points on the two edges of the strip which both map to the thick parts of their triangles-provided the e for defining thick is small enough). Since M is acylindrical M is doubly incompressible. It follows from Theorem 5.7, Uniform injectivity that the two edges of each of these strips are close together in the domain of the pleated surface, that is, in the k-th hyperbolic structure on aM. In particular, the transverse arcs to these strips are homotopic to (unique) arcs on aM. Thus, we can construct a map gI: C I ~ M to take aC I - "1 to aM, and so that the image fk("I) in Nk has small area. For any component C of "1' gI*'lTiC) is abelian. If C is closed, glue on a 3-manifold Me and map it to M, as we did in Section 4, The deformation space for closed manifolds is compact. If gl*'lT I( aC) C 'lTi aM) is trivial, add a 2-handle to each component of ac, map it to aM, and finish as in the closed case. In the remaining case, there is some component of C which is non-trivial in 'lTiM). Byacylindricity, gl*'lTI(C) can only be Z, and C is homotopic reI ac to aM. Let Me = C X I, and map it into M to realize such a homotopy. Define C 2 = C I U e Me, and l~t

a

a

be a map extending g l' As in Section 4, C 2 has the property that the sequence of representations of its fundamental group converges. We will prove that the degree of g 2 is 1 by proving the equivalent statement that the degree of ag 2 : aC 2 ~ aM is 1. This degree is the same for each component of aM. The original surface " c M has one boundary component for each nondegenerate triangle T of aM. The image by i k of each boundary component (in the

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hyperbolic structure for aM) is a curve which goes around near the boundary of an ideal triangle of aM - A. The total geodesic curvature of such a curve is approximately 3'17, concentrated at three left U-turns. (See Figure 6.1, Boundary K.)

behavior

FIGURE 6.1. Boundary K. Top. The boundary of K on the surface Sk has one component going around the boundary of each non-degenerate ideal triangle of Sk. Bottom. The boundary of K1 is obtained from that of K by surgeries which correspond to the thin strips of K. The surgeries are" short" because of the uniform injectivity theorem, 5.7.

When we modified G and

K

to obtain Gland

K l'

and constructed the

a

map g 1: G 1 - M,ihe curves on M changed. The 2-handles which were added did not affect the boundary curves, but when the neighborhood of a transverse arc from each thin strip of " was deleted and pushed out to the boundary (adding the "semi-2-handles"), this operation did · affect the boundary curves. The effect was to pair each long straight segment of a boundary curve with an opposite long straight segment and perform a 0surgery, thereby· creating two new U-turns. The curve system after these operations is likely not to be embedded.

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The new U-turns are in an opposite sense from the U-turns coming from comers of triangles, so that the components of aK 1 have total geodesic curvature near O. We claim that each component of aK is homotopic on Sk to its geodesic on Sk by a homotopy of small mass (as k ~ 00). Perhaps the clearest way to establish this claim is to make use of the thin strips of K to define an alternate set of identifications of the ideal triangles of Sk' That is, the two sides of each of these thin strips run along near two sides of ideal triangles; identify these two sides by a map, approximately determined by the geometry of the strip, to define a new surface Rk (see Figure 6.2, Alternate identifications). The surface Rk

6.2. Alternate Identifications. An alternate set of identifications for the ideal triangles of Sk is approximately determined by the geometry of the thin strips of This gives a new surface Rk which is the interior of a surface with geodesic boundary components; on the new surface, the system of boundary curves of "1 becomes embedded. FIGURE

a".

might not be closed, and it might not be metrically complete. Topologically, Rk is equivalent to the surface obtained by identifying true 2-simplices in the given combinatorial pattern, then removing the vertices. Its metric completion Rk in general is a hyperbolic surface with either a cusp or a geodesic boundary component for each missing vertex (see [Th1, pp. 3.17-3.22]). Since we are free to make slight adjustments in the gluing patterns, we may assume that all missing vertices correspond to geodesics. In R k , the components of aK l map to simple, disjoint curves. Topologically, each of these curves circles around a neighborhood of one of the missing vertices. By the Gauss-Bonnet theorem, the punctured disk which gives the homology of any of these curves to its boundary geodesic has small area on Rk • For large k, there is a map of Rk to Nk, constructed by mapping neighborhoods of the edges of the ideal triangulation over the thin strips of K. This map can be made an approximate isometry without difficulty except perhaps on the union of the thin parts of the points of ideal triangles. By Theorem 5.7, Uniform

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injectivity, the edges of the thin strips are close to each other on the pleated surface 5 k; so this defines a map of Rk to Sk as well, also an approximate isometry except possibly in the union of the thin parts of points of ideal triangles. Each component of the union of the thin parts of the pOints of the ideal triangles is topologically an open cylinder, and the only invariant up to approximate isometry is the translation length of its holonomy. Therefore, by easy adjustments' the map can be made an approximate isometry everywhere. Within the union of the thick parts of the triangles of R k , each of the images of K 1 has a homology of small mass to a curve with uniformly small curvature. This homology pushes forward to a similar homology on 5 k. from K 1 to some curve of small absolute curvature on 5 k. Any curve of uniformly small curvature on a complete hyperbolic surface of finite area is homotopic to its geodeSiC or cusp by a homology whose mass is a constant near 0 times the length of the curve. Putting these two homologies together, we establish the claim. Finally, we add certain submanifolds Me to C 1, yielding C z, and we extend the map g1 over the Me's to construct a map g2: C z - M. The image of the fundamental group of each Me in N is abelian; so the map of ac z - ag 1 = aMe - K to Sk is homotopic to a geodeSic of Sk. The image 2-chain is therefore a sum of chains as constructed above. Consequently, the local degree of the map of aC 2 to Sk is approximately 1, therefore exactly 1. It follows that the degree of gz is 1. The proof of Theorem 1.2, AH( acylindncal) is compact is completed by an application of Lemma 4.3, Finite index convergence implies convergence, as in Section 4, The deformation space for closed manifolds is compact.

a

a

7. Manifolds with designated parabolic loci Often in the study of hyperbolic manifolds, it is important to specify information about cusps. In this section, we will generalize Theorem 1.2, AH( acylindncal) is compact to take into account such specifications. Data concerning cusps for hyperbolic structures on a manifold M n can conveniently be encoded by gi~ng a submanifold P n - 1 caM. P should have the follOWing properties: (a) The fundamental group of each of its components injects into the fundamental group of M. (b) The fundamental group of each of its components contains an abelian subgroup with finite index. (c) Any cylinder

c: (51 X I, as 1 X 1) - (M, P) such that

'IT 1( C)

is injective is homotopic rel boundary to P.

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(d) P contains every component of finite index.

aM which has an abelian subgroup of

A manifold M equipped with a sub manifold P of its boundary satisfying these conditions is a pared manifold. Condition (d) is in a sense unnecessary, since a component of the boundary of M which has an abelian subgroup of finite index is necessarily parabolic for any element of H(M). For the same reason, it is convenient to include it. In the three-dimensional oriented case, P is a union of tori and annuli. Define H(M, P) to be the set of complete hyperbolic manifolds M together with a homotopy equivalence of (M, P) to (N, cusps), where we represent cusps by disjoint horoball neighborhoods. Since H(M, P) c H(M), there are induced topologies AH(M, P), GH(M, P) and QH(M, P). A pared 3-manifold (M, P) is acylindrical if aM - P is incompressible and if every cylinder

c: (SI X I,

as l X 1)

~

(M, aM - P)

such that '/Tl(C) is injective is homotopic rel boundary to aM. With this definition, it follows that aM - P is doubly incompressible. Our main theorem generalizes to: THEOREM

7.1. AH(pared acylindrical) is compact. If (M 3 , p 2 ) is an

acylindrical pared manifold, then AH(M, P) is compact. This proof is practically identical to the proof in Section 6, Proof that AH( M) is compact, but we will repeat it anyway. This theorem is definitely not contained in the previous version. For example, an important case is when M is a handlebody or a surface X interval. In either case, it is easy to see that AH(M) is noncompact; yet even for these manifolds, there are many choices of Pea M such that (M, P) is an acylindrical pared manifold.

Proof Let {Nd be a sequence of elements of AH(M, P). As in Section 6, we can choose a triangulation and ideal simplicial maps fk: M ~ Nk such that fklaM - P factors as an ideal simplicial map composed with a pleated surface, fklaM - P = Pk 0 i k. The domain of the pleated surface Pk is a complete hyperbolic structure of finite area on aM - P, and after passing to a subsequence if necessary, the pleating locus is always contained in a fixed lamination A which cuts the surface into ideal triangles. Pass to a subsequence, as described in Section 3, Shapes and limits, so that edge invariants for non degenerate simplices all converge (perhaps to degenerate values). The good submanifold G then intersects aM as in Section 6, only in the

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complement of P. The image under i k of this intersection is the union of the main parts of the ideal triangles of aM - P - A. As before, a limiting gluing diagram for these ideal triangles is obtained by gluing according to the thin strips of the surface K, yielding a hyperbolic surface Soo ' By the uniform injectivity theorem, Soo has an approximate isometry to the k-th hyperbolic structure on aM - P, for large k. A manifold C I with a distinguished submanifold K I C C I is constructed just as in Section 6 by gluing on 2-handles and semi-2-handles to each portion of K which is the preimage of an edge of y - Yo' We can map C 1 to M so that a C 1 - K I maps to aM, and so that only a small change is made in the area of ike I - K1)· Each component of K 1 now corresponds to a curve circumnavigating a missing vertex of Soo' and the image of the fundamental group of each component of K 1 in the fundamental group of M is abelian. If a component C of K 1 is closed, it can be capped off with a manifold Me as before. Otherwise, C is homotopic to an annulus on aM. A product manifold Me realizing a homotopy can be glued on, giving a new manifold C 2 on whose fundamental group the sequence of representations converges, with a map (C 2 , aC 2 ) -+ (M, aM). The only difference in the picture is that aC 2 may hit annuli in P. Nonetheless, the area argument works precisely as before, to show that the map (C 2 , aC 2 ) -+ (M, aM) has degree 1. Therefore, '7T 1(C 2 ) -+ '7T I (M) is surjective; so the sequence of representations of '7T I( M) converges. This proves that AH( M, P) is compact.

a

aC

a

PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY

REFERENCES [Bon] [DH] [FLP] [Mor] [Mos] [M-Sl] [M-S2]

FRANCIS BONAHON, Bouts des varietes hyperboliques de dimension 3, Ann. of Math. 124 (1986), 71-158. A. DOUADY and J. HUBBARD, A proof of Thurston's topological characterization of rational functions, preprint Institut MiUag-Leffier, 1985. A. FATHI, F. LAuoENBACH and V. POENARU et al., Travaux de Thurston sur les surfaces, Asterisque 66-67 (1979), 1-284. JOHN MORGAN, On Thurston's uniformization theorem for three-dimensional manifolds, in The Smith Conjecture, Academic Press, 1984. C. D . MosTOw, Strong Rigidity of Locally Symmetric Spaces, Annals of Math. Studies 78 (1973), Princeton University Press. JOHN MORGAN and PETER SHALEN, Valuations, trees, and degenerations of hyperbolic structures, I, Ann. of Math. 120 (1984), 401 - 476. _ _ _ , Degenerations of hyperbolic structures, II: Measured laminations, trees, and 3-manifolds, to be published.

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246 [Seo) [Thl] [Th2) [Th3)

[Th4) [Th-K]

WILLIAM THURSTON

PETER Scon, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), 401 - 487. W . P . THURSTON, The geometry and topology of three-manifolds, Princeton Math. Dept. , 1979. W. P. THURSTON, On the geometry and dynamics of homeomorphisms of surfaces, preprint. W. P . THURSTON, ThretXIimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. AMS 6 (1982), 357-381. W. P . THURSTON, On the combinatorics and dynamics of iterated rational maps, preprint. W . P . THURSTON and S. P . KERCICHOFF, Non-continuity of the action of the modular group at Bers' boundary of Teichmilller space, pre print. (Received July 10, 1981) (Revised March 27,1986)

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HYPERBOLIC STRUCTURES ON 3-MANIFOLDS, II: SURFACE GROUPS AND 3-MANIFOLDS WHICH FIBER OVER THE CIRCLE WILLIAM P. THURSTON

ABSTRACT. The main result (0.1) of this paper is that every atoroidal three-manifold that fibers over the circle has a hyperbolic structure. Consequently, every fibered three-manifold admits a geometric decomposition. The main tool for constructing hyperbolic structures on fibered three-manifolds is the double limit theorem (4.1), which is of interest for its own sake and lays out general conditions under which sequences of quasi-Fuchsian groups have algebraically convergent subsequences. The main tool in proving the double limit theorem is an analysis of the geometry of hyperbolic manifolds that are homotopy equivalent to a surface. This analysis is also of interest in its own right. This eprint is based on the August 1986 version of this preprint, which was submitted, refereed, and accepted for publication; for reasons that are hard to fathom, I never returned a corrected version to the journal. I apologize for my long neglect of its publication, and I want to thank the referee for detailed comments which have been incorporated into the present eprint. No other significant changes have been made, except conversion to Jb.TEX, which has resulted in changes in numbering. The 1986 preprint was in turn a revision of a 1981 preprint; the various versions were fairly widely circulated in the early 1980's, and the results became widely known and used. Too many developments have intervened to be easily summarized, except for pointers particularly to the works of McMullen [McM96] and Otal [Ota96] that give alternative proofs for the main results of this paper and contain other interesting material as well.

1. This is a portion of the limiting sphere-filling curve for a fiber of the punctured torus bundle over 8 1 with gluing map R4 L, where R is a right-handed Dehn twist about a (1, O)-curve and L is a left-handed Dehn twist around a (0, I)-curve.

FIGURE

Date: August 1986 preprint --+ Jan 1998 eprint. This project has been supported by the NSF, currently # DMS-9704135. WILLIAM P. THURSTON

79

WILL IAM p, T HURSTON CONTENTS

O. Introduction 1. Quasi-Fuchsian groups 2. Geodesic laminations and pseudo- Anosov homeomorphisms 3. An estimate for the shapes of certain pleated surfaces 4. Existence of double limits of quasi-Fuchsian groups 5. Hyperbolic structures for mapping tori 6. On limits and limiting behavior of surface groups 7. Infinitely generated geometric limits References

2 4

7 11

16 18

19 22 31

O. INTRODUCTION

This is the second in a series of papers dealing with the conjecture that all compact 3- manifolds admit canonical decompositions into geometric pieces. The main purpose of the current paper is to prove Theorem 0.1 (Mapping torus hyperbolic). Let M3 be a compact 3-manifold (possibly 'With boundary) which fibers over 8 1 , and whose fiber is a compact surface of negative Euler characteristic. Then the interior of M either

(i): has a complete IHI2 X lR structure of finite volume, and can be described as a Seifert fibration over some hyperbolic 2-orbifold,

(ii): contains an embedded incompressible torus not isotopic to a boundary component, and splits along this torus into two simpler three-manifolds, or

(iii): (generic case) has a complete hyperbolic structure of finite volume. Cases (i) and (ii) are not mutually exclusive, but (iii) excludes the other two cases. If M fibers over the circle and has non-empty boundary, then the boundary is a union of tori and (in the non-orientable case) Klein bottles. A statement equivalent to the main theorem is that a 3-manifold admits a hyperbolic structure if and only if it is homotopically atoroidaL A 3-manifold is homotopically atoroidal if every map of a torus into the manifold which is injective on fundamental groups is homotopic to the boundary. This is not quite the same as the condition that every embedded incompressible torus is isotopic to the boundary; a manifold with the latter property is geometrically atoroidal. The product of a three-punctured sphere with a circle is an example which is geometrically atoroidal, but not homotopically atoroidaL The complement of an open regular neighborhood of any torus knot is another example. There are other results of independent interest in this paper. In particular, Theorem 4.1 is a strong general existence theorem for limits of surface groups acting in hyperbolic three-space, and it is the main ingredient in the proof of Theorem 0.1. After that proof is complete, futher results are proven concerning limits of surface groups. For example, Theorem 7.2 shows how to construct infinitely generated groups as geometric limits of surface groups of a fixed genus. It illustrates the fact that the measured lamination is not precisely the right structure for controlling limits of Kleinian groups. This paper depends directly on Theorem 5.7 of [Thu86], but it is otherwise independent of [Thu86]. Certain information about geodesic laminations, homeomorphisms of surfaces and limits of Kleinian groups will be assumed. This information is summarized in §1 and §2. The proofs can be gleaned from chapters 8 and 9 of [Thu79] together with [Fet a1.79], but an exposition will also be given in parts V and VI of this series. Theorem 0.1 was proven for the case when the fiber is a torus minus an open disk by Troels J¢rgensen. The proof of Theorem 0.1 is considerably different from the proof of the related result for other Haken manifolds. It is essentially a proof concerning surface groups. The existence of a hyperbolic structure on any 3-manifold which fibers over the circle is paradoxical: in the universal covering space of such a 3-manifold, the covering space of a fiber is necessarily a uniformly bounded distance from any image of itself by a covering transformation. This would at first seem to be inconsistent with hyperbolic geometry: two distinct hyperbolic planes, for instance, cannot have a uniformly

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bounded separation. Two horospheres can have uniformly bounded separation, but they have the intrinsic geometry of the Euclidean plane, which is not possible for the universal covering space of a fiber. A striking image is obtained by adjoining S~, the sphere at infinity for hyperbolic space. The action of the fundamental group of a hyperbolic 3-manifold of finite volume is minimal on S~, that is, it admits no closed proper invariant subsets. The closure of the universal covering of any fiber, intersected with S~, is a closed invariant subset, since the fundamental group of a fiber is normaL Consequently, the closure of the cover of any fiber contains the entire sphere at (Xl! (See figures 1 and 2.) The universal covering space of the interior of any compact surface of negative Euler characterist ic has a canonical compactification as a disk. Here is a more delicate fact about the fibers ( [CT85]; see also [Fen92]' [Min94] and [Thu9?] for generalizations of this theorem):

FIGURE 2. An approximation to a sphere filling curve which arises from the fiber of a hyperbolic three-manifold which fibers over the circle. The three-manifold in this case is the complement of the figure eight knot, and the fiber is the punctured torus bounded by the figure eight knot.

Theorem 0.2 (Sphere filling curve; Cannon and Thurston). Let M3 be a hyperbolic 3-manifold which fibers over S l with fiber F. Then the map i : P -----1- M extends continuously to a map

z: D2

-----1-

D3.

The boundary of D2 thus gives a sphere filling curve, or Peano curve, on S~. A second purpose of this paper is to develop some of the general theory of surface groups, especially those which share with the fibers of fibered hyperbolic 3-manifolds the property that their limit set is the entire sphere. The result of Cannon and Thurston above is not known in the general case. Theorem 0.1 will finally be an easy corollary of a general result (Theorem 4.1) which constructs limiting surface group actions. Precise statements along with their proofs will be given in §4, after some background material is reviewed in §1 and §2 and a key technical theorem, 3.3, is proven in §3. Theorem 0.1 will be derived in §5 from the result of § L

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4

After the proof of the main theorem, there are two more sections, which take up about half the length of this paper. These two sections, §6 and §7 continue with the analysis of the limiting behaviour of surface group actions. Some of this information will be used later in this series, but the real thrust of this material is toward the goal of a complete understanding of finitely generated Kleinian groups. The first examples of hyperbolic 3-manifolds which fiber over a circle were constructed and recognized by Troels J¢rgensen through his deep study of limits of quasi-Fuchsian actions of the fundamental group of the punctured torus acting on hyperbolic 3-space. See [J¢r77] for a description of some examples. An earlier attempt I made, before I met J¢rgensen, to wrestle with the question of the existence even of a Riemannian metric of negative curvature on any such manifold, led me to really look at homeomorphisms of surfaces and develop a geometric theory of their classification: see [Thub], a detailed exposition of which appears in Fathi, Laudenbach, POEmaru et al. [Fet al.79]. I would like to thank Troels J¢rgensen for opening up this subject, and for sharing with me his many visions and insights. I would also like to thank Dennis Sullivan for his relentless attack on the original unwritten, soon-to-be-forgotten proof of Theorem 0.1, from which he cut out parasitical cusps and other undesirable elements. See his Bourbaki seminar exposition [SuISO] for the first written account of Theorem 0.1. Much of the further refinement of the current version also was inspired by conversations with Sullivan. An earlier version of this paper was circulated in preprint form in 19S1. The outline of the proof of the main theorem is substantially the same as in the 19S1 version, but the execution has been streamlined, especially in the proof in §3 of Theorem 3. The last section of the current version, §7, is entirely new, and its main result is new; §6 also contains new material. The current version is essentially the same as a version from 19S6 that was refereed and accepted for publication. I dropped the ball and did not correct and return a final copy to the journaL This is the 19S6 version, translated to Jb.'I'EX, modified according to the referee's comments, but with little change otherwise. 1. QUASI- FUCHS IAN GROUPS We recall some notation from [ThuS6]. Let Mn be any oriented manifold and pn-l C aM be any submanifold such that the fundamental group of each component of P contains an abelian subgroup of finite index. Then the set H(M, P) (or simply H(M) when P = 0) is the set of complete hyperbolic n-manifolds, equipped with a homotopy equivalence f M ---+ N which sends Pinto horoball neighborhoods of cusps of N. Two such objects are equivalent if there is an orientation-preserving isometry between them in the homotopy class which makes the diagram commute. There are three significant topologies on H(M, P): the algebraic topology AH(M, P), the geometric topology GH(M, P), and the quasi-isometric topology QH(M, P). The maps

QH(M, P) --+ GH(M, P) --+ AH(M, P) are continuous. In the present paper we shall be dealing mainly with the case that (M, P) = (S X I, x I) where S is x I) is Fuchsian if its limit a compact surface, and S will denote its interior. An element N E H(S X I, set (that is, the limit set of the group of covering transformations of 1HI 3 over N) is a geometrical circle. An equivalent condition is that N contains a totally geodesic oriented surface homeomorphic to S. The subset of Fuchsian elements F( S) c H( S X I, as x I) is thus identified with H( S, as); the three topologies restricted to this set agree, and are the same as the Teichmiiller space T(S). The classical uniformization theorem further identifies Teichmiiller space with the space of conformal classes of Riemannian metrics of divergence type on S, that is, metrics for which every harmonic function is a constant. An element N E AH(S X I, x I) is quasi-Fuchsian if N is quasi-isometric to a Fuchsian manifold. This is equivalent to the condition that the limit set for N is homeomorphic to a circle. The three topologies also agree on the set QF(S) consisting of quasi-Fuchsian elements of AH(S X I, as x I). There is a fourth definition of a topology on H(M, P), namely the quasiconformal topology Q'H(M, P). A quasiconformal map between two metric spaces X and Y is a map f X ---+ Y for which there exists a constant K such that for all x EX,

as

as

.

hmsup r--+O

(SUPX'ES.(X)

where Sr(x) denotes the sphere of radius r about x.

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If f is a quasiconformal map and if K l is a constant which works in the above inequality for almost all x on S2, then f is a K 1-quasiconformal map. An t:-neighborhood of N E H(M, P) in the quasiconformal deformation space consists of those N' E H( M, P) such that the two actions of 7rl (M) on S~ are conjugate by an orientation preserving t:-quasiconformal homeomorphism of S~. The topologies Q/H(M, P) and QH(M, P) are homeomorphic, by a result proved independently by Tukia [TV82], [Tuk85[ and myself (cf. [Thu79]' chapter 11 for a partial discussion). The current paper does not depend logically on the homeomorphism of Q/H(M, P) with QH(M, P); we will work with Q/H(M, P). Nonetheless, to get a good intuitive feeling it is important to think about the quasi-isometric structure of hyperbolic manifolds. When M is a complete hyperbolic manifold, let Q/HO( M ) denote the component of M in Q/H(Mo, P), where M o is M minus horoball neighborhoods of its cusps and P is its boundary. We recall the fundamental deformation theorem for Kleinian groups; the strong version as stated here is due to Sullivan [SuI83]. For any Kleinian group f, we denote its limit set by L r and its domain of discontinuity by Dr.

Theorem 1.1 (Quasiconformal deformations; Ahlfors, Bers, ... , Mostow,. ,Sullivan). Let M be any complete hyperbolic 3-manifold 'IlJith finitely generated fundamental group. Suppose that every component of the domain of discontinuity of7rl(M) is simply connected. The conformal invariant of the quotient of the domain of discontinuity defines a homeomorphism conf: Q'Ho(M) --+ D ,, (M )/7rl (M). Theorem L 1 gives a canonical isomorphism of QF( S) with T( S) X T( S) since the quotient of the domain of discontinuity for a quasi-Fuchsian group has two components, each homeomorphic to S. If g, hE T(S), we denote as qf(g, h) the group determined (up to conjugacy) by 9 and h in this parametrization. Inversely, if f is a quasi-Fuchsian group, we denote the two conformal structures on the quotient surfaces of the domain of discontinuity by Cl (f), C2 (f) E T( S). The two components are distinguished by the orientation induced from S~: the first has positive and the second negative orientation. (The semantic convenience of orientation as a distinction between the two halves is one reason we are sticking for now with oriented manifolds.) It is easy to see that the subspace F(S) c AH(S X I, iJS X I) is closed. Note that F(S) is the diagonal in the product structure for QF(S). The whole of QF(S), on the other hand, is not closed. For example,

Theorem 1.2 (Bers slice). The closure in AH(S X I, for QF( S) is compact.

as X I)

of any slice x

X

F(int S) of the product structure

In this paper, we will find algebraic limits for sequences of quasi-Fuchsian groups which are tending to 00 not just in one factor, but in both factors. We will prove that such limits exists, provided the two coordinates go to 00 in directions far enough apart (thereby avoiding, for example, a sequence of different "markings" of a fixed group). The analysis will involve the geometry of the quotient manifold, so we need to relate the conformal structure at infinity to the geometry of the interior. We will now give two such relations; either of these is sufficient as a starting point for the rest of the paper. For any element IE 7rl (S) and any N E QF(S), let lb) denote the length of t he closed geodesic in N homotopic to I' Let length+ oo and length_ oo denote the lengths of the closed geodesics homotopic to I on the two quotient surfaces at infinity, using their Poincare metrics.

Proposition 1.3 (Poincare length bounds hyperbolic length [Ber70]).

and in particular Remark. Note that the first inequality becomes an equation when N is Fuchsian. A sequence of examples can be constructed to show that the constant, 2, in the second inequality is sharp. The idea is to "bend" a Fuchsian group along a closed geodesic whose length is near zero, with the bending angle near 7r. See Theorem 4.1 or §7 for constructions which shows that there are no inequalities of this form in the opposite sense.

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Proof of 1.3. Let D+ and D_ denote the positive and negative components of the domain of discontinuity. Then D+/ < I > and D_/ < I > are cylinders. Calculation (say in the upper half-plane) shows these cylinders are conformally constructed from rectangles whose dimensions are 7r X length+ oo (,) and 7r X length_ oo (,), by isometrically gluing the sides of height l L These two annuli fit inside the torus (S2 - L' This torus is conformally constructed from a 27r X l(f)-rectangle by isometrically gluing first the two sides of length lb), then the two circles of length 27r of the resulting cylinder (with an arbitrary twist). A standard extremal length argument (see for example [AS60]) gives the proposition. D A subset A of a complete Riemannian manifold is convex if every geodesic arc with endpoints in A is contained in A. Note that this definition depends strongly on the topology of the ambient manifold: for instance a closed manifold has no proper convex subsets. (This follows from the fact that the geodesic flow for such a manifold is recurrent). Clearly the intersection of any collect ion of convex subsets is convex. Any non-empty convex set must contain a possibly broken geodesic in every homotopy class. From this it follows that a convex set in a complete hyperbolic manifold contains every closed geodesic, since the broken geodesics in homotopy classes an, from no matter what basepoint, wind arbitrarily near the closed geodesic in the free homotopy class of a, if such a closed geodesic exists. It follows that any complete hyperbolic manifold N whose fundamental group contains at least one hyperbolic 1 element has a minimal non-empty subset which is convex; this set, C(N ), is called the convex core of N. C(N) is a co dimension 0 sub manifold except in degenerate circumstances, when it may be a submanifold of any lower dimension, possibly with boundary.

FIGURE 3. This is an approximate drawing of the convex hull of the limit set of a quasiFuchsian group. The group is a punctured torus group, that is, it is generated by two elements whose commutator is parabolic. This picture is in true perspective as viewed by an observer on the sphere at infinity of hyperbolic space. It may also be thought of as a more standard picture of the projective ball model for hyperbolic space, where the ball is fair ly large compared to t he field of vision, and t he limit set has been transformed by Mobius transformation so that it fits within the frame.

An alternate description of C(N) for a hyperbolic manifold is that it is the quotient of the convex hull of L 7rdN ) in the projective ball model for hyperbolic space intersected with hyperbolic space and quotiented by the action of 7rl (N). When N is 3-dimensional and C(N) is non-degenerate, the boundary of C(N) is a developable surface: it is a hyperbolic surface, homeomorphic to the quotient of the domain of discontinuity, and isometrically embedded in N. Since aC(N) is often not a C 1 submanifold, we cannot take the definition of an isometric embedding from standard differential geometry, and we should clarify what definition we use: A definition l We use the term 'hyperbolic' here in its inclusive meaning, to include the loxodromic case.

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which serves well is that an embedding is isometric if every geodesic in the surface is mapped to a rectifiable path of the same arc-length. See [Thu79]' chapter S, for more background.

Proposition 1.4 (Bounded distortion to infinity; Sullivan). There is a constant 1 < K < 00 such that for every complete hyperbolic 3-manifold N and any incompressible component S of aC(N), there is a K-quasiisometry (in the correct homotopy class) from S to the corresponding surface at 00 equipped 'Illith its Poincare metric. See [SuISO] for a brief proof, or [EM9S] for a detailed writeup which produces a concrete value for K. The reasonable conjecture seems to be that the best K is 2, but it is hard to find an angle for proving a sharp constant. 2. GEODESIC LAMINATIONS AND PSEUDO-ANOSOV HOMEOMORPHISMS

In this section we will review some background concerning the geometry of hyperbolic surfaces near 00 in Teichmiiller space, and the effect of a homeomorphism of a surface on its geometry. For details, the reader is referred to [Fet al.79[' [CBSS[ and chapters Sand 9 of [Thu79[. We will use geodesic laminations on surfaces, objects which are generalizations of simple closed curves in much the same way that real numbers are generalizations of the rational numbers. In fact, on the torus, a simple closed curve is described by its slope, which is a rational number, while the slope of a geodesic lamination is a real number. An alternative structure which serves much the same purpose is the measured foliation. Another very closely related object is the quadratic differential on a Riemann surface. These notions are also closely related to (and partly inspired by) the work of Nielsen on homeomoprhisms of surfaces. See [Nie86a] and [Nie86b], and [HTS5] or [GilS2] for a discussion of Nielsen's work and its relation to mine. Let S be any complete hyperbolic surface of finite area. A geodesic lamination A on S is a closed subset of S which is the disjoint union of simple geodesics. These geodesics are called the leaves of A. They may be either infinite simple geodesics, or simple closed geodesics. One way to think of a geodesic lamination is to pass to S = 1HI 2. Each leaf of 5. C 1HI 2 ~s determined by its two ends on S~. Therefore A is determined by the collection of pairs of endpoints of ..\, which is a closed subset G A of the open Moebius band M = (S~ X S~ - diagonal)/ 2 2, where 2 2 acts by interchanging coordinates. The condition that the set G A comes from a lamination is equivalent to two closed conditions:

(a): G" is invariant by 7rj (S). (b): For any two points (a l, a2) and (b 1 , b2 ) in G A , a l and a2 do not separate b1 and b2 on S~. A geodesic lamination is defined by a subset of the projectivized tangent bundle of a surface, namely the t angent spaces to the leaves of the lamination. Any Hausdorff limit of the tangent spaces of the leaves of a family of geodesic lamination itself comes from a geodesic lamination; we call this lamination the Hausdorff limit of the family. Every geodesic lamination on a hyper bolic surface is a Hausdorff limit of a sequence of geodesic laminations which have only finitely many leaves. By forming limits of geodesic laminations with only one leaf, it is not hard to see that if S is more complicated than a 3-punctured sphere, it has uncountably many geodesic laminations. A geodesic lamination on a hyperbolic surface always has zero Lebesgue measure on S (in contrast to the situation on a torus). A transverse measure for a geodesic lamination is a measure on G A invariant by 7rl (S). Described directly in terms of S, a transverse measure assigns a measure to each curve 0: in S transverse to A, in such a way t hat for any two curves 0: and j3 and any homeomorphism f : 0: ----7 j3 which takes In 0: to In j3 for every leaf I of ..\, f preserves the measure. One thinks of the measure as measuring the quantity of leaves crossed by the 0:. A geodesic lamination equipped with a transverse measure of full support is a measured lamination. The set of measured laminations has a topology, coming from the weak topology on measures on the Moebius band M. Denote this space, including the empty lamination with the trivial measure, by ML(S). The subspace of compactly supported laminations is denoted by MLo(S).

Theorem 2.1 (ML Euclidean). MLo(S) is homeomorphic to Euclidean space of dimension equal to that of T(S).

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Two nOll- trivial measured laminations /11 and /12 are projectively equivalent if their underlying laminations agree, and one measure is a constant times the other. The space of projective classes of measured laminations is denoted PL(S), and the projective classes of compactly supported measured laminations is PLo(S). As expected from Theorem 2.1, PLo is a sphere. Each simple closed curve defines a point in PLo(S). These points are dense. Laminations and measured laminations can easily be transferred from one hyperbolic surface to any homeomorphic surface by means of the set G A, using the fact that S~ is a topological invariant of a surface. Thus, the spaces ML(S), PL(S) and variations really depend only on the topological surface S. The notion of length of a simple closed geodesic extends readily to a continuous function

length: T(S)

X

MLo(S) --+ lR.

One way to define this extension is to define the length lengthsCu) of a measured lamination P on a hyperbolic surface S to be the total mass of the "product" of transverse measure with I-dimensional Lebesgue measure of the leaves of A. More precisely, the product measure is defined by its rule of integration, which is an iterated integral: in any small coordinate patch, first integrate along the leaves of A with respect to Lebesgue measure, then with respect to the transverse measure. When a simple closed curve is given its tranverse counting measure, this definition agrees with the length for the curve. The geometric intersection number i(a,,B) of two simple closed geodesics a and,B is the total number of intersection points, unless a and,B coincide, in which case i(a,,B) = O. This also extends to a continuous function i : ML( S) X ML( S) --+ lR:. The definition for i(Pl,P2) on S is defined as the total mass of a measure PI X P2 on S, where PI X P2 is the product of the two transverse measures in any small open set where the laminations are transverse to each other, and zero on any leaves which the two laminations have in common. Transverse intersections are automatically confined to a compact subset of S, so this intersection number is always finite. It depends only on the topological surface S.

Theorem 2.2 (Laminations compactify Teichmuller space). The union T(S) = T(S)UPLo(S) has a natural topology homeomorphic to a disk. In this topology, a sequence {9i} of hyperbolic structures in F(S) converges to a lamination P E PLo(S) if and only if there is a sequence {Pi} ----7 00 of measured laminations converging projectively to P such that for all 1'/ E MLo(S) for which i(I",I') '" 0,

.

hm i--+oo

length" (1") = L i(pi' pi)

Furthermore, lengthgo(Pi) ----7 00, but lengthgi (Pi) remains bounded. Moreover, there is a constant C such that

i(I", I'i) S length" (1") S i(I", I'i)

+ C length" (1").

In other words, intersection number with an appropriate lamination is quite a good approximation to hyperbolic length near infinity in Teichmiiller space. It is not always the case that for a measured lamination p; in the projective class of P lengthgi (p;) itself goes to zero. For example, consider a sequence of metrics obtained by the sequence of ith powers of Dehn twists about a single geodesic ,. These metrics converge to " yet, has constant length. The convergence of the 9i in this case is "tangential" to the boundary of Teichmiiller space, so that a sequence of Pi satisfying the conditions converge to, rather slowly in PLo(S). Without the conditions in the last two paragraphs of the theorem, the sequence of measured laminations i lengthgO (, h would serve. If ,B is any simple closed curve which intersects " then a certain constant multiple of the sequence of images of,B by the ith power of the inverse Dehn twist will satisfy the conditions for Pi except for the last paragraph. The laminations Pi are constructed in [Thua], in the course of development of cataclysm coordinates. This is quite related to the proof of Theorem 3.3 as well; in cataclysm coordinates, given a lamination A and a metric 9, a measured foliation :F is constructed such that the length along A in 9 exactly agrees with its transverse measure. The first stage of the polygonal approximations made in the proof of 3.3 show that the measured lamination v defined by :F serves well to estimate length.

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9

Note that ifthe Pi are normalized to converge to P as measured laminations, rather than simply projective laminations, then lengthqi(Pi) ----+ O. Pairs of laminations are often important in the theory of surfaces. A pair (p, v) of laminations is called binding if (a): they have no leaves in common, and (b): for each component U of the complement of the union of P and v, the metric completion of U is either a compact polygon, a polygon with one ideal vertex which tends to a cusp, or a punctured polygon, where the puncture is a cusp of the surface.

Proposition 2.3 (Lamination crosses binding). A pair of laminations (p, v) on a surface S is binding if and only if every simple geodesic on S has at least one transverse intersection 'IlYith a leaf of p or a leaf of v. A pair of compactly supported measured laminations (p, v) is binding if and only if for every compactly supported measured lamination A,

i(>., 1') + i(>., v) > 0 Proof of 2.3. If (p, v) is binding, then every leaf of p crosses at least one leaf of v (and vice versa). For suppose, on the contrary, that (p, v) is a pair of laminations such that p has a leaf I which doesn't meet v. Then the closure of I is a sublamination A of Pi if A meets v, then the two laminations have leaves in common, so the pair is not binding. Otherwise, since v is closed, A has a neighborhood not meeting v. This violates (b), so the pair is not binding. If (p, v) is binding, and if 9 is any simple geodesic on the surface, then if 9 intersects some component of the complement of the union of p and v, it is clear that 9 intersects one of the two laminations transversely, by (b). The only other possibility is that 9 is a leaf of p or of v, in which case it intersects the other lamination transversely, by the preceding paragraph. On the other hand, suppose that (p, v) is a pair of laminations such that every simple geodesic intersects one or the other transversely. Then, in particular, they have no leaves in common. If any region of the complement is not simply-connected, then its fundamental group must be parabolic, otherwise it would contain a simple closed geodesic. The metric completion of a region of the complement has boundary made up of segments of leaves of p and of v, so it is polygonal with possibly some missing vertices. There can be no more than one missing vertex, for otherwise, there would be a simple geodesic connecting one ideal vertex to the other. The neighborhood of any ideal vertex must tend toward a cusp of S, because if it recurred in compact subsets of S, one could form a limit, and construct a new simple geodesic with no transverse intersections with p or v. If the boundary has any missing vertices, then the region is simply-connected, or again there would be a simple geodesic tending to the ideal vertex at either end, looping around the fundamental group. The only remaining possibilities are those mentioned in (b). This establishes the claim of the first paragraph of the proposition. Now let us suppose that p and v are compactly supported measured laminations. From the previous condition, it follows that if A is a non-trivial measured lamination such that i(A, p) = i(A, v) = 0, then (p, v) is not binding. To establish the converse, recall (for instance, from Proposition 5.4 of [Thu86]) that every measured lamination is the finite union of minimal sublaminations. Therefore, if p and v have any leaves in common, the set of common leaves constitutes a finite union of minimal sets of both. It inherits a transverse measure say from p, to give it the structure of a measured lamination A for which i(..\, p) = i(A, v) = O. If a region of the complement of p U v violates condition (b), there are two possibilities. If the region has a fundamental group not consisting of parabolic elements, there is a measured lamination A supported on a simple closed geodesic and not intersecting p or v. If the boundary of the region has an ideal vertex, it cannot tend to a cusp, since p and v have compact support. Therefore, its limit set gives a minimal set in p or v, or both. It inherits a transverse measure from p or from v, thereby defining a measured lamination having intersection number 0 with p and v. D For later use, we need:

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Proposition 2.4 (Binding confinement). Let /11 and /12 be any two measured laminations which have no leaves in common, and for which S \ (P I U /12) consists of pieces which are simply connected or neighborhoods of cusps. Then for any constant C > 0, the set of hyperbolic structures 9 on S for which

and

lengthg (1' - 2) S C is compact.

Proof of 2.4. Two such laminations /11 and /12 have the property that for any measured lamination v, i(Pl ) v) > 0 or i(P2) v) > O. Applying this to the case that v ranges over a sequence Pi from 2.2, we see that no sequence of metrics for which both /11 and /12 have bounded lengths can approach the boundary of Teichmiiller space. D For a different, directly geometric proof of a more general proposition, see Theorem 6.3. Here is another fact for future reference: A homeomorphism ¢ 8 -----1- 8 is reducible if ¢ permutes some finite system of disjoint simple closed geodesics. The terminology is based on the fact that by cutting 8 along such a system of curves, one can reduce the study of ¢ to the study of a homeomorphism of a simpler surface. This is analogous to the notion of a reducible three-manifold, or (closer to home) a torus-reducible three-manifold.

Theorem 2.5 (Classification of surface homeomorphisms). Every homeomorphism ¢' of 8 is isotopic to a homeomorphism ¢ which either

(i): has finite order (ii): is reducible, or (iii): does not satisfy (i) or (ii), and preserves a unique pair of projective classes of measured laminations, /1s and /1u ' In the case (iii), for any point x E T(8), if xi- /1s and x

i- /1u ,

lim n(x)

~

1',

lim n(x)

~

I'n-

n--++oo

then

and n--+-oo

There is a constant ..\ > 1 such that ¢ multiplies the transverse measure for /1u by..\ and the transverse measure for /1s by 1/..\. Case (iii) is both most common and most interesting. The isotopy class of ¢ in case (iii) is called pseudoAnosov. There is a more geometric form for ¢, in this case, if the measured laminations are replaced by measured foliations. When this is done, ¢ becomes almost an Anosov homeomorphism: it is Anosov in the complement of a finite collection of points. From any homeomorphism ¢ of a surface 8 to itself, a 3-manifold called the mapping torus of ¢ can be constructed by first forming the product 8 X I, and then gluing 8 X {I} to 8 X {O} using ¢. The resulting 3-manifold M cf fibers over 81, and ¢ is called the monodromy of the fibration of M cf . Actually, only the isotopy class of ¢ is determined by the fibering M cf -----1- 8 1, and the isotopy class of ¢ determines M cf up to homeomorphisms which commute with M cf -----1- 8 1 .

Proposition 2.6 (Non-pseudo-Anosov mapping torus). If ¢ is a homeomorphism of a compact surface of negative Euler characteristic, then

S

(i): ¢ is isotopic to a homeomorphism of finite order if and only if the interior of M cf admits a complete ill[ 2 X

lR structure of finite volume

(ii): ¢ is isotopic to a reducible homeomorphism if and only if Mcf admits an embedded incompressible torus not isotopic to 8M, and

(iii): otherwise ¢ is pseudo-Anosov.

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Proof of 2.6. If ¢ is isotopic to a homeomorphism of finite order, then there is a hyperbolic structure on S which is invariant by ¢ up to isotopy. From this, a 1HI 2 X 1Ft structure is constructed on M cf . If the interior of M cf admits a 1HI 2 X IR structure, that structure defines on M cf a codimension two transversely hyperbolic foliation. It was shown in [Thu79] that such a foliation either describes a Seifert fibration, or the manifold fibers over the circle with fiber a torus. Clearly the former case obtains. Consider the projection of the fiber S of the fibration over S l to the base of the Seifert fibration. There is an induced map from the fundamental group of S to the fundamental group of the base orbifold. The image is a normal subgroup. The quotient group must be finite, since there is no finitely generated normal subgroup of the fundamental group of a hyper bolic 2-orbifold with quotient group Z. The map is also injective on the level of fundamental group. This implies it is homotopic to a covering map over the base orbifold. Then ¢ is isotopic to a deck transformation, so it has finite order. If ¢ is isotopic to a reducible homeomorphism, then the trace of a reducing curve, dragged around by ¢ until it comes back to itself, with the same orientation, gives an incompressible torus. Conversely, if there is an incompressible torus, consider the projection of the torus to the circle. It can be perturbed to have Morse-type singularities. A standard 3-manifold argument shows that the singularities can all be eliminated by an isotopy of the torus, so that the torus becomes transverse to each fiber. The intersection with one of the fibers defines a family of reducing circles for some homeomorphism isotopic to ~

D 3. AN ESTIMATE FOR THE SHAPES OF CERTAIN PLEATED SURFACES

Let S be a compact surface and N E AH(S X I, as X I) be a hyperbolic 3-manifold. For purposes of reference, fix a complete hyperbolic structure of finite area on S. Most non-trivial simple closed curves (except parabolics) are realized as closed geodesics in N. Similarly, most geodesic laminations on Shave realizations in N. In particular, the realizable laminations contain an open dense subset of PL(S). This theory is developed in chapters 8 and 9 of [Thu79]. In the special case N E QF(S), all geodesic laminations on S are realized in N. The quantity lengthN(p) of a realizable measured lamination p in a hyperbolic 3-manifold N is defined similarly to t he length of a laminat ion on a surface, as the total mass of a "product" measure formed from I-dimensional Lebesgue measure along the leaves with transverse invariant measure. To put it another way, in a local coordinate system, the transverse invariant measure is a measure on the space of leaves of Pi the integral of a continuous function f with respect to the product measure is defined by first integrating f with respect to Lebesgue measure on each local leaf, then integrating with respect to the transverse measure. A further complication is that p is only mapped into N, but not necessarily embedded: one should do the computation in the domain, by pulling back Lebesgue measure from N .

Proposition 3.1 (Length continuous). The length of laminations is a continuous function on the set R PL(S) X AH(S X I , iJS X I) of realizable laminations.

c

The details of this and other basic facts about measured laminations in two and three dimensions will be proven in part VI of the series. A geodesic lamination ..\ of a surface S is maximal if it is maximal among all geodesic laminations, which is equivalent to the condition that each region of S - ..\ is isometric to the interior of an ideal triangle. Every lamination is contained in a maximal lamination, although a lamination may be maximal among measured laminations without being maximal in the current sense: to extend it to a maximal lamination, one must often make do with a lamination admitting no transverse measure of full support. Every surface admits maximal laminations which have only finitely many leaves. See for example Figure 2.1 of [Thu86] for a way to construct laminations by "spinning" triangulations. If as i- 0, the situation can be simplified by allowing no closed leaves on S. Then..\ becomes the I-skeleton of an "ideal triangulation" of S. A lamination ..\ with a finite number of leaves is realizable provided all its closed leaves are realizable. If ..\ is maximal, then it determines a pleated surface

fA: FA --+ N, where FA is a complete hyperbolic structure on S and fA is an isometric map of FA into N which folds or pleats, at most, along ..\.

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If A is a maximal lamination with only finitely many leaves and p is a measured lamination of compact support, we will define a quantity

a(>.,I') which in some sense measures the complexity of p relative to A. We will define a(>.) p) first in the case that p is a simple closed curve. If p is a leaf of A, then we define a( '\ p) = o. Otherwise, there is at most a countable set of intersections of the two laminations. If x and yare two intersection points with no intervening intersections along 11, then the leaves of A through x and yare two sides of an ideal triangle of A (since A is maximal, by hypothesis). These leaves are therefore asymptotic on either one side of 11, or the other. If x, y, and z are three successive intersection points along 11, and if the leaves of A through x and yare asymptotic on the opposite side of 11 to the leaves of through y and z, then let us call y a boundary intersection.

A are marked in the universal cover, 1HI 2 • Even where leaves of A accumulate, the boundary intersections are isolated. FIGURE 4. The boundary intersections of a geodesic 11 with a lamination

It can also happen that a point y of intersection of A and 11 is an accumulation point of leaves of A. In such a case, the leaf of A through y is a closed leaf, and the leaves of A spiral to the closed leaf on both sides. We will call such an intersection point a boundary intersection if the directions of spiraling on the two sides are different. Note that both kinds of boundary intersection are isolated, and so there is a finite number of boundary intersections in alL We define a(A, 11) (in the case that 11 is a simple closed curve) to be the total number of boundary intersections of A and 11. In other words, a(A, 11) measures the total number of times the direction of asymptoticity of the leaves of A changes as you go around 11. When 11 is a general measured lamination, the situation is much the same. Designate a point y of transverse intersection of A and 11 a boundary intersection if the direction of asymptoticity of the leaves of A changes at y. Then compute a(.-\, 11) as the total l1-transverse measure of the set of boundary intersection points.

Proposition 3.2 (Continuity of alternation number). For a fixed maximal finite lamination A, the alternation number a(.-\, 11) is a finite-valued, continuous function of the measured lamination 11. Proof of 3.2. Associated with a measured lamination 11 is a measure M(I1) on the projective bundle W(S) of S, defined as the product of I-dimensional Lebesgue measure with the transverse measure. We will construct a continuous function C(A) such that for all 11,

a(>',I')~

r

JeIS)

C(>.)dM(I').

This implies that a(A,I1) is continous in 11. The idea for constructing C(A) is that instead of counting boundary intersections directly, we can spread the contribution out so that it becomes an integral over a larger portion of W(S).

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Every isolated leaf I of>., lifted to the universal cover of the surface, separates two ideal triangles. The union of the two triangles forms an ideal quadrilateraL If a geodesic m crosses I, it is a boundary intersection iff m passes through opposite sides of this quadrilateraL Choose a continous function on the unit interval with integral 1 which is 0 at the two endpoints of the intervaL By scaling, this transfers to a function whose integral is 1 on an arbitrary intervaL For each quadrilateral obtained by removing an isolated leaf of A in the universal cover, and for each geodesic m which intersects opposite sides of the quadrilateral, scale this function to the intersection of m with the quadrilateral, and lift to W(1HI2). Define a function in W(1HI2) by adding up over all quadrilaterals and all geodesics. For any point of W(1HI2), there are at most 3 contributions, so this gives a well-defined function on W(1HI2). It is continuous, because when a sequence of geodesics m which cross opposite sides of a quadrilateral converge to one that doesn't, the length goes to 00 so the function tends to O. This function is invariant by deck transformations, so it gives a continous function on S. Similarly, if a is a closed geodesic of A for which the spiraling is in opposite directions on its two sides, we add a contribution for each geodesic m which crosses a, supported on the intersection of m with an f neighborhood of a. The result, after adding all the contributions, is a continuous function C(A), with the desired properties. D The following theorem gives an estimate of how efficiently certain homotopy classes can be represented on a pleated surface with finite pleating locus:

Theorem 3.3 (Efficiency of pleated surfaces). Let S be a fixed compact surface, possibly 'UJith boundary. For any f > 0 there is a constant C < 00 such that the follo'Wing holds: Let A be any finite maximal lamination on S. Let N be any element of AH(S X I , x I). Suppose that no closed leaf of A has length less than f in N, so that in particular, no such leaf is parabolic, and cons equently a surface f A : FA -----1- N exists uli,ich is pleated along A. Let p E MLo(S) be any compactly-supported measured lamination which is realizable in N. Then

as

This inequality does not generalize to an inequality for lengths on more general pleated surfaces. The finite combinatorial complexity of A is cruciaL When p is a not necessarily realizable lamination in MLo(S), define lengthN (p) to be the lim inf of lengths of nearby realizable laminations. If p has only one component, then it follows from [Thu79] or [Bon86] that lengthN (I') ~ o.

Corollary 3.4 (Efficiency for unrealizable laminations). Theorem 3.3 works for arbitrary p E MLo(S) if lengthN (I') is replaced by lengthN (I'). Proof of 3.4. Apply the theorem to an appropriate sequence of realizable laminations converging to p.

D

Proof of Theorem 3.3. In view of proposition 3.1, it suffices to prove 3.3 in the case that p is a simple closed curve, since simple closed curves are dense among realizable laminations and the inequality is homogeneous in p. The idea of the proof is to represent a simple closed curve p by a polygonal path on F A whose number of sides is O(a(\ 1')) and which follows along the leaves of ,\ except for O(a(\ 1')) of its length. By an application of the uniform injectivity theorem [Thu86, Thm. 5.7], one finds that this polygonal path on FA, when mapped to N, cannot "double back" very much, so that it cannot be too inefficient. From now on in t he proof, p will be a simple closed geodesic on FA' We shall homo tope p to a polygonal curve. We may assume that p is not a leaf of >., for in that case there is nothing to prove: its length in N equals its length in the pleated surface. We may also assume that p is not a geodesic shorter than f on S, for if it is not a leaf of A, then a( >., p) :2: 2, and the inequality is trivial.

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Otherwise, a(.-\ !1) > 0, and there is a chain of length a(.-\ p) consisting of the leaves of A on which there are boundary intersections with p. Successive leaves in the chain are asymptotic) and if the asymptotic pairs of ends of leaves are replaced by short jumps, one obtains a curve homotopic to p. It is easiest at this point to describe the picture in the universal cover of the surface, ill[ 2. Let R be a strip of constant width, say .5, about the geodesic fl. For concreteness, we can consider the picture in the Poincare disk model, with jJ being the horizontal diameter of the disk. We have an infinite chain gi [i = -00 . . . (Xl] of geodesics connecting the two endpoints of jJ, with endpoints of successive geodesics meeting at S~. Each 9i crosses R. Let ai be the lower endpoint of R n gi, and bi the upper endpoint. Let Ai and Bi be the (hyperbolic) perpendicular projections of ai and bi to ji. Then Ai < A i+1 , Bi < B i+1 , and IA i + 1 - B i l is bounded by an a priori constant. If B i > A i+ 1 , let Xi+ l be the point of gi+ l whose perpendicular projection to ji is Bi; otherwise, let Xi+ l = ai+l·

.. FIGURE 5. An arbitrary simple closed geodesic p on a pleated surface can be approximated

by a polygonal path with 2a(A,p) sides, which follows leaves of A except on O(a(\p)) of its length, and whose length exceeds that of p only by O(a(A,p)). The path simply follows appropriate segments of the sequence of asymptotic geodesics given by its boundary intersections with A. The possibly long segments of this polygonal path which follow leaves of A are mapped efficiently in any A-pleated surface. By making further adjustments to avoid problems with the thin parts of the surface, the polygonal path can be forced to map efficiently as a whole on a A-pleated surface, up to an additive constant which is a bounded multiple of a(,\, /1,). Define a polygonal path

mc

JH[ 2 to consist of the the geodesic segments

This polygonal path consists of intervals of the leaves gi, interspersed with jumps of bounded length from one leaf to another. The projection to FA is a polygonal curve m whose length exceeds the length of p only by a constant times a(A, p). Furthermore, m lies on leaves of A except for a portion of its length which is a constant times a(A , p). Next, we will make a homotopy of m to a new curve n wit h similar properties to m, but which also has the property that none of the jumps between leaves of A occur in the thin part of FA. We may assume that p is not a short geodesic The curve m enters only those thin parts of FA which are neighborhoods of short geodesics. Since, by hypothesis, no closed geodesics of A are short, each leaf of A which enters such a thin part exits on the opposite side. All leaves of A which enter the thin part remain fairly close together and roughly parallel for the entire intersection with the thin part. For any interval of m which crosses a thin part of FA, we can inductively replace the first segment on a leaf in the thin part and the first jump in the thin part by a jump just outside the thin part, and a longer segment on a leaf. Each such step increases the length of the curve by at most a constant. The total number of steps is at most a( A, p), so in the end we obtain a polygonal curve n with the desired properties. The image FA(n) of n in the three-manifold N is probably not polygonal, but we can homotope it to a polygonal path p by homotoping the image of each jump segment of n to its geodesic in N. The length of

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p differs from the length of p by at most a constant times a(,\ p). The polygon p has the same number of sides as n: 2a( '\ p). Let p' be the geodesic of N homotopic to FA (p). Construct a pleated annulus A representing the homotopy from p to p' by spinning around p': concretely, for each side of p, form a triangle based on that side with third "vertex" spiralling infinitely around pl. Form A by gluing all these triangles together, and completing with pl. The area of each triangle is less than 7r, so the area of A is less than 27ra(.-\, p). In the intrinsic hyperbolic metric of A, the portion of the p boundary component of A which admits a regular neighborhood of width f in A is bounded by area(A)/f, which is less than a constant t imes a(.-\, p). Any remaining length of p consists of segments on leaves of .-\ in N which run close and nearly parallel to some other portion of the boundary of A. We claim that the nearby portion of the boundary of A is part of p', not part of p, except possibly for a total length bounded by a constant times a(.-\, p).

FIGURE 6. A pleated annulus A constructed between a polygonal curve m and its closed geodesic has area bounded by the number of sides. If the polygonal boundary component is large compared to the number of sides, this means that much of its length must be nearly parallel to another portion of the boundary of A. We have constructed the polygon so that the length of m is in excess of a constant times a( \ m) consists of leaves of'-\. The leaves of .-\ are forced apart in light of the the uniform injectivity theorem so the lengths of the two boundary components differ by only O(a(\ m)).

For any portion of p which is not in FA thin' this will follow fairly directly from Theorem 5.7 of [Thu86]. In fact, if there is any short arc a on A ~onnecting two segments of p on leaves of .-\ but not in FA t hin' a would be homotopic to a short arc on S, by the uniform injectivity theorem. From the picture on' A, a would also be homotopic to an interval of p and of n, so it gives a way to shorten n. This is possible at most for a part of n of length less than a constant times a(.-\, p). The polygonal curve n on FA has the property that if there is any interval I of n with endpoints in FA thin which is homotopic reI endpoints to FA thin' then this interval is contained on a single leaf of'-\. The 'map from components of the thin set of FA to the thin set of N is injective. Therefore, two distinct segments of p which are in the thin part of FA cannot be close together on A. Therefore, all but a constant times a(.-\, p) of the length of p runs close to p' on A. This implies that

lengthp, (I') S length(p) S lengthN (I') + Ca(>., 1'), D

as desired.

Theorem 3.3 is false without the stipulation that closed leaves of .-\ have length at least f in FA, or equivalently, in N. It is possible to construct examples where there is a pleated surfaces which has a nearly

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180 0 fold along a very short closed leaf of A. Any geodesic of FA which intersects such a fold can be shortened considerably in N. The limiting case of complete inefficiency is for surface groups which have an accidental parabolic. Take A to be a lamination having a closed leaf whose homotopy class is parabolic. Then there is no actual pleated surface FA, but there is still a type of "pleated surface with nodes", which goes off to 00 at the parabolic curve. Any homotopy class passing through the parabolic curve has infinite length on FA, but finite length in N. Here is a slightly stronger version of the theorem, which might be useful sometime:

Theorem 3.5 (Restricted efficiency of pleated surfaces). Let S be a fixed compact surface, possibly 'IlYith boundary. For any E > 0 there is a constant C < 00 such that the follo'llJing holds: Let A be any finite maximal lamination on S. Let N be any element of AH(S X I) x I). Let Al be the measured lamination which is the sum of the closed leaves of A of length less than E in N, and let A2 be the union of closed leaves of length zero in N, that is, closed leaves which are parabolics. Then there is a pleated surface fA FA -----1- N pleated along A \ A2, where FA is a complete hyperbolic structure on S - A2' For any P E MLo(S) such that i(p) AI ) = 0,

as

lengthN (I') S lengthp, (I') S lengthN (I') + Ca(>., 1'). Proof of 3.5. The proof of 3.3 works also for this statement. The absence of short geodesics of A was used only in the construction of a polygonal approximation to the curve p) such that no jumps take place inside N thin . The difficulty is averted if p does not enter the components of FA thin surrounding short closed leaves of A. But any simple closed geodesic remains entirely outside such a thin neighborhood) unless it crosses it. D Remark. In fact) both Theorem 3.3 and the more general version) Theorem 3.5 can be extended to curves on the surface which are not simple) and to more general measured laminations tangent to the geodesic flow in W(S)) which do not project to simple laminations on S. The alternation number makes just as much sense) and the polygonal approximations work equally welL A non-simple geodesic may enter the neighborhood of a short geodesic or a neighborhood of a cusp) wind around many times) and then exit through the boundary component of the thin set where it entered. In this case) one can again push all the jumps between leaves of A in a polygonal approximation out of the thin set) one by one) adding only a bounded multiple of the alternation number. 4. EXISTENCE OF DOUBLE LIMITS OF QUASI-FuCHSIAN GROUPS

Using the results of §3) we can now prove a good existence theorem for limits of surface group actions in 1HI 3 In particular) the theorem will show the existence of many doubly degenerate groups) that is) surface groups whose limit sets constitute the entire sphere. We will construct such groups as limits of quasi-Fuchsian groups where the conformal structures on the two components of the domain of discontinuity go to infinity in different directions.

Theorem 4.1 (Double limit theorem). Let p and p' be any two laminations in MLo(S) satisfying the condition that for all v E MLo(S), i(l', v) + i(I" , v) > o. Then for any sequence {(9i, hi)} zn T(S) X T(S) converging to (1',1") in 1'(S) X 1'(S), the sequence of quasi- Fuchsian groups {qf(9i, hi)} has a subsequence whose associated manifolds 1HI 3 / qf(gi ) hi) converge algebraically to a point Noo E AH(S

I, iJS

X

X

I)).

Proof of 4.1. First we will prove the theorem in the case that S has non-trivial boundary) since this lacks some of the complications. In this case) we can choose a lamination A of S which is the I-skeleton of an ideal triangulation. Represent the projective classes of p and p' by measured laminations of the same names. By Theorem 2.2) there are sequences of measured laminations Pi -----1- P and pi' -----1- p' such that

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and

lengthc,(I., 1') ~ 0 for all >. E K(f). We can use a normalized version of this to define neighborhoods of ,6,(f). That is, for any lamination >. E K(f), the function n,,(I') ~ i(>., 1')/ length s (l') depends only on the cla.ss II'I E PLo(S). A neighborhood for ,6,(f) is defined by picking a finite collection of such .-\, and requiring that the functions n A have value less than f. The proof works by induction. Let f1 be as hypothesized. We start with the last i such that i(ai' f1) > o. The image of f1 by a large power of this Dehn twist is near ai, by 7.3. Define fj to be the set of ak where j :s; k, and for no j :s; 1< k is i(al, ak) > o. Assume by induction on i - j that the image f1j of f1 by the j through n terms of the composition is near ,6,(fj ). Consider the result of of applying the q - 1st power of the Dehn twist about aj- l to f1j. If the intersection number of f1j with aj_l is not too small compared with the length of f1j, then according to 7.3 the image will be close to aj_l. Otherwise, f1j is already near the simplex f j _ 1 . The length of a lamination can be estimated to within a constant factor as the sum of the intersection numbers with any finite collection of simple closed curves C such that Is(C) = S. Choose C so that it has at least one element which intersects any component of f j _ 1 but none of the others. Then, if f1j is close to the subsimplex of f j - 1 opposite aj, we see that Dehn twists about aj cannot diminish the length of f1j beyond a bounded factor, since these twists do not affect certain of the intersection numbers which at the beginning contribute a significant fraction of the estimate of the length of f1j. Dehn twists about aj do not affect the intersection numbers with>' E K(f), so they cannot increase the functions n A beyond a bounded factor. We conclude that f1j-l is near ,6,(f j _ 1 . The proof of the assertion for a metric 9 is similar, but slightly simpler, since we can start the induction at the last term in the composition. The initial step follows form 7.3. One can measure a neighborhood of a lamination in compactified Teichmiiller space in the analogous way. The easiest way to deal with the normalization is to choose a set of curves whose least subsurface is all of S, and normalize a metric by multiplying with a constant which makes the sum of their lengths equal to L A more elegant way is to extend the function lengthq (>.) to a function lengthq(h), where h E T(S), with the aid of "random geodesic" for 9 and for h. The length of the h-random geodesic in the metric 9 is the desired function. This quantity is also the intersection number between the random geodesics in the h metric and the 9 metric, so it is symmetric in 9 and h. See [Thua] for details. The subsequent steps are identical to the case for a lamination, since the image of 9 is near a lamination.

D The first phase of the proof of Theorem 7.2 will be to show that there is a subsequence such that Pk converges. Before proceeding in general, it is worth describing how this phase of the proof goes in some relatively simple cases. First consider the case that S is a punctured torus. Then -< is actually a linear ordering. Consider the surface at height n + 0.5 -label this cut Kn. By 7.4, the conformal structure of the top component of the domain of discontinuity for f(qj, Kn) converges to a(n + 1), provided we choose qj appropriately, and the conformal structure on the bottom component converges to a(n). The double limit theorem applies, yielding a subsequence such that the representations restricted to S Kn converge. By a diagonal process, we reach a subsequence so that these groups converge, for all n. For any two consective integers, the image of the fundamental group of the two surfaces in Me intersect on the fundamental group of a thrice-punctured torus. This implies that the fundamental group of their amalgamation also converges (since the intersection is non-elementary). Inductively, it follows that the representations of the entire fundamental group of Me converges.

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Another special case which is fairly easy to handle is the case of an arbitrary surface 5 when the the sequence of curves C is either finite or semi-infinite. Suppose, for example, that the index set is the positive integers. Let m be the set of minimal elements of - t > to, the probability for the possible choices of tangent vector Xto at time to is proportional to the degree of Z at the base point of X to . This transverse measure gives rise also to a measure on GF, defined locally as the product of Lebesgue measure along geodesics with the transverse measure. Proposition 1.1 (Exponential map injective). Let BS(M, P) be the covering space induced from the universal covering if of M. Let x E BS(M, P) be an arbitrary point. The exponential iimap )) at x is injective, in the sense that if [1 and [2 are any two geodesic intervals beginning at x which have the same endpoints, then they coincide. Proposition 2.1 is a related statement of a more topological form. Proof of 1.1. If there are geodesics 1, and 12 on BS(M,P) which have identical endpoints, but are not identical, push them forward to KS(M, Pl. Pick a maximal interval of the image of 1, whose interior does not intersect the image of 12 . This configuration gives rise to a rectangle in (M - W), with two of its sides on 8M and the other two on the characteristic

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----.--~~

-' FIGURE 4. Geodesics on non-Hausdorff surfaces can coincide for a time until they reach a branch point, then separate. The space GF is the global description of this behavior, consisting of all possible unit tangent vectors to BS(M, P) together with a bi-infinite geodesic through the vector. Proposition 1.2 limits the rate of branching by the growth of volume in He. This limit in turn limits the total length of the branch locus. submanifold W. Every such rectangle can be deformed, reI W, into W, by the theory of the characteristic submanifold. (See [Joh79] or [JS79].) This contradicts the fact that 1, is a D hyperbolic geodesic. The locus of endpoints of all geodesics of length R emanating from a point x E BS is a branched i-manifold CR(x). In an ordinary hyperbolic surface, this curve is the sphere of radius R, and its total length is 211" sinh R. For a hyperbolic structure on BS, CR ( x) will tend to grow faster the more the surface branches. To make a formal statement, let (3 c BS be any component of the branch locus. For any fundamental cycle Z, there is a constant value degree((3) to the degree of Z along (3. Choose an arbitrary orientation for (3, so we can talk about its left side and its right side. The degree of Z just to the left of (3 is also a constant, dl((3), and the degree just to the right is another constant dr((3). Proposition 1.2 (Growth proportional to branching). Let Z as above be afurulamental cycle for BS, and let 9 be a hyperbolic structure on BS. Let Y be the set of branch curves on BS(M, Pl. For each (3 E Y, define

1((3)

=

-1

og

(de gree((3)) -1 (de gree((3)) dl((3) og dr((3) .

Then the average A(R), averaged over x E BS, of length(CR(x) degreex(Z), is at least

. ( '\' .51engthg ((3) degree((3) ) 211" smh(R) exp R Lmass(Z) 1((3) ~EY

Remark. We are really making an estimate for the entropy of the geodesic flow with respect to the measure determined by Z. The quantity 1((3) is minus the log of the probability that a geodesic coming along on the right chooses to cross (3, plus a similar term for the left. In the summation, this is multiplied by the ratio of the flux of the geodesic flow through either side of (3 to the total measure of GF. Froof of 1.2. For each x in BS and for each R

> 0, we will define a certain function F(x,R)

on the part GF x of GF above x, whose integral is the total length of CR(x). F(x,R) will be constant over each set of (X, I) E GFx such that the geodesics I agree for time 0 S t S R.

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These conditions determines F: the value F(x,R)(X,I) is 27rsinh(R)/ degreex(Z) times the product, over all choices which l makes during time 0 ~ t :s; R, of the reciprocal of the probability of making that choice. The average A( R) of C R(X) degree x ( Z), as x ranges over BS, is the average of F(x,R)(X, I) degree x ( Z). The products in the formula for F(x,R)(X, I) can be thought of as exp of the sum of minus the logarithms of the probabilities. Since exp is convex upward, A( R) is greater than or equal to 27rsinh(R) exp(B(R)), where B(R) is the expected value, among all geodesics of length R in BS, of the sum of minus the logarithms of the probabilities of the choices it makes. There is an exact formula for B(R). Each branch curve fJ on BS contributes -log(degree(fJ)/ dl(fJ) each time a geodesic flows through from the left, and the total volume of the flow through (3 in time R is .5R length(fJ) degree(fJ). There is a symmetrical formula for flow from the right. The contribution to B( R) coming from flow through fJ is therefore .5Rlength g (fJ) area(BS) l(fJ), so

A( R) ;:> 27r sinh( R) exp (

~5R leng~:~~~egree(fJ) l(fJ)) .

The proposition follows.

D

Theorem 1.3 (Window frame bounded). For any pared manifold (M, P) such that oo(M) is incompressible, there is a constant C such thai among all elements N E AH(M, P), the length in N of owb(M, P) is less than C.

Proof of 1.3. The surface area of a sphere of radius R in 1HI 3 is 47rsinh 2 (R), so both the volume of the ball of radius R and the area of the sphere of radius R have exponent of growth 2R. (The exponent of growth of a function f(R) is limsuplog(J); in this case, the lim sup is the limit.) We can represent the surface BS as a pleated surface in N, by mapping all the branch curves to their corresponding geodesics (if they have geodesics), and then extending to the remaining parts of BS, which are ordinary surfaces with boundary. If some of the branch curves are parabolic elements, so that they have zero length, then we can similarly construct a pleated surface based on BS minus a union of branch geodesics; it is still a branched surface of a similar type. This imparts a complete hyperbolic structure of finite area to BS or to some subsurface Q. Fix a fundamental cycle Z on BS, as described. It defines also a fundamental cycle on any of the subsurfaces Q which might arise. From proposition 1.2, we see that the exponential growth rate of A( R) is 1 plus a positive linear combination of the lengths of the curves (3. (If some branch curves were removed from (3, then they are not included in the linear combination, but the inequality still holds since their length is zero.) If Q is compact, the argument is direct. In that case, there is an upper bound to the area of intersection of Q with a ball of radius say .1 in IBI3. The volume of a ball of radius R in IBI3 is foR 47r sinh 2 (t) dt, so it grows with exponential growth rate 2R. Therefore, the image of the exponential map at any point in Cd can have area at most a constant times exp(2R) , so the total length of the branch set must be bounded. In general, each cusp of Q is contained in a subsurface bounded by branch curves, with no branch curves in the interior. Construct horoball neighborhoods of the cusps. If x E Q

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is any point, and if x is in a horoball H embedded in Q which covers one of the horoball neighborhoods of cusps, then the area of intersection of the ball of radius R about x with H is less than 211" sinh(R). Whether or not x is in a horoball, if HI is one of the horoballs not containing x, then the area of the intersection of the ball of radius R about x with HI is no more than a constant C~,) times the area of intersection with an outer ring of HI of width 1. Consider Q minus smaller horoball neighborhoods of the cusps, shrunk a distance of 1. This subset K of Q is compact, so there is a supremum to the ratio of volume in N with area of intersection with K. Therefore, the length of the intersection of the ball of radius R about any point in Q with K is no more than a bounded multiple of exp(2R). By 1.2, this implies that the total length of the branch set is bounded. D A more careful analysis would show that the geodesic flow for Q is ergodic, and that the area of the image of the exponential map in Q at any two points has a bounded ratio (with bound depends on the pair of points, though), so that in particular, the exponential growth rate of area is independent of x. Note that if the hyperbolic structure is turned into a metric on the Hausdorffication of Q, then the growth rate of the ball of radius R in the metric might be considerably more than the growth rate of the image of the exponential map. This is hard to exploit, however, because it is hard to know what cover of Q is induced from the universal cover of M. The exponential map, in effect, picks out a subset of the fundamental group of the universal covering of the fundamental group of Q which injects no matter which 3-manifold it arose from. As an example of how the constants work out in Theorem 1.3, consider the three-manifold of figure 1 obtained by gluing three thickened punctured tori to a solid torus. In this case, BS has three branch curves, each of degree 1, and the rest of the surface has degree 2. For any hyperbolic structure, the three branch curves have the same length, a. If (3 is any of these curves, then 1((3) = 210g(2) ~ 1.38629. The exponential growth rate of A(R) is therefore 1 + 1.5Iength((3)1 ((3) / (1211") ~ 1 + .05516Iength((3). Consequently, length((3) < 18.1294. If n punctured tori are glued to a solid torus, then there are n(n-l)/2 branch curves in BS, and we can choose Z to give each of them degree 1/ (n - 1), and the rest of the surface degree 1. Then 1((3) = 210g(n - 1), mass(Z) = 211"n, so the exponential growth rate of A(R) is 1 + (n - 1) log(n - 1) length((3), 811" so length((3) < (811")/((n-l) log(n-l)). For n = 4, the length is less than 7.6256; for n the length is less than 2.3378.

=

7,

It would be interesting to work out extensions of this growth rate argument to more general contexts. The argument certainly applies directly in one dimension lower, to train tracks on surfaces, and gives an estimate for the minimum length of a train track, over all hyperbolic structures on its surface and all maps of the train track into the surface; this is not very exciting. It also applies to faithful representations of a surface group in an arbitrary homogeneous space. What could be very useful would be the extension of this analysis to general incompressible branched surfaces in a 3-manifold which carries at least one surface with positive weights. The problem is that it does not seem possible to make a hyperbolic pleated surface to represent a branched surface which has vertices where branch curves cross. It can be given a

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hyperbolic structure in the complement of such points, but the exponential map is no longer injective. Perhaps by estimating the entropy for the geodesic flow and correcting for the rate of occurrence of multiple counting because of geodesics which go in opposite ways around vertices where positive curvature is concentrated, an estimate could be worked out. A related problem is to extend the growth rate estimates in terms of the length of the branch locus to the case of hyperbolic branched surfaces with geodesic boundary. The estimate should involve the length of the boundary, and degrade as the boundary becomes longer. In the case of a Hausdorff hyperbolic surface with geodesic boundary, the growth rate was studied by Patterson and Sullivan; the exponent of growth is the same as the Hausdorff dimension of the limit set for the group. 2. RELATIVE INJECTIVITY OF PLEATED SURFACES

A 3-manifold which admits essential cylinders decomposes into its window, some miscellaneous solid tori, and an "acylindrical" part Acyl(M). It is not really acylindrical, however, once the windows are removed: it is just that cylinders in M cannot cross Acyl(M) in an essential way. To express this in a general way, let N be a 3-manifold, f : 5 -+ N a compact surface mapped into N, and X c 5 a system of non-trivial and homotopically distinct simple closed curves on 5 including all components of 8(5). Then (5, X, 1) is incompressible in N if (a): there is no compressing disk for (5,1) whose boundary is a curve on 5 which intersects X in one or fewer points. The triple (5, X, 1) is doubly incompressible in N if in addition (b): there are no essential cylinders with boundary in 5 - X, ( c ): there is no com pressing disk for (5, 1) whose boundary is a curve on 5 which intersects X in two or fewer points, and (d): Each maximal abelian subgroup of 7rl(5 - X) is mapped to a maximal abelian subgroup of 7rl(N). There is also a weaker form of (d); (5, X, 1) is weakly doubly incompressible if it satisfies (a), (b), (c), and (d1): Each maximal cyclic subgroup of 7rl(5- X) is mapped to a maximal cyclic subgroup of 7rl(N). This weakening of (d) allows for simple closed curves on 5 to be homotopic to a (Z + Z)cusp of N. The significance of this is that in geometric limits, cyclic subgroups can turn into Z+Z.

Proposition 2.1 (Acylindrical part doubly incompressible). The triple (Acyl(M), 8wb(M) n Acyl(M), c) is doubly incompressible.

Compare 1.1.

Froof of 2.1. This is part of the basic theory of the characteristic submanifold. See [Joh79] D or [JS79]. We will extend the results of §5 of [Thu86] to apply to doubly incompressible triples

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Theorem 2.2 (Relative injectivity). Let N be a hyperbolic 3-manifold, and (8, X, 1) weakly doubly incompressible in N. If A is any maximal lamination on 8 containing all curves in X as leaves, and if j". : P A -+ N is a A-pleated surface, then A injects into lP'(N). This remains true if degeneracies of the pleated surface are allowed, where closed curves of A map to cusps of N. If the recurrent part of A consists only of closed curves, then at least a degenerate pleated surface always exists which represents 8 -+1 X. Froof of 2.2. Let p be the recurrent part of A. According to 5.5 of [Thu86]' the map p -+ lP'(N), restricted to non-degenerate leaves, is a covering map to its image, and it extends to a map on a small neighborhood of p in 8 which is a covering map to its image. It follows easily from the weak double incompressibility of (8, X) that this can only be the trivial covering, so that at least p embeds in lP'(N): condition (d1) guarantees that a closed leaf can map only as a trivial covering to its image, condition (b) guarantees that no two closed leaves are mapped to a single closed leaf in the image, and condition (b) also prevents components of p with more complicated neighborhoods to map by non-trivial coverings. Each end of each leaf I of A - p is asymptotic with either one or two leaves of p - one if it is asymptotic with a closed leaf, two if it is asymptotic with a non-compact leaf. We will refer to these two types of ends as type 1 ends and type 2 ends. Suppose 1, has the same image in lP'(N) as 12. Since p embeds, it follows that 1, and 12 are asymptotic (on 8) at both ends. A closed loop can be formed, by bridging between the two leaves at their two ends along short arcs. If an end is of type 2, then the short arc will not to intersect any closed leaves of p. If an end is of type 1, then the arc can be chosen so that it intersects a closed leaf of p in at most one point. In particular, the closed loop on 8 intersects X in at most two points. It follows from condition (c) that the image of the loop is null-homotopic in N. It follows from hyperbolic geometry that 1, = 12 . There remains still the possibility that a leaf I of A - p could have image in lP'(N) which is a circle. Such a situation would force both ends of I to be of type 2, since the set of identifications under A -+ lP'(N) is closed. This also forces the image of I to coincide with the image of the circle at either end - which must be a single circle - and it forces the closed leaf to be non-degenerate. Construct a loop on 8 which follows along I, crosses a short bridge to the closed leaf, then unwinds on the closed leaf, finally crossing a bridge to the other end of I. Since the amount of unwinding is adjustable, it can be chosen so that the entire configuration maps to a null-homotopic curve in N. This contradicts incompressibility, (a), since a small homotopy of the loop makes it intersect a closed leaf of A in at most one point. To prove that a possibly degenerate pleated surface always exists provided p consists only of closed leaves, we must show that each leaf I of A - p, when pushed forward to N by a continuous map, can be straightened out to a geodesic without changing the asymptotic behavior of its ends. In!HI3 , the closed leaves at either end of I either are covered by geodesics, or they map to cusps. If one of the curves is parabolic and one hyperbolic, the endpoints are automatically distinct, so I can be straightened. If the end curves are both hyperbolic, then it follows from incompressibility of (8, X), as above, that the endpoints are distinct. Finally, suppose that I is asymptotic to parabolic curves at both ends. Map I to Nand lift to!HI'- If the two endpoints are the same, then I can be retracted into an arbitrarily small neighborhood of a cusp. Form a subsurface of 8 from a neighborhood of I together

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with the circle or circles it is asymptotic to. Choose two non-commuting elements of the fundamental group of this surface which are represented by loops, such that one loop (say, parallel to one of the closed leaves) does not intersect X, and the other loop intersects X in at most one point. (Make it from I, with a short bridge between its ends if they are asymptotic, or a bridge to the second closed leaf, a traversal of this leaf, a bridge back to I, and a return journey along I back to the basepoint, otherwise.) The commutator of the two D loops is null-homotopic in N; it intersects X in at most two points, violating (c). Theorem 2.3 (Relative uniform injectivity). Let 5 be a compact surface, and X a collection of non-trivial, homotopically distinct simple closed curves on 5 which includes all boundary components. Let Band fO be positive constants. Among all pleated surfaces f : 5 -+ N (N a variable hyperbolic manifold) pleated along laminations A containing X, where (5, X, 1) is doubly incompressible and the total length of X in N is less than B, the associated maps

9 : A -+ lP' (N) are uniformly injective on the fo-thick part of 5. That is, for every f > 0 there is a 0 > 0 such that for any such 5, N, A and f and for any two points x and yEA whose injectivity radii are greater than fO, if d(x,y);:> f then d(g(x),g(y));:> 0. The same statement holds true when degenerate pleated surfaces are allowed, in which certain closed leaves of A may be parabolic.

The uniform injectivity theorem of [Thu86] is the special case of this when X where the constant B = 0 - that is, where any boundary curves are parabolic.

=

a5, and

Proof of 2.3. As in [Thu86], the main step of the proof is to establish that the geometric limits of surfaces which satisfy the hypotheses are at least weakly doubly incompressible. Consider a sequence fi : 5 -+ Ni of pleated surfaces, pleated along Ai ::J X, such that (5, X, J,) are doubly incompressible, and for which the total length of the curves of X in Ni is less than B. Let gi be the metric induced on 5 by k For each i, let Bi be a collection of points, one in each component of the fo-thin part of 5 with respect to gi. and let Ei be a collection of orthonormal frames at the elements of B i . There is some subsequence such that sequence of hyperbolic surfaces 5 i defined by the metric gi on 5 with respect to the collection of base frames Ei converges to a geometric limit. By this we mean that, first, if the universal cover of 5 is developed into 1HI2 , where any of the frames ei E Ei is sent to some fixed base frame in hy2, then the sequence of images of 11"1(5) converge in the Hausdorff topology for Isom hy2: this is the geometric limit from the point of view of ei. Second, the Hausdorff limit of the image of Ei from the point of view of ei must exist, that is, any other frames which stay within a bounded distance of ei should converge. When these conditions are met, there is a well-defined, possibly disconnected, geometric limit with respect to Ei . We can pass to a further subsequence so that the system of geodesics X and the laminations Ai also converges geometrically, that is, in the Hausdorff topology when they are developed onto JHr using frames in Ei to get started. Let H be the topological surface which is the geometric limit. On H, denote the limiting hyperbolic metric gl, curve system yl and lamination N. Each component of 5 - X is incompressible in N i . Since the boundary components of each such component have bounded length with respect to 9i, there is a non-elementary subgroup of 11"1(5) - X based at any point x E Bi generated by loops through x of bounded

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length. Therefore, the injectivity radius of Ni at the image of x is bounded abvoe zero. The image of each ei E Ei in Ni can be extended uniquely to an orthonormal frame j, in N i , and there is a further subsequence so that the Ni with collection of base frames Uh converges geometrically, to a hyperbolic manifold N and so that the maps Ii : 5 -+ Ni also converge R' -+ N. geometrically to a .v-pleated surface It may be that there are pairs of cusps of R' which came from the two sides of a sequence of geodesics on 5 which either were degenerate, or grew shorter and shorter in the sequence. Form a new topological surface R by joining the two cusps together along a simple closed curve; we can think of I as a degenerate pleated surface for R, if we extend .\ by adding a closed leaf for each parabolic curve we adjoin, and spinning the ends of leaves which tend to these cusps around the new closed leaves. Label the new curves as belonging to Y according to whether they were limits of elements of X, possibly after passing to a subsequence. Pick a homotopy class I of maps of the new surface R into N, which agrees with the previous on R'. For each degenerate curve, this involves a free choice of a power homotopy class of the Dehn twist about the curve, and if the cusp is a Z + Z cusp, the choice of how the annulus wraps around the torus. We verify the conditions for double incompressibility of the limiting pleated surface, with curve system Y. Condition (a) is contained in condition (c) that there are no compressing disks which intersect Y in two or fewer points. Suppose there were an essential disk for I : R -+ N with boundary a curve on R meeting Y in two or fewer points. The map I is approximated by a map of R to the N i , for sufficiently high i. The approximation at least restricted to R' factors (up to homotopy) through an embedding ji of R as a subsurface of 5 with incompressible boundary. This factorization extends over the degenerate curves of R as well, since the annulus of R and the annulus of 5 both map into the thin set of N i ; the fundamental group of this component of the thin set is cyclic, and generated by the core curve of the annulus, so the two annuli must be homotopic rei boundary. The compressing disk would be approximated by a compressing disk for 5 whose boundary intersects X in only two points, contradicting the hypothesis that (5, X, Ii) is doubly incompressible. Condition (b) says that there should be no essential annulus for I : R -+ N whose boundary is disjoint from X. Any essential annulus would be approximable by an annulus for j, : 5 -+ N, for i sufficiently high. We need to check that the approximating annulus is essential, that is, that its two boundary components are not homotopic on 5. If the length of either of the boundary curves on R is greater than zero, then that boundary curve can be represented by a geodesic on R. Two geodesics are isotopic on R if and only if they coincide; the same criterion carries over to the approximating surfaces, so an essential annulus which has a boundary curve of positive length would carryover to an essential annulus for the approximations, contradicting the hypotheses. If there were an essential annulus whose boundary components have zero length on R, then when it is carried over to the approximations, it would still essential for otherwise we would have added a degenerate curve joining the two cusps in question when we formed R. Again, this contradicts the hypothesis that (5, X, j,) is doubly incompressible. Condition (dl) also persists in a geometric limit. Let H be a maximal cyclic subgroup of the fundamental group of R - Y. The embedding ji of R as an incompressible subsurface of 5 for high i carries H to a maximal cyclic subgroup of "1(5 - X). Therefore, its image is

t :

t

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a maximal cyclic subgroup of 7rl(N)i. If Q is any element of 7rl(N) such that some power is a cyclic generator of the image of H in 7rl(N), then Q is approximated in Ni by an element with a power approximately, and therefore exactly, equal to the cyclic generator of the image of H in N i ; the power can only be 1. We have established that any sequence of pleated surfaces satisfying the hypotheses of the theorem, has a subsequence converging to a weakly doubly incompressible pleated surface in a hyperbolic 3-manifold. Theorem 2.3 follows now from 2.2. D

3.

RELATIVE BOUNDNESS FOR

AH(M)

The relative uniform injectivity theorem of the last section has a direct application to the boundness of deformations of a hyperbolic 3-manifold when the total length of a sufficiently complicated system of curves on its boundary is held bounded. Theorem 3.1 (Relative boundedness). Let M be a 3-manifold, and X a collection of nontrivial, homotopically distinct curves on 8M such that (8M, X, c) is doubly incompressible. Then for any constant A > 0, the subset of AH(M) such that the total length of X not exceeding A is compact. Note that by setting X = 0 we obtain 1.2 of [Thu86]. By letting X be the set of core curves of the parabolic annuli of a pared manifold and setting A = 0 we obtain 7.1 of [Thu86]. Froof of 3.1. The proof is much the same as the proof of the main theorem of [Thu86]' but we will go through the details for the sake of completeness. We may as well assume that 8M is non-empty. Let Z :J X be a collection of curves which has at least one element on each boundary component of M. Choose a triangulation T of M, with one vertex on each element of Z and such that each element of Z is formed by one edge of T n 8M. Choose an orientation for each element of Z. Denote the closed unit ball !HI 3 U S!, = !HI'- For any element N of AH(M), an ideal simplicial map fN : M -+ N can be defined in the standard way: let iN : M -+ !HI3 :J N by map a vertex v of T to the positive fixed point at infinity for the covering transformation which generates forward motion along the component of von Z, and extend fN to a piecewisestraight map. There is a canonical factorization of fNI8M as an ideal simplicial map iN to a possibly degenerate hyperbolic structure on 8M, followed by a possibly degenerate pleated surface PN. The hyperbolic structure g(N) on 8M is a metric on 8M minus any torus components, and possibly with certain curves deleted whose two sides become cusps in the hyperbolic structure. Associated with fN is a cert",in infinity sub complex CN of T, ~onsisting of those simplices such that any copy of them in M is mapped to a single point by fN. The degenerate simplices for an ideal simplicial map are those simplices of which at least one edge is contained in c. Thus, a degenerate triangle collapses either to a line, or to a point at infinity. A degenerate 3simplex collapses to an ideal triangle, a line, or a point. A 3-simplex is not called degenerate if it merely flattens to a quadrilateral, or maps with reversed orientation. Let N(i) be any infinite sequence of elements of AH(M) such that the total length of X is bounded by A. We will extract a convergent subsequence. For notational simplicity, each time we pass to a subsequence, we implicitly relabel it by the full sequence of positive integers.

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First, observe that 'N(i) = , does not depend on i, for it can be determined homotopically: a simplex is in , if and only if it is homotopic, rei vertices, either to a torus boundary component of M, or to a curve in the collection Z. Recall from [Thu79], [Thu97] or [Thu86] that to each edge of each non-degenerate simplex is associated an edge invariant in t - {O, 1, co}. The edge invariants of opposite edges of a three-simplex are equal, and the three values tend to 0, 1 and co if the shape of the simplex degenerates. We may pass to a subsequence of N (i) such that the edge invariants of each edge of each nondegenerate 3-simplex of T converge in Following §3 of [Thu86], we define a "bad complex" B for the sequence N(i), which is formed as the union of all i-simplices (which are bad only because they do not yield information), all degenerate simplices, and within each degenerating 3-simplex, a quadrilateral attached along the four edges whose edge invariants are tending to and co. The good submanifold G for the sequence N(i) is obtained by deleting a regular neighborhood of B, and the "cutting surface" I< for {N}(i) is defined as the internal boundary of G, I< = aG - aM. An ideal triangle in N (i) serves like a base frame: a certain "view" of N (i) is obtained by sending its three vertices to 0, 1, and co. If Q is any path in G beginning and ending on a triangle of T, it defines a sequence of views of N(i), which are related by a sequence of isometries of !HI'- The submanifold G was defined in such a way that this sequence of isometries converges. Note in particular that when a path in G enters a degenerating 3-simplex through one triangle and exits through the other accessible triangle, the isometry that relates the two views converges to the identity. Therefore, the sequence of representations p(i) for 7rl(M) based at any point pEG converges when restricted to 7rl((G,P)). As in the earlier paper, we will augment G together with its map to M, in several steps gi : Gi -+ M, in such a way that the sequence of composed representations Gi -+ M -+ Isom(!HI 3 ) continues to converge, until finally the map to M is a proper map (one which takes boundary to boundary) of degree 1. We start with Go be G, and go the inclusion. The key point is that the surface I< is in a certain sense becoming smaller and smaller in N (i), as the sequence progresses, so that we can think of it as if it were only i-dimensional. If we proceed far out in the sequence, we can homotope the map of I< to N(i) so that it lies close to the i-skeleton of the image ideal 3-simplices, so that the area of the image of I< tends to zero, and it geometrically it looks like a i-complex. In fact, we will analyze in terms of a map to a certain graph I defined by this geometry. Toward this end, define a subsurface 1 O. Then we see that Hl(H) is non-zero if and only if Hl(K) is. As long as U is non-trivial, the index [G : K] = [Q : U] is smaller than [G: H] = #Q, so computing Hl(K) is easier that computing Hl (H). Returning to the example of PSL 21Fp, there is such a U of index about p2, whereas the order of PSL 21Fp is about p3 /2. Looking at a matrix with side O(p2) is a big savings over one of side O(p3). Moreover, finding such a U given Q is easy. First compute the character table of Q and the conjugacy classes of subgroups of Q (these are both well-studied problems). For each subgroup U of Q compute the character XU of the permutation representation of Q on iC[Q/U]. Expressing XU as a linear combination of the irreducible characters tells us exactly what the ei are. Running through the U, we can find the subgroup of lowest index where all of the ei > O.

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When we were searching for positive betti number covers, we used this method of replacing H with K = j-1(U) and computed the ranks of the resulting matrices over a finite field IFp. Once we had found an H with positive IFp-betti number, we did the following to check rigorously that H has infinite abelianization. First, we went through all the subgroups U of Q, till we found the U of smallest index such that j-1(U) has positive IFp-betti number. For this U, we computed the Q-betti number of j-1(U) using one of the methods described in Section 4. Doing this kept the matrices that we needed to compute the Q-rank of small, and was the key to checking that the covers really had positive Q-betti number. For instance, for the PSL 2 IF 101 -cover of degree 515,100 there was a U so that the intermediate cover j-1(U) with positive betti number had degree "only" 5,050. It's worth mentioning that the rank over Q was very rarely different than that over a small finite field. Initially, for each manifold we found a cover where the IF 31991 -betti number was positive. All but 3 of those 10,986 covers had positive Q-betti number.

4

Computing the rank over Q

Here, we describe how we computed the Q-rank of the matrices produced in the last section. Normally, one thinks of linear algebra as "easy", but standard row-reduction is polynomial time only if field operations are constant time. To compute the rank of an integer matrix A rigorously one has to work over Q. Here, doing row reduction causes the size of the fractions involved to explode. There are a number of ways to try to avoid this. The first is to use a clever pivoting strategy to minimize the size of the fractions involved [33,32,311. This is the method built into GAP, and was what we used for the covers of degree less than 500, which sufficed for 99.2% of the manifolds. For all but about 7 of the remaining 94 manifolds, we used a simplified version of the p-adic algorithm of Dixon given in [171. Over a large finite field IFp, we computed a basis of the kernel of the matrix. Then we used "rational reconstruction", a partial inverse to the map Q -+ IF p to try to lift each of the IFp-vectors to Q-vectors (see [17, pg. 139]). If we succeeded, we then checked that the lifted vectors were actually in the kernel over Q. For 7 of the largest covers (degree 1,000-5,000), this simplification of Dixon's algorithm fails, and we used the program MAGMA [4], which has a very sophisticated p-adic algorithm, to check the ranks of the matrices involved.

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5

Simple covers

To gain more insight into this problem, we looked at a range of simple covers for a randomly selected 1,000 of the census manifolds which have 2-generator fundamental groups. For these 1,000 manifolds we found all the covers where the covering group was a non-abelian finite simple group of order less than 33,000. For each cover we computed the homology. We will describe some interesting patterns we found. First, look at Table 2. There, the simple groups are listed by their ATLAS [111 name (so, for instance, Ln(q) = PSLnlFq ), together with basic information about how many covers there are, and how many have positive betti number. There is quite a bit of variation among the different groups. For instance, only 11.3% of the manifold groups have L2(16) quotients but 42.8% have L3(4) quotients. Moreover, there are big differences in how successful the different kinds of covers are at producing homology. Only half of the L2(37) covers have positive betti number, but almost all (97.5%) of the U4 (2) covers do. There are no obvious reasons for these patterns (for instance, the success rates don't correlate strongly with the order of the group). It would be very interesting to have heuristics which explain them, and we will explore these issues in [211. In terms of showing manifolds are virtually Haken, even the least useful group has a Hit rate greater than 10%. That is, for any given group at least 10% of the manifolds have a positive betti number cover with that group. So unless things are strongly correlated between different groups, one would expect that every manifold would have a positive betti number simple cover, and that one would generally find such a cover quickly. Let Q(n) denote the nth simple group as listed in Table 2. Set V (n) to be the proportion of the manifolds which have a positive betti number Q(k)-cover where k S n. We expect that the increasing function V (n) should rapidly approach 1 as n increases. This is born out in Figure 4. Figure 4 shows that the groups behave pretty independently of each other, although not completely as we will see. Let H(n) denote the hit rate for Q(n), that is the proportion of the manifolds with a Q(n) cover with positive betti number. If everything were independent, then one would expect

V(n) '" V(n - 1)

+ (1- V(n -

l))H(n).

If we let E(n) be the right-hand side above, and compare E(n) with V(n) we find that E( n) - V (n) is almost always positive. To judge the size of this

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Quotient

As L2(7) A6 L2(8) L2(11) L2(13) L2(17) A7 L2(19) L2(16) L3(3) U3(3) L2(23) L2(25) Mll

L2(27) L2(29) L2(31) As

L3(4) L2(37) U4 (2) Sz(8) L2(32)

Order 60 168 360 504 660 1092 2448 2520 3420 4080 5616 6048 6072 7800 7920 9828 12180 14880 20160 20160 25308 25920 29120 32736

Hit 14.0 17.8 21.6 15.4 24.1 29.4 29.4 41.1 28.2 11.3 19.2 16.4 32.7 24.7 14.6 14.2 42.0 38.1 18.7 42.8 24.9 26.6 26.9 12.4

HavCov 26.9 28.2 31.4 21.7 32.8 41.1 43.1 45.8 44.4 18.3 28.0 18.0 47.6 33.0 17.1 26.6 57.1 56.5 20.7 50.2 54.2 27.8 43.9 17.9

SucRatl 52.0 63.1 68.8 71.0 73.5 71.5 68.2 89.7 63.5 61.7 68.6 91.1 68.7 74.8 85.4 53.4 73.6 67.4 90.3 85.3 45.9 95.7 61.3 69.3

SucRat2 52.9 66.3 68.7 72.6 71.8 77.8 69.6 90.9 65.7 65.3 76.5 92.8 70.1 75.5 88.8 57.1 74.1 70.9 92.3 89.1 50.5 97.5 73.1 72.1

Table 2: Hit is the percentage of manifolds having a cover with this group which has positive betti number. HavCov is the percentage of manifolds having a cover with this group. SucRatel is the percentage of manifolds having a cover with this group which have such a cover with positive betti number. SucRate2 is the percentage of covers with this group having positive betti number.

deviation, we look at

E(n) - V(n) 1-V(n-l)

which lies in [-0.007,0.13]'

and which averages 0.022. In other words, V (n) - V (n - 1) is usually about 2% smaller as a proportion of the possible increase than E( n) - V (n - 1).

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100,-----,-----,-----,-----,-----~~~~--_,

80 ~

a

60

E w

40

.0 ~

a

"" ~

~

20

2?

o

L -_ _ _ _

1.5

~

____

2

~

_ _ _ __ L_ _ _ _

2.5

3

~

_ _ _ _~_ _ _ _ _ _L __ _~

3.5

4

4.5

5

x ~ Log(Order of group)

Figure 4: This graph shows how quickly simple group covers generate homology. Each

+ plotted is the pair (log(#Q(n)), V(n)), where the log is base 10. Thus the leftmost + corresponds to the fact that 14% of the manifolds have an A5 cover with positive betti number. The second leftmost + corresponds to the fact that 29% of the manifolds have either an A5 or an L2(7) cover with positive betti number, and so on.

For a graphical comparison, define V!(n) by the recursion

V!(n)

=

V!(n - 1) + (1- V!(n - l))H(n),

and compare with V(n) in Figure 5. Asymptotically, every non-abelian finite simple group is of the form L 2 (q), and so it's interesting to look at a modified V(n) where we look only at the Q(n) of this form. This is also shown in Figure 5. 5.1 Amount of homology Suppose we look at a simple cover of degree d, what is the expected rank of the homology of the cover? The data suggests that the expected rank is linearly proportional to d. For the simple group Q(n), set R(n) to be the mean of Pl(N), where N runs over all the Q(n) covers of our manifolds (including

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~

v

•~

l

80

'"8•

60

"

E

a

E •

40

~

~

o

~

•

20

OL-____L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _ 1.5

2

2.5

3

3.5

~

_ _ _ _ ~ _ _~

4

5

4.5

x = Log(Order of group)

Figure 5: The top line plots (log(#Q(n)), P(n)), the middle line (log(#Q(n)), V(n)) (as in Figure 4), and the lowest line plots only the groups of the form L2(q). those where (3i(N) = 0). Figure 6 gives a plot of logR(n) versus log(#Q(n)). Also shown is the line y = x - 1.3 (which is almost the least squares fit line y = 1.018x - 1.303). The data points follow that line, suggesting that: logR(n) ~ log(#Q(n)) -1.3

and hence

R(n)

~

#Q(n)

20.

(1)

Now each of the 3-manifold groups we are looking at here are quotients of the free group on two generators F2 . Let G be fundamental group of one of our 3manifolds, say G = F2/N. Given a homomorphism G -+ Q(n), we can look at the composite homomorphism F2 -+ Q(n). Let H be the kernel of G -+ Q(n) and K the kernel of F2 -+ Q(n). Then the rank of Hi (K) is #Q(n) + 1. As Hi (H) is a quotient of Hi (K), Equation 1 is says that on average, 5% of Hi (K) survives to Hi (H). This amount of homology is not a priori forced by the high hit rate for the Q(n). For instance, L2 (p) has order (p3 - p) /2 but has a rational representation of dimension p. Thus it would be possible for L2(p) covers to have log(R(n)) ~ (1/3)log(#G(n))

+ C,

even if a large percentage of these covers had positive betti number. This data suggests that on a statistical level these 3-manifold groups are trying to behave

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The virtual Haken conjecture: Experiments and examples

• 2

0.5

OL-~~__~~__~__~~~~~

1.5

2

2.5

3

10g(#Q(n)) Figure 6: This plot shows the relationship between the expected rank and the degree of the cover. The line shown is y = x - 1.3.

like the fundamental group of a 2-dimensional orbifold of Euler characteristic -1/20. Caveats The data in Figure 6 is not based on the full Q(n) covers but on subcovers coming from a fixed subgroup U(n) < Q(n), chosen as described in Section 3. The degree plotted is the degree of the cover that was used, that is [Q(n) : U(n)] not the order of Q(n) itself, so the above analysis is still valid. Also, throughout Section 5 having positive betti number really means having positive betti number over lF31991. Also, we originally used a list of the Hodgson-Weeks census which had a few duplicates and so there are actually 12 manifold which appear twice in our list of 1000 random manifolds. 5.2 Homology of particular representations As discussed in Section 3, if we look at a cover with covering group Q, the homology of the cover decomposes into

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Partition 7 1,6 2,5 1,1,5 3,4 1,2,4 1,1,1,4 1,3,3

Dim. of rep 1 6 14 15 14 35 20 21

Success rate 2% 22% 63% 64% 41% 70% 61% 61%

Mean homology 0.0 1.5 19.8 21.8 11.0 101.6 20.7 33.9

Table 3: The Q-irreducible representations of A 7 . Success Rate is the percentage of covers where that representation appeared. Mean Homology is the average amount of homology that that representation contributed (the mean homology of an A7 cover was 210.3).

where G is the fundamental group of the base manifold and the Vi are the irreducible Q-modules. For Q an alternating group, we looked at this decomposition and found that the ranks of the H 1 (G, Vi) were very strongly positively correlated. This contrasts with the relative independence of the ranks of covers with different Q(n). We will describe what happens for A 7 , the other alternating groups being similar. The rational representations of A7 are easy to describe: they are the restrictions of the irreducible representations of 57. They correspond to certain partitions of 7. Table 3 lists the representations and their basic properties. Table 4 shows the correlations between the ranks of the H 1 (G, Vi). Many of the correlations are larger than 0.5 and all are bigger than 0 (+ 1 is perfect correlation, -1 perfect anti-correlation and 0 the expected correlation for independent random variables). Figure 7 shows the distribution of the homology of the covers. 5.3 Correlations between groups In the beginning of Section 5 we saw that the two events (having a Q(n)-cover with (31

> 0, having a

Q(m)-cover with (31

> 0)

were more or less independent of each other, though overall there was a slight positive correlation which dampened the growth of V (n). In the appendix, there is a table giving these correlations, was well one giving those between the events: (having a Q(n)-cover,having a Q(m)-cover).

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7 16 25 115 34 124 1114 133

7 1.00 0.01 0.11 0.08 0.15 0.17 0.02 0.13

16 0.01 1.00 0.22 0.09 0.23 0.19 0.18 0.19

25 0.11 0.22 1.00 0.63 0.65 0.79 0.37 0.61

115 0.08 0.09 0.63 1.00 0.52 0.80 0.75 0.78

34 0.15 0.23 0.65 0.52 1.00 0.73 0.50 0.65

124 0.17 0.19 0.79 0.80 0.73 1.00 0.65 0.89

1114 0.02 0.18 0.37 0.75 0.50 0.65 1.00 0.66

133 0.13 0.19 0.61 0.78 0.65 0.89 0.66 1.00

Table 4: Table showing the correlations between the ranks of H, (G, Vi) where the Vi are indexed by the partition of the corresponding representation. 1 75 1 50 1 25 1 00 75 50 25

Figure 7: Plot showing the distribution of the ranks of the homology of the 964 covers with group A 7 . The x-axis is the amount of homology and the y-axis the number of covers with homology in that range.

Some of these correlations are much larger than one would expect by chance alone-for instance the correlation between ( having a L 2 (7)-cover with

(31

> 0, having a L 2 (8)-cover with

(31)

0)

is 0.38. Moreover, there are very few negative correlations and those that exist are quite small. Overall, the average correlation is positive as we would expect from Section 5. One way of trying to understand these correlations is to observe that almost all of these manifolds are Dehn surgeries on the minimally twisted 5-chain. Let us focus on the simpler question of correlations between having a cover with group Q(n) and having a cover with group Q(m). Let M be the complement of the 5-chain. Consider all the homomorphisms h: 7r1M -+ Q(n). Supposes

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X is a Dehn filling on M along the five slopes (/1, /2, /3, /4, /5) where /i is in

7r1(aiM). The manifold X has a cover with group Q(n) if and only if there is an !k where each /i lies in the kernel of !k restricted to 7r1 (aiM). Thus having a cover with group Q(n) is determined by certain subgroups of the groups 7r1(aiM) = z2 If we consider a different group Q(m) we get a different family of subgroups of the 7r1 (aiM). If there is a lot of overlap between these two sets of subgroups, there will be a positive correlation between having a cover with group Q(n) and having a cover with group Q(m). If there is little overlap then there will be a negative correlation. However, even looked at this way there seems to be no reason that the average correlation should be positive. If we look at the same question for manifolds which are Dehn surgeries on the figure-8 knot (a simplified version of this setup) there are many negative correlations and the overall average correlation is O. If we look at the question for small surgeries on the Whitehead link, the overall average correlation is positive and of similar magnitude of that for the 5-chain. If we also look at larger surgeries on the Whitehead link the average correlation drops somewhat. By changing the link we get a different pattern of correlations, and so it is unwise to attach much significance to these numbers.

6

Further questions

Here are some interesting further questions related to our experiment. (1) What happens for 3-manifolds bigger than the ones we looked at? Do the patterns we found persist? It is computationally difficult to deal with groups with large numbers of generators, which would limit the maximum size of the manifolds considered. But another difficulty is how to find a "representative" collection of such manifolds. (Some notions of a "random 3-manifold", which help with this latter question, will be discussed in [21]). (2) How else could the virtually Haken covers we found be used to give insight into these conjectures? For instance, one could try to look at the virtual fibration conjecture. While there is no good algorithm for showing that a closed manifold is fibered, one could look at the following algebraic stand-in for this question. If a 3-manifold fibers over the circle, then one of the coefficients of the Alexander polynomial which is on a vertex of the Newton polytope is ±1 (see e.g. [18]). One could compute the Alexander polynomial of the covers with virtual positive betti number and see how

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often this occurred. As many of our covers are quite small, computing the Alexander polynomial should be feasible in many cases. (3) One could use our methods to look at the Virtual Positive Betti Number conjecture for lattices in the other rank-1 groups that don't have Property T. This would be particularly interesting for the examples of com plex hyperbolic manifolds where every congruence cover has (31 = o. These complex hyperbolic manifolds were discovered by Rogawski [47, Thm. 15.3.11 and are arithmetic.

7

Transferring virtual Haken VIa Dehn filling

In the rest of this paper, we consider the following setup. Let M be a compact 3-manifold with boundary a torus. The process of Dehn filling creates closed 3-manifolds from M by taking a solid torus D2 x 51 and gluing its boundary to the boundary of M. The resulting manifolds are parameterized by the isotopy class of essential sim pIe closed curve in 8M which bounds a disc in the attached solid torus. If Q denotes such a class, called a slope, the corresponding Dehn filling is denoted by M (Q). Though no orientation of Q is needed for Dehn filling, we will often think of the possible Q as being the primitive elements in H 1 (8M,Z) and so H 1 (8M,Z) parameterizes the possible Dehn fillings. If you have a general conjecture which you can't prove for all 3-manifolds, a standard thing to do is to try to prove it for most Dehn fillings on an arbitrary 3-manifold with torus boundary. For instance, in the case of the Geometrization Conjecture there is the following theorem:

7.1 Hyperbolic Dehn Surgery Theorem [531 Let M be a compact 3manifold with 8M a torus. Suppose the interior of M has a complete hyperbolic metric of nnite volume. Then all but nnitely many Dehn nllings of Mare hyperbolic manifolds.

For the Virtual Haken Conjecture there is the following result of Cooper and Long. A properly embedded compact surface 5 in M is essential if it is incompressible, boundary incompressible, and not boundary parallel. Suppose 5 is an essential surface in M. While 5 may have several boundary com ponents, they are all parallel and so have the same slope, called the boundary slope of 5. If Q and (3 are two slopes, we denote their minimal intersection number, or distance, by 6(Q,(3).

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7.2 Theorem (Cooper-Long [12]) Let M be a compact orientable 3-manifold with torus boundary which is hyperbolic. Suppose S is a non-separating orientable essential surface in M with non-empty boundary. Suppose that S is not the fiber in a fibration over Sl. Let .\ be the boundary slope of S. Then there is a constant N such that for all slopes a with L'l(a,.\) 2' N, the manifold M(a) is virtually Haken.

Explicitly, N = 12g - 8 + 4b where 9 is the genus of Sand b is the number of boundary components. This result differs from the Hyperbolic Dehn Surgery Theorem in that it excludes those fillings lying in an infinite strip in H 1 (8M) , instead of only excluding those in a compact set. Here, we will prove a Dehn surgery theorem about the Virtual Positive Betti Number Conjecture, assuming that M has a very simple Dehn filling which strongly has virtual positive betti number. Our theorem is a generalization of the work of Boyer and Zhang [5], which we discuss below. The basic idea is this. Suppose M has a Dehn filling M(a) which has virtual betti number in a very strong way. By this we mean that there is a surjection 7r1(M(a)) -+ r where r is a group all of whose finite index subgroups have lots of homology. In our application, r will be the fundamental group of a hyperbolic 2-orbifold. Given some other Dehn filling M(;3) , we would like to transfer virtual positive betti number from M(a) to M(;3). Look at 71'1 (M)/ (a,;3) which we will call 7r1(M(a,;3)). This group is a common quotient of 7r1(M(a)) and 7r1(M(;3)). Choose, E 71'1 (8M) so that {a,,} is a basis of 7r1(8M). Then ;3 = am,n. If we think of 71'1 (M(a, ;3)) as a quotient of 71'1 (M(a)) we have:

7r1(M(a, ;3))

= 71'1 (M(a))/

(;3)

=

7r1(M(a))/ (rn).

Thus 71'1 (M(a, ;3)) surjects onto r / (,n), where here we are confusing, and its image in r. So 7r1(M) surjects onto r/ (,n). If r has rapid homology growth, one can hope that r n = r / (,n) still has virtual positive betti number when n is large enough. This is plausible because adding a relator which is a large power often doesn't change the group too much. If there is an N so that r n has virtual positive betti number for all n 2' N, then M(;3) has virtual positive betti number for all ;3 with n = L'l(r, a) 2' N. Our main theorem applies when M(a) is a Seifert fibered space whose base orbifold is hyperbolic: 7.3 Theorem Let M be a compact 3-manifold with boundary a torus. Suppose M(a) is Seifert fibered with base orbifold Z; hyperbolic. Assume also

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that the image of 7rl (8M) under the induced map 7rl (M) -+ 7rl (Z;) contains no non-trivial element of ilnite order. Then there exists an N so that M(;3) has virtual positive betti number whenever 6( a,;3) 2' N. If Z; is not a sphere with 3 cone points, then N can be taken to be 7.

In light of the above discussion, if we consider the homomorphism 7rl(M(a)) -+ 7rl (Z;) = r, Theorem 7.3 follows immediately from: 7.4 Theorem Let Z; be a closed hyperbolic 2-orbifold without mirrors, and r be its fundamental group. Let I E r be a element of inilnite order. Then there exists an N such that for all n 2' N the group

has virtual positive betti number. In fact, surjects onto a free group of rank 2.

rn

has a ilnite index subgroup which

If Z; is not a 2-sphere with 3 cone points, then N this case, N is at most 7.

=

max{1/11

+ x(Z;)I, 3}.

In

In applying Theorem 7.3, the technical condition that the image of 7rl(8M) not contain an element of finite order holds in many cases. For instance, Theorem 7.3 implies the following theorem about Dehn surgeries on the Whitehead link. Let W the exterior of the Whitehead link. Given a slope a on the first boundary component of W, we denote by W (a) the manifold with one torus boundary com ponent obtained by filling along a. Theorem (9.1) Let W be the exterior of the Whitehead link. Then for all but ilnitely many slopes a, the manifold M = W (a) has the following property: All but ilnitely many Dehn illlings of M have virtual positive betti number. In fact, our proof of this theorem excludes only 2S possible slopes a (see Section 9). The complements of the twist knots in 53 are exactly the W(l/n) for n E Z. Theorem 9.1 applies to all of the slopes l/n except for n E {D, 1} which correspond to the unknot and the trefoil. Thus we have: 7.5 Corollary Let K be a twist knot in 53 which is not the unknot or the trefoil. Then all but ilnitely many Dehn surgeries on K have virtual positive betti number. For the simplest hyperbolic knot, the figure-S, we can use a quantitative version of Theorem 7.4 due to Holt and Plesken [351 which applies in this special case. We will show:

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7.6 Theorem Every non-trivial Dehn surgery on the flgure-8 knot in 53 has virtual positive betti number.

As we mentioned, Theorem 7.3 generalizes the work of Boyer and Zhang [5]. They restricted to the case where the base orbifold was not a 2-sphere with 3 cone points. In particular, they proved: 7.7 Theorem [5] Let M have boundary a torus. Suppose M(ex) is Seifert flbered with a hyperbolic base orbifold Z; which is not a 2-sphere with 3 cone points. Assume also that M is small, that is, contains no closed essential surface. Then M(;3) has virtual positive betti number whenever 6(ex,;3) ;:> 7.

The condition that M is small is a natural one as if M contains an closed essential surface, then there is a ex so that M(;3) is actually Haken if 6(ex,;3) > 1 [15, 57]. Boyer and Zhang's point of view is different than ours, in that they do not set out a restricted version of Theorem 7.4. While the basic approach of both proofs comes from [2], Boyer and Zhang's proof of Theorem 7.7 also uses the Culler-Shalen theory of SL21C-character varieties and surfaces arising from ideal points. From our point of view this is not needed, and Theorem 7.7 follows easily from Theorem 7.3 (see the end of Section 8 for a proof). In Section 11, we discuss possible generalizations of Theorem 7.3 to other types of fillings. In a very special case, we use toroidal Dehn fillings to show (Theorem 12.1) that every Dehn filling of the sister of the figure-8 complement satisfies the Virtual Positive Betti Number Conjecture.

8

One-relator quotients of 2-orbifold groups

This section is devoted to the proof of Theorem 7.4. The basic ideas go back to [2] which proves the analogous result for r = Zip * Zlq. Fine, Roehl, and Rosenberger proved Theorem 7.4 in many, but not all, cases where Z; is not a 2-sphere with 3 cone points [22, 23]. In the case Z; = 5 2(al,a2,a3), Darren Long and Alan Reid suggested the proof given below, and Matt Baker provided invaluable help with the number theoretic details. Proof of Theorem 7.4 Let Z;n be the 2-complex with marked cone points consisting of Z; together with a disc D with a cone point of order n, where the boundary of D is attached to Z; along a curve representing ,. Thus r n =

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7rl(Z;n). Now the Euler characteristic of Z;n is X(Z;) + lin, which is negative if n> 1/Ix(Z;)I. From now on, assume that n> 1/Ix(Z;)I. Suppose rn contains

a subgroup r~ of finite index such that if Q is a small loop about a cone point then Q rt r~. For instance, this is the case if r~ is torsion free. Let Z;~ be the corresponding cover of Z;n, so r~ = 7rl(Z;~). Then Z;~ is a 2-complex without any cone points. Since Z;~ has negative Euler characteristic and there is no homology in dimensions greater than two, we must have Hl (Z;~, Q) fc o. Thus r n has virtual positive betti number. One can show more: Let d be the degree of the cover Z;~ -+ Z;n. The complex Z;~ is a smooth hyperbolic surface S with din discs attached. From this description it is easy to check that r~ has a presentation where .

(# of generators) - (# of relatIOns)

=

(lx(S)1

=

1+ d

+ 1) -

d n

(lx(Z;)I-~)

2> 2

By a theorem of Baumslag and Pride [1], the group r~ has a finite-index subgroup which surjects onto Z * Z. So it remains to produce the subgroups r~. First, we discuss the case where Z; is not a sphere with 3 cone points. A homomorphism f: r -+ Q is said to preserve torsion if for every torsion element Q in r the order of f (Q) is equal to the order of Q. (Recall that the torsion elements of r are exactly the loops around cone points.) The key is to show: 8.1 Lemma Suppose Z; is not a 2-sphere with 3 cone points, and that I E r has inilnite order. Given any n > 2, there exists a homomorphism p: r -+ PSL 2 1C such that p preserves torsion and ph) has order n.

Suppose we have p as in the lemma, which we will regard as a homomorphism from r n to PS~IC. By Selberg's lemma, the group p(I') has a finite index subgroup A which is torsion free. We can then take r~ to be p-l(A). Because the lemma only requires that n > 2 and the preceding argument required that n > 1/Ix(Z;) I, in this case we can take the N in the statement of Theorem 7.4 to be max{3,1 + 1/Ix(Z;)I}. A case check, done in [5], shows that N is at most 7. As we will see, the proof of Lemma 8.1 is relatively easy and involves deforming FUchsian representations r -+ Isom(JHI 2 ) to find p. The harder case is when Z; is a 2-sphere with 3 cone points, which we denote S2(al, a2, a3). Here the fundamental group r can be presented as a, a2 a3 1). ( Xl, X2, X3 Xl = X2 = X3 = X1X2X3 = I

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Geometrically,

Xi

is a loop around the

ith

cone point. We will show:

8.2 Lemma Let r =7rl(S2(al,a2,a3)) where 1/al+1/a2+1/a3 < 1. Given an element, E r of inilnite order, there exists an N such that for all n 2> N the group r has a ilnite quotient where the images of (Xl, X2, X3, ,) have orders exactly (al' a2, a3, n) respectively. With this Lemma, we can take r~ to be the kernel of the given finite quotient. The proof of Lemma 8.2 involves using congruence quotients of r and a some number theory. Unfortunately, unlike the previous case, the proof of Lemma 8.2 gives no explicit bound on N. In any event, we've established Theorem 7.4 modulo Lemmas 8.1 and 8.2.

D

The rest of this section is devoted to proving the two lemmas. Proof of Lemma 8.1 Because Z; is not a 2-sphere with 3 cone points, the Teichmiiller space of Z; is positive dimensional. Thus there are many representations of r into Isom(1HI 2). We can embed Isom(1HI 2) into Isom+(1HI3 ) = ps~e as the stabilizer of a geodesic plane. We will then deform these FUchsian representations to produce p. Pick a simple closed curve (3 which intersects, essentially. There are two cases depending on whether a neighborhood of (3 is an annulus or a Mobius band. Suppose the neighborhood is an annulus. First, let's consider the case where (3 separates Z; into 2 pieces. In this case r is a free product with amalgamation A * ((3) B. Let Pl: r -+ PSL2e be one of the FUchsian representations. Conjugate Pl so that Pl ((3) is diagonal. Then Pl ((3) commutes with the matrices Ct

=

(t rl0) 0

for

t

in ex.

For t in ex, let Pt be the representation of r whose restriction to A is Pl and whose restriction to B is CtP1Ct- 1 Consider the function f: ex -+ e which sends t to tr 2(pt(r)). It is easy to see that f is a rational function of t by expressing, as a word in elements of A and B. We claim that f is nonconstant. First, suppose that neither of the two components of Z; \ (3 is a disc with two cone points of order 2. In this case, (3 can be taken to be a geodesic loop. If we restrict t to IR then the family {pt} corresponds to twisting around (3 in the Fenchel-Nielsen coordinates on Teich(Z;). As , intersects (3 essentially, the length of , changes under this twisting and so f is non-constant. From

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this same point of view, we see that that j has poles at 0 and co. If one of the pieces of Z; \ (3 is a disc with two cone points of order 2, then (3 naturally shrinks not to a closed geodesic, but to a geodesic arc joining the two cone points. There is still a Fenchel-Nielsen twist about (3, and so we have the same observations about j in this case (think of Z; being obtained from a surface with a geodesic boundary component by pinching the boundary to a interval). Since the rational function j has poles at {O,co}, we have j(C X ) = C. So given n> 1, we can choose t E C X so that tr 2 (pth)) = ((2n + (in1 where (2n = e Ki / n . Then pth) has order n. Moreover, Pi preserves torsion because P1 does, and so we have finished the proof of the lemma when (3 is separating and has an annulus neighborhood. If (3 has an annulus neighborhood and is nonseparating, the proof is identical except that r is an HNN-extension instead of a free product with amalgamation.

?

Now we consider the case where the neighborhood of (3 is a Mobius band. The difference here is that you can't twist a hyperbolic structure of Z; along (3. To see this, think of constructing Z; from a surface with geodesic boundary where the boundary is identified by the antipodal map to form (3. Instead, we will deform the length of (3 in Teich(Z;). Here we will need the hypothesis that n > 2, as you can see by looking at IR P2(3, 5) with I a simple closed geodesic which has a Mobius band neighborhood. The only quotient of 71"1 (IR P2(3, 5)) where I has order 2 is Z/2 and this doesn't preserve torsion. The underlying surface of Z; is non-orientable. We can assume that Z; has at least one cone point since every non-orientable surface covers such an orbifold. Pick an arc a joining (3 to a cone point p. Let A be a closed neighborhood of (3 U a. The set A is a Mobius band with a cone point. Let B be the closure of Z; \ A. Let a be the boundary of A. A small neighborhood of a is an annulus, so if I intersects a essentially, we can replace (3 with a and use the argument above. So from now on, we can assume that I lies in A. Let 1jJ: r -+ PS~C be a Fuchsian representation. Suppose we construct a representation p: 71"1 (A) -+ PS~C so that P preserves torsion, ph) has order n, and tr2(p(a)) = tr 2 (1jJ(a)). Then as r = 71"1 (A) *(a) 7I"l(B) and P and 1jJ are conjugate on (a), we can glue P and 1jJ restricted to 7I"l(B) together to get the required representation of r. Thus we have reduced everything to a question about certain representations of 71"1 (A). The group 71"1 (A) is generated by a and (3. Choosing orientations correctly, a small loop about the cone point p is 6 = (32a. If p has order r, then 71"1 (A) has the presentation \a,(3,6

16 = (32a, 6"

=

1).

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Given any representation ¢ of 71"1 (A) , we will fix lifts of ¢(Q) and ¢(j3) to SL21C. Having done this, any word w in Q and 13 has a canonical lift of ¢( w) to SL 21C. We will abuse notion and denote this lift by ¢( w) as well. In this way, we can treat ¢ as though it was a representation into SL 21C so that, for instance, the trace of ¢( w) is defined. Define a i-parameter family of representations Pt for t E ICx as follows. Set -1 t

1j!(j3) =

(0 1),

and

1j!(Q) =

(~ e~l)

where e+e- 1 =tr(1j!(Q)) and s = t(e- 1t 2 - (e+e- 1) -tr(1j!(S)). This gives a representation of 71"1 (A) because s was chosen so that tr(pt(S)) = tr(1j!(S)) and so Pi (S) also has order r in PSL2iC. Let Teich(A) denote hyperbolic structures on A with geodesic boundary where the length of the boundary is fixed to be that of the FUchsian representation 1j!. This Teichmiiller space is IR with the single Fenchel-Nielsen coordinate being the length of 13. Note that any irreducible representation of 71"1 (A) is conjugate to some Pt, and so each point in Teich(A) yields a FUchsian representation Pt. As 13 gets short in Teich (A) , the curve I gets long. Thus if we set f = tr(pt(r)) , then f is a non-constant Laurent polynomial in t. Let v = (2n + (in1 . To finish the proof of the lemma, all we need to do is find atE ICx so that f(tJ2 = v 2 . As a map from the Riemann sphere to itself, f is onto and there are t1 and t2 in C so that f(t1) = v and f(t2) = -v. As n > 2, v is not 0 and so t1 and t2 are distinct. As f is non-constant and finite on IC x, it has a pole at at least one of 0 and co. Therefore, at least one of t1 and t2 is in IC x and we are done. D Proof of Lemma 8.2 The group r is naturally a subgroup of PSL2IR. Set bi = 2ai. Let Xi be the matrix in PS~ 1R corresponding to the generator Xi. As Xi has order ai, it follows that tr(Xi) = ±((bi + (i;1) where (b i is some primitive bi th root of unity. Any irreducible 2-generator subgroup of PSL21C is determined by its traces on the generators and their product, and so we can conjugate r in PSL 21C so the Xi are:

_ ( 0 1 ) _ ( (be + (i;1 X1-1 (h+(b;l ,X2(bs1

-(b,) 0

_ ( )-1 . , andX3 - X1 X 2

Henceforth we will identify r with its image. The entries of the Xi lie in Q( (h, (b2' (b,) , and moreover are integral, so r is contained in the subgroup PSL20(Q((h, (b2' (bs))· Let G be a matrix in PSL21C representing,. Let a

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be one of the eigenvalues of G. Note that a is an algebraic integer, in fact a unit, because it satisfies the equation a 2 - (tr G)a + 1 and tr G is integral. Let K be the field Q( (h, (b2' (bS' a). From now on, we will consider r as a subgroup of PSL 20(K). We will construct the required quotients of r from congruence quotients of PS~O(K). Suppose r is a prime ideal of O(K). Setting k = O(K)/r, we have the finite quotient of r given by

r

-+ PSL20(K) -+ PSL2k.

What conditions do we need so that (Xl, X2, X3, ,) have the right orders in PSL 2k? Well, the eigenvalues of Xi are ±{(b" (b:, l } , so as long as (b i has order bi in k X , the matrix Xi in PSL 2k also has order bi . Similarly, if we set m = 2n, then G in PS~k has order n if a has order m in P. Thus the following claim will complete the proof of the lemma: 8.3 Claim There exists an N such that for all n ;:> N there is a prime ideal r such that if k = O(K)/r then the images of ((h,(b2'(bs,a) in k X have orders (b l , b2 , b3 , m).

Let's prove the claim. The idea is to show that am - 1 is not a unit in O(K) for large m, and then just take r to be a prime ideal dividing am -1. We have to be careful, though, that ((h, (be, (bs, a) don't end up with lower orders that expected in k x . A prime ideal is called primitive if it divides am - 1 and does not divide a r - 1 for all r

< m. Postnikova and Schinzel proved the following theorem:

8.4 Theorem [48, 461 Suppose that a is an algebraic integer which is not a root of unity. There there is an N such that for all n ;:> N the integer an - 1 has a primitive divisor.

The proof of Theorem 8.4 relies on deep theorems of Gelfond and A. Baker on the approximation by rationals of logarithms of algebraic numbers. Because , has infinite order, we know that a is not a root of unity. Thus Theorem 8.4 applies, and let N be as in the statement. By increasing N if necessary, we can ensure that the primitive divisor r given Theorem 8.4 does not divide any element of the finite set R = { (bi

-

1

I

1 S r < bi } .

Thus for all m ;:> N, we have a prime ideal r which divides am - 1 but does not divide a r - 1 for r < m. Thus a has order m in k x. As r does not divide any element of R, the element (b, has order bi in kX. This proves the claim and thus the lemma. D

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It would be nice to have given a proof of Lemma 8.2 which gave an explicit bound on N. The number theory used gives "an effectively computable constant" for N, but doesn't actually compute it. Perhaps there are other proofs of Lemma 8.2 more like that of Lemma 8.1. While 7I"1(S2(a1,a2,a3)) has only a finite number of representations into PSL21C, if one looks at representations into larger groups there are deformation spaces where you could hope to play the same game. For instance, if one embeds 1HI2 as a totally geodesic subspace in complex hyperbolic space IC H2, then a Fuchsian representation deforms to a one real parameter family in Isom+(ICH 2) = PU(2, 1). One could instead consider deformations in the space of real-projective structures, which gives rise to homomorphisms to PGL31R [10]. In general, the structure of the space representations of 71"1 (S2(a1' a2, a3)) -+ SLnlC is closely related to the Deligne-Simpson problem [51].

We end this section by deducing Boyer and Zhang's original Theorem 7.7 from Theorem 7.3.

Proof of Theorem 7.7 Let M be a manifold with torus boundary which is small. Suppose that M(a) is Seifert fibered with hyperbolic base orbifold Z; which is not sphere with 3 cone points. We need to check that Theorem 7.3 applies. Let (3 be a curve so that {a,(3} is a basis for 7I"l(oM). It suffices to show the image of (3 does not have finite order in r = 71"1 (Z;). Suppose not. Then there are infinitely many Dehn fillings M(ri) of M where 7I"l(M('/i)) surjects onto r. The orbifold Z; contains an essential simple closed curve which isn't a loop around a cone point. Therefore, r has non-trivial splitting as a graph of groups and so acts non-trivially on a simplicial tree. Then each 7I"l(M('/i)) act non-trivially on a tree and so M(ri) contains an essential surface. As infinitely many fillings contain essential surfaces, a theorem of Hatcher [3~] implies that M contains a closed essential surface. This is contradicts that M is small. So the image of (3 has infinite order and we are done. D

9

Surgeries on the Whitehead link

Consider the Whitehead link pictured in Figure 8. Let W be its exterior. We will denote the two boundary components of W by 00 Wand 01 W. For each Oi W, we fix a meridian-longitude basis {ILi,.\d with the orientations shown in the figure. With respect to one of these bases, we will write boundary slopes as rational numbers, where PIL + q.\ corresponds to p/q. We will denote Dehn filling of both boundary components of W by W(po/qO;pl/q1). Dehn filling

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Figure 8: The Whitehead link, showing our orientation conventions for the meridians and longitudes.

on a single component of W will be denoted W(po/qO;·) and W( ·;Pl/ql). As W(p/q; .) is homeomorphic to W(· ;p/q), we will sometimes denote this manifold by W(p/q). With our conventions, W(l) is the trefoil complement, and W(-l) is the figure-8 complement. The manifold W(p/q) is hyperbolic except when p/q is in {co, a, 1, 2,3,4}. The point of this section is to show: 9.1 Theorem Let W be the complement of the Whitehead link. For any slope p/q which is not in E = {co, a, 1, 2, 3, 4, 5, 5/2, 6, 7/1, 7/2, 8, 8/3,9/2,10/3,11/2,11/3,13/3,13/4,14/3,15/4,16/3, 16/5, 17/5, 18/5, 19/4, 24/5, 24/7} the manifold W (Q) has the property that all but tlnitely many Dehn tlllings have virtual positive betti number. Proof The proof goes by showing that except for p/q in E, the manifold W(p/q) has at least 2 distinct Dehn fillings which are Seifert fibered and to which Theorem 7.3 applies. The reason that W(P/q) has so many Seifert fibered fillings is because the manifolds W(l), W(2), and W(3) are all Seifert fibered with base orbifold a disc with two cone points. In particular, the base orbifolds are D2(2, 3), D2(2, 4), and D2(3,3) respectively. Therefore, all but one Dehn surgery W(l;p/q) on W(l) is Seifert fibered with base orbifold a sphere with 3 cone points. Similarly for W(2) and W(3). In fact, you can check that

Ip- 6ql) 5 (2,4, Ip - 4ql) 5 2 (3,3, Ip - 3ql)

• W(l;p/q) Seifert fibers over 5 2 (2,3, • W(2;p/q) Seifert fibers over • W(3;p/q) Seifert fibers over

2

fc if p/q fc if p/q fc if p/q

6. 4. 3.

Now fix a slope p/q, and consider the manifold M = W( .;p/q). We want to know when we can apply Theorem 7.3 to M(l), M(2), or M(3). First, we

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need the base orbifold to be hyperbolic, i.e. that the reciprocals of the orders of the cone points sum to less than 1. This leads to the conditions: For M(l) that For M(2) that For M(3) that

Ip Ip Ip -

6ql 4ql

3ql

> 6. > 4. > 3.

(2)

We claim that as long as the base orbifold is hyperbolic then Theorem 7.3 applies. Consider the map 71"1 (M) -+ r where r is the fundamental group of one of the base orbifolds. Let I-' in 8M be the meridian coming from our meridian 1-'0 of W. Since I-' intersects any of the slopes 1,2,3 once, its image in r generates the image of 71"1 (8M). Thus we just need to check that the image of I-' is an element of infinite order in r. One can work out what the image in r is explicitly (most easily by with the help of SnapPea [56]): For M(l), I-' -+ aba- 1 b- 1 where

r

=

(a,b

I

a 2 = b3 = (ab)P-6q =

1).

For M(2), I-' -+ ab 2 where

r

=

(a,b

I

a 2 = b4 = (ab)p-4 q =

1).

(3)

For M(3), I-' -+ ab- 1 where

r

=

(a,b

I

a 3 = b3 = (ab)P-3q =

1).

It remains to check that the images of I-' above always have infinite order in r. This is intuitively clear for looking at loops which represent these elements. The suspicious reader can check that this is really the case by using, say, the solution to the word problem for Coxeter groups [6, § 11.3].

Thus, Theorem 7.3 applies whenever one of the conditions in (2) holds. If p/q is such that two of (2) hold, then all but finitely many Dehn surgeries on M have virtual positive betti number. The set in H 1 (8M,ITI!.) = ITI!.2 where any one of the conditions fails is an infinite strip. So the set where a fixed pair of them fail is compact, namely a parallelogram. Hence, outside a union of 3 parallelograms, at least two of the conditions hold. These 3 parallelograms are all contained in the square where Ipl, Iql :S 100. To complete the proof of the theorem, one checks all the slopes in that square to find those where fewer that two of (2) hold. D For most of the slopes in E, one of (2) holds, and so one still has a partial result. The slopes where none of the conditions in (2) hold are

{oo,O, 1,2,3,4,5,6, 7/2,9/2}. One interesting manifold among these exceptions is the sister of the figure-8 complement W(5). We will consider that manifold in detail in Section 12.

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10

The figure-8 knot

Here we prove: 10.1 Theorem Every non-trivial Dehn surgery on the tlgure-8 knot has virtual positive betti number.

Proof Let M be the figure-8 complement. As the figure-8 knot is isotopic to its mirror image, the Dehn filling M(P/q) is homeomorphic to M(-p/q). Now, if W is the Whitehead complement as in the last section, M = W(-l). Hence M has at least 6 interesting Seifert fibered surgeries namely M(±l), M(±2) and M(±3). In (3), we saw exactly which orbifold quotients r / (fIn) arise when we try our method of transferring virtual positive betti number. By a minor miracle, Holt and Plesken have looked at exactly these quotients and shown: 10.2 Theorem [351 Let

r; =

(a,b

I

r~ = (a,b

I

a 2 = b3 = (ab? = (aba-1b-1)n = a 2 = b4 = (ab)5 = (ab 2 )n),and

r~ = (a,b

I

a 3 = b3 = (ab)4 = (ab-1)n =

1),

1).

These groups have virtual positive betti number if n ;:> 11 for for r~ and r~.

r;

and n ;:> 6

Thus M (Q) has virtual positive betti number if any of the following hold:

L'I.(Q, ±1) ;:> 11, L'I.(Q, ±2) ;:> 6,

or

L'I.(Q, ±3) ;:> 6.

It's easy to check that the only slopes Q for which none of these hold are {oo, D, ±1, ±2}. Since Hl (M(D)) = Z and the Seifert fibered manifolds M(±l) and M(±2) have virtual positive betti number, we've proved the theorem. D

11

Other groups of the form tions

r / (r't)

and further ques-

As we have seen, groups of the form r/ (,n), where r is a Fuchsian group, are very useful for studying the Virtual Haken Conjecture via Dehn filling. So it is natural to ask: what other types of r give similar results? In this section, we consider r which are free products with amalgamation of finite groups. The

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key source here is Lubotzky's paper [40], which gives a number of applications of these groups to the Virtual Positive Betti Number Conjecture. For convenience, we will only discuss free products with amalgamation, but there are analogous statements for HNN extensions. Let r = A *c E be an amalgam of finite groups where C is a proper subgroup of A and E. The group r acts on a tree T with finite point stabilizers. By [50, § 11.2.6]' r has a finite index subgroup A which acts freely on T. The subgroup A has to be free, and so r is virtually free. It is not hard to show that if one of [A : C] and [E : C] is ;:> 3 then r is virtually a free group of rank;:> 2 [40, Lemma 2.2]. From now on, we will assume [A : C] ;:> 3. Because r is virtually free, it is natural to hope that the answer to the following question is yes: 11.1 Question Let r be an amalgam ofilnite groups, and ilx I E r ofinilnite order. Does there exist an N such that for all n ;:> N, the group r n = r / (,n) has virtual positive betti number?

Note that by Gromov, there is an N such that hyperbolic group for all n ;:> N.

rn

is a non-elementary word

Now consider these groups in the context of Dehn filling. Suppose M is a manifold with torus boundary, and suppose a is a slope where 7I"l(M(a)) surjects onto r, an amalgam of finite groups. Choose I in 71"1 (8M) so that {a, I} form a basis. The proof of Theorem 7.7 shows that if M does not contain a closed incompressible surface, then the image of I in r has infinite order. There are candidate a where one expects that 7I"l(M(a)) will surject onto an amalgam of finite groups. Suppose that N = M(a) contains a separating incompressible surface 5. Then 7I"l(N) splits as 7I"1(N1) *K l (S) 71"1 (N2), where the Ni are the components of N \ 5. Recall that 71"1 (5) is said to separable in 71"1 (N) if it is closed in the profinite topology on 71"1 (N). Lubotzky showed [40, Prop. 4.2] that if 71"1 (5) is separable then there is a homomorphism from 71"1 (N) to an amalgam of finite groups r, which respects the amalgam structure. Provided that 5 is not a semi-fiber (that is, the Ni are not both I-bundles), then r = A *c E can be chosen so that [A: C] ;:> 3. In general, we will say that 71"1 (5) is weakly separable when there is such an amalgam preserving map from 71"1 (N) to an amalgam of finite groups. A priori, this is weaker than 71"1 (5) being closed in 71"1 (N), which is in turn weaker than 7I"l(N) being subgroup separable (aka LERF). Note that if 71"1 (5) is weakly separable, then N has virtual positive betti number as 71"1 (N) virtually maps onto a free group. If N is hyperbolic, it seems quite

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possible that the fundamental group of an embedded surface is always weakly separable. If this is the case, there is no difference between being virtually Haken and having virtual positive betti number. Subgroup separability properties for 3-manifold groups have been difficult to prove even in special cases. Weak separability also seems quite difficult to show even though the surface 5 is embedded. Let M be a manifold with torus boundary which is hyperbolic. Assume that M does not contain a closed incompressible surface. Then there are always at least two Dehn fillings of M which contain an incompressible surface [14, 151. If embedded surface subgroups are weakly separable, we would expect that for most M, there are at least two slopes where 7I"l(M(a)) surjects onto an amalgam of finite groups. One has to say "most" here because M (a) might be a (semi- )fiber or the Poincare conjecture might fail. This makes it plausible that, regardless of the truth of the virtual Haken conjecture in general, for a fixed M all but finitely many Dehn fillings of M have virtual positive betti number. In this context, it is worth mentioning the result of Cooper-Long [131 which says that for any such hyperbolic M all but finitely many of the Dehn fillings contain a surface group. If fundamental groups of hyperbolic manifolds are subgroup separable, then this result would also imply that all but finitely many fillings of M have virtual positive betti number. One case where weak separability is known is when N = M(a) is irreducible and the incompressible surface 5 in N is a torus. Then N is Haken and, by geometrization, 71"1 (N) is residually finite. Using this it's not too hard to show that 71"1 (5) is a separable subgroup. So in this case 71"1 (N) maps to a amalgam of finite groups. In the next section, we will use these ideas in this special case to show that all of the Dehn filings on the sister of the figure-8 complement satisfy the Virtual Haken Conjecture.

12

The sister of the figure-8 complement

Let M be the sister of the figure-8 complement. The manifold M is the punctured torus bundle where the monodromy has trace -3, and is also the surgery on the Whitehead link W(5). We will use the basis (/-', >.) of 71"1 (8M) coming from the standard basis on W. We will show: 12.1 Theorem Let M be the sister of the ilgure-8 complement. Then every Dehn illling of M which has inilnite fundamental group has virtual positive betti number.

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Proof The manifold M has a self-homeomorphism which acts on 71"1 (8M) via (fJ,A) c-+ (fJ+ A,-A). Let N be the filling M(4)"" M(4/3). The manifold N contains a separating incompressible torus. It turns out that this torus splits N into a Seifert fibered space with base orbifold D2(2,3) and a twisted interval bundle over the Klein bottle. Rather than describe the details of this splitting, we will simply exhibit the final homomorphism from 71"1 (N) onto an amalgam of finite groups. In fact, 71"1 (N) surjects onto r = 53 *C 2 C 4 where Cn is a cyclic group of order n.

According to SnapPea, the group 71"1 (N) has presentation: (a,b I ab 2ab- 1a 3 b- 1 = ab 2a- 2b2 = 1) where fJ E 7I"(8M) becomes ab in 7I"l(N). If we add the relators a 3 the presentation of 71"1 (N), we get a surjection from 71"1 (N) onto

r

=

(a,b

I

a 3 = b4 = (ab 2 )2 =

=

b4 = 1 to

1).

As 53 has presentation (x,y I x 3 = y2 = (xy? = 1), we see that r is where the first factor is generated by {a, b2 } and the second by b.

53*C2C4

We will need: 12.2 Lemma Let

r

be

53 *C 2

C 4 and let I E r be abo The group

has virtual positive betti number for all n ;:> 10. For n < 10, the group ilnite.

rn

is

Assuming the lemma, the theorem follows easily. Given a slope a in 7I"1(8M), if either L'l(a,4) ;:> 10 or L'l(a,4/3) ;:> 10 then M(a) has virtual positive betti number. The only a which satisfy neither condition are E = {D, -1, co, 1, 1/2, 2, 3, 3/2, 4, 4/3, 5/2, 5/3, 7/3, 7/4}. One can check that the fillings along these slopes either have finite 71"1 or have virtual positive betti number (the 6 hyperbolic fillings in E are all among the census manifolds which we showed have virtual positive betti number in the earlier sections). Now we will prove the lemma. Proof of Lemma 12.2 As in the case of a FUchsian group the key is to show: 12.3 Claim Let n ;:> 12. Then there is a homomorphism f from r to a ilnite group Q where f is injective on the amalgam factors 53 and C4 and where I has order n.

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Assuming this claim, we will prove the theorem for n 2' 12. The Euler characteristic (in the sense of Wall [55]) of I' is 1/6 + 1/4 - 1/2 = -1/12. Let K be the kernel of j. The subgroup K is free, and from its Euler characteristic we see that it has rank 1 + #Q /12. Let KI be the kernel of the induced homomorphism from I'n -+ Q. Then H1(KI,Z) is obtained from H1(K, Z ) by adding #Q/n relators. As n 2' 12, this implies that H1(KI, Z ) is infinite and I'n has virtual positive betti number. To prove the rest of the theorem, one can check that I'lO and I'll have homomorphisms into S12 and PSL 2 lF23 respectively whose kernels have infinite H 1 . Using coset enumeration, it is easy to check that I'n is finite for n < 10. Now we establish the claim. For each n, we will inductively build a permutation representation j: I' -+ Sn where j (,) has order n. We will say that j: I' -+ Sn is special if it is faithful on the amalgam factors, j (,) is an n-cycle, and j (b) fixes n. If j satisfies these conditions except for j (b) fixing n, we will say that j is almost special. Our induction tool is: 12.4 Claim Suppose that j is a special representation of I' into Sn. Then there exists a special representation of I' into Sn+6' Also, there exists an almost special representation of I' into Sn+7.

To see this, let j be a special representation. First, we construct the representation into Sn+6' Let L

=

{1, 2, ... , n} U {P1,P2,P3,P4,PS,P6}.

We will find a special representation into SL. Let g: I' -+ S{n,PI"P6} be the special representation given by

g(a)

= (P1P2P3)(P4PSP6)

and

g(b)

= (np1)(P2P4P3PS).

It's easy to check (using that j(a) commutes with g(b 2 ), etc.) that h(a) = j(a)g(a) and h(b) = j(b)g(b) induces a homomorphism h: I' -+ SL. Moreover, h(ab) = j(a)g(a)j(b)g(b) = j(a)j(b)g(a)g(b) = j(ab)g(ab). Thus h is the product of an n-cycle and a 7-cycle which overlap only in n, and so is a n + 6 cycle. So h is special. To construct the almost-special representation, do the same thing, where 9 replaced is now defined by

This establishes the inductive Claim 12.4.

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Using the induction, to prove Claim 12.3 it suffices to show that there are special representations for n = 6,7,15,17, and that there is an almost-special representation for n = 16. These are n = 6

a c-+ (1,2,3)(4,5,6)

b c-+ (2,4,3,5) n = 7

a c-+ (2,3,4)(5,6,7)

b c-+ (1,2)(3,5,4,6) n = 15

a c-+ (2,3,4)(5,7,9)(6,8,11)(12,13,15)

b c-+ (1,2)(3,5,4,6)(7,10,11,14)(8,12,9,13) n = 16

a c-+ (2,3,4)(5,7,9)(6,8,11)(12,13,15)

b c-+ (1,2)(3,5,4,6)(7,10,11,14)(8,12,9,13)(15,16) n = 17

a c-+ (2,3,5)(6,8,11)(7,10,9)(12,15,13)(14,16,17)

b c-+ (1,2,4,7)(3,6,9,12)(5,8,10,13)(11,14,15,16).

This completes the proof of the claim, the lemma, and thus the theorem.

D

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237

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Dunfield and Thurston

Appendix

As

L2 ( 'r )

1.00 0 ,02

0,0 2 1.00

0. 13 0,0 4

0,01S 0. 2 3

0. 1 'r 0,01S

Ae

0. 13

0,0 4

1.00

-0,0 4

0. 13

L2 (8 )

0 ,0 1S 0. 1'1

0.23 0,01S 0. 16

0. 20

0,03 0. 1 2 0 .15

0,01S 0,06 -0,0 2

0,0 4 0. 1 3 -0,0 '1 0,0 2 0. 10 0. 11

1.00 0,02

0,03

S Z(8)

Q,Og 0 ,02 0,0 2 0 ,0 1 0,0 4 0 ,1 6 0,0 1 0 ,0 1 0,0 8 0 ,11 0, 15 0,02 0, 18 -0,00

L 2 (32)

0 ,0 '1

0,0 4 0 .12 O,Og 0 ,1 0 0,06 0,0 3 O, W 0, 13 0,08 0, 14 0,03 0,0 1 0,0 2 0,02 0 ,06

Q,Og 0,04 0,00 0,03 0, 15 0,2 1 0,0 5 0,0 1 0, 18 0, 12 0, 13 0,06 0, 24 0, 11 -0,02

L2 ( 11 ) L 2( 13)

L2 ( 1'r ) A'r

L 2( W ) L 2( 16) L S(3)

U S (3) L 2( 2 3) L 2 (25) Mll

L 2(2 'r ) L2 (2 g ) L 2(3 1 ) As Lg (4) L 2(3 'r ) U4 ( 2 )

As L 2( 'r )

Ae L2 (8) L 2 ( 11 ) L 2 13) L2 ( 1'r A'r L2 ( W ) L 2 (1 6) Lg (3) US(3) L 2 (23) L 2(2 5 ) Mll

L2 (2 'r ) L 2(2 g ) L 2 (3 1 ) As L S( 4 )

L 2(3 'r ) U 4(2) S Z(8)

L2 ( 3 2)

0,0 1 0, 10 0,03

0,0 '1 0,0 5 0,0 3 0, 12 -0,04 0,03 0,03 0, 15 0,0 4 1.00 O,O g 0,0 4

0,0 '1 0,02 0,0 8 0,02 0,0 1 0,00 0,0 1 -0,0 4 0,0 8

0,0 4 0,06 0, 15 0,06 0, 14 0,03 0, 13 O,Og 0, 10 0 ,1 0 0,2 1 0,08 O,O g 1.00 0,05 0 ,1 5

0,0 '1 0, 14 0, 12 0,0 5 0,06 0, 10 0,03 0,03

0 ,1 6 0,03 0, 21 0,00 O,O g -0,02 O,O g 0 ,1 2 0,0 1 0 ,0 5 0,0 5 0,06 0,0 4 0,0 5 1.00 0,0 1 -0,00 0, 14 0, 14 O, lg 0 ,00 0,2 1 O,Og 0,0 4

0,0 2 1.00

0,03 0. 16 -0,0 '1 0.20 -0,0 1

0,06 0,03

-0,0 1

1.00

0,00

0,06 0,08 0,05

0,03 0. 11 0 .11

0,00 -0,0 1 0,04

1.00 0,0 1 0,01S

0,00 -0,00 0, 11

0. 14

0 ,0 '1

0,0 4 0,06 O,O g 0,0 3 0,03 -0,0 2 0, 15 0, 10 -0,05 0,08 -0,0 4 0,02 0,0 4 -0,03 0,0 3

0,0 5 0 ,0 5 0, 14 O,Og 0,0 2

0,0 '1 0,06 -0,00 0, 2g 0, 14 0,00 0, 11 0,02 0,0 2 0,00 -0,0 1 -0,02

0 ,1 '1 0, 10 0,0 8 0, 11 0,06

0,0 '1 0,03 0 ,0 1

0,0 1 O, W -0,0 5 0,2 g 0,02 0, 15 0,04 O,O g 0,04 0,00 0,06 0, 10

0,0 '1 0, 15 -0,0 1 1.00 O, W 0,0 1 0, 11 0,02 0,0 3 0,0 1 -0,0 4 0,05

0,0 1 0, 13 0,0 1 0, 14

0, 1'1 0, 10 0,00 O, W 0, 15 0,06 0,00 0,0 1 0,02

0,0'1 -0,00 O, W 1.00

0,0 '1 0, 12 0, 11 0,0 8 0,03 -0,0 1 -0,02

0,0 3 0,01S

0. 12 0,06

0. 115 -0,02

Q,O g -0,04

0,02 0 .1 2

0,0 2 Q,O g

0,02

0. 10

0. 11

Q,O g

0,04

0,00

0,08 0. 11

0,01S 0. 11

0,00 0. 14

0,00 0,0 '1

0 .11 0,01S

-0,0 1

0,0 4

0,0 4

0,06

Q,Og

0,0 1 1.00 0 ,08

0,01S 0,08 1.00

0,03 0. 10 0. 11

0. 11 0,03 0,03

0,03 0. 11 0, 12 0, 12 0, 13 O,Og 0,0 4 -0,00 0, 11

0. 10 0 ,03 0, 11 0 ,0 4 O,O g 0 ,1 2 O,O g O,W 0,0 4

1.00 -0,0 2

0,0 '1

0 ,1'1

0,02 1.00 0, 10 0 ,15 0,2 1 0,05 0,06 0,00 O,O g 0,04

0 ,1 0 1.00 0 ,0 4 0,08 0 ,06 0, 10 0 ,0 1 0,06 0 ,11

0,03

0,23 0,04 0, 13 -0,02 -0 ,02

0. 11 0,0 3 0,03 0,0 3 0, 10 0,0 1 0,04 0, 15 0, 10 0, 10 0,05 0,08 0,05 O,O g 0,0 1

0. 12 0. 11 0 ,03 0,0 '1

0,0 '1

0,03

0,00 0,0 2 -0,0 1 -0,00

0,02 0,0 1 -0,03 0 ,05

0,0 '1 0,0 1 0,00 0,00

0 ,08 0,0 8 0, 18 0,00 0, 10 -0,0 5 0, 11 0 ,0 4 0, 10 O,Og O,Og 0 ,06 0,0 8 0 ,1 4 0, 14 0 ,0 1

0,0 '1 1.00 O,O g 0, 10 0 ,02 0, 13 0 ,08 0,08

0, 11 0, 14 0, 12 0, 11 0,0 8 0,0 8

0,0 '1 0, 1'1 0, 10

0,0'1 0,04 0, 11 0,0 2 0, 12 0, 14 0, 11 0, 12 O,O g 1. 00 0, 15 -0,0 1 0, 14 0,0 8 -0,03

0, 15 0,03 0, 13 0,0 2 0, 11 -0,0 4 0,03 0, 2 3 0,0 5 0,0 1

0,0 '1 0,03 0,0 1 0,0 5 O, W 0,02 0, 11 0, 10 0, 15 1.00 -0,00 0, 21 0, 26 -0,0 4

0,02 0,0 1 0,06 0,02 0,06 0,0 2

0,0 '1 0,0 4 0,08 0 ,1 3 0,00 0,02 0,00 0,06 0,00 0,03 0,08 0,02 0,0 1 -0,00 1.00 0,0 1 -0,03 0,06

0,0 '1 0,0 3 0, 10 0,0 5 0,00 0,06 O,Og

0,0 '1 0,0 1 0, 13 0,0 5 -0,03 0,0 2

0 ,1 8 0,0 2 0, 24 0,00

0,0 '1 -0,0 4 0,0 1 0 ,13 0,0 5 0 ,0 5 0,0 2 -0,0 1 0,0 1 0 ,1 0 0,2 1 0,0 1 0,03 0, 13 0, 14 0, 21 0 ,0 1 1.00 0 ,02 0,0 4

0,00 0,0 2 0, 11 0,0 1 0,03 -0,03 0,00 -0,0 2 O,O g 0,03 -0,0 1 -0,03 0,0 4 0,03 O,Og 0,0 4 -0,0 1 0,08 0,08 0, 26 -0,03 0,02 1.00 -0,03

0,0 '1 0,06 -0,02 0,0 2 0,0 1 0,03 0,00 -0,02 0,0 1 0 , 02 -0,00 0,0 5 0,0 8 0,03 0,0 4 0,0 5 -0,0 2 0,0 8 0,03 -0,0 4 0 ,06 0,0 4 -0 ,03 1.00

Table 5: This table gives the correlations between: (having a cover with group 1, having a cover with group 2). The average off-diagonal correlation is 0.06.

238

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The virtual Haken conjecture: Experiments and examples

AS

L2('r) Ae L2(8) L2(11) L2(13) L2(1'r) A'r L2(19) L2(16) Ls(3) Us 3

L2(23 L2(2.'5) Mll

L2(2'r) L2(29) L2(31) As Ls(4)

L2(3'r) U4(2) SZ(8)

L 2 (32)

As

L2('r) Ae L2(8) L2(11) L 2 (13) L2(1'r) A'r L2(19) L2(16) Ls(3)

Us(3)

L 2 (23) L2(2.'5) Mll

L2(2'r) L2 (29) L2(31) As Ls(4)

L2(3'r) U4(2) SZ(8)

L2(32)

1.00 -0,0 1 0,28 0,0.'5 0,2.'5 0,01 0,06 0,11 0,23 0,1 1 0,02 0,02 0,06 0, 12 0,19 0,03 0,08 0,22 0, 11 0,2 1 0,09

0, 1'1 0,08 0,06

0,06 0, 13 0,02 0,04 0,06 0,0.'5 0, 13 -0,01 0,06 0,0.'5 0, 1.'5 0,09 1.00 0, 11 0,0.'5 0,0 4 0,04 0,08 0,0 1 0,0 4 0,09 0,02 -0,0.'5 0, 1.'5

0,0 1 1.00 0,0.'5 0,38 0,0 4 0,2.'5 0, 14 0, 11 0,02 0,0 1

0,28 0,0.'5 1.00 0,00 0,22

-0,0'1 0, 12 0, 13

0, 1'1 0, 10 0,08 0,02 0,02 0,20 0,33 0,06 0,0 4 0,30 0, 1.'5

0, 1'1 0, 13 0, 13 0, 13 0,04 0,38

0, 1'1 0,08 0, 15 0,08 0,03 0,03 0,08 0,0.'5

0,2'1 0, 12 0,34

0, 1'1 -0,0 1

0, 12 0, 13 0,20 0, 14

0, 1'1 0,06

0, 1'1 0, 12 0, 14 0, 1.'5 0,2 1 0, 13 0, 11 1.00 0, 12 0, 1.'5 0, 15 0, 18 0,20 0,06 0, 16 0, 14 0,0 4 0,02

0,19 0,0 4 0,33 0,03 0, 12 0,00 0, 11

0, 1'1 0,0'1 0,08 0,06

0,0'1 0,0.'5 0, 12 1.00 0,04 0,02 0,22 0,14 0,24 0,0.'5 0,2.'5 0,08 0,0.'5

0,0.'5 0,38 0,00 1.00 0,0.'5 0,36 0, 11 0, 12 0,06 0,06 0,03 0, 12 0,0 4 0, 14 0,03 0,4.'5 0,2 4 0,02 0, 14 0,04 0,0.'5 0,0 1 0,03 0,0 1

0,03 0,38 -0,06 0,45 0,0.'5 0,3.'5 0,06 0, 10 0,0 1 0,0 1 0,09 0,16 0,04 0, 1.'5 -0,04 1.00 0,2.'5 -0,03 0, 10 0,02 0,03 0,0 1 0,0 1 0,08

0,2.'5 0,0 4 0,22 0,0.'5 1.00 0,03

0,0 1 0,2.'5

-0,0'1

0,06 0,1 4 0, 12 0,1 1

0, 11 0, 11 0, 13 0, 12 0,06 0,0 1

0,23 0,02

0, 1'1

0,36 0,03 1.00

0,0'1 0,0'1

0,0'1

0,0'1

1.00

0,0'1

0,06 0, 18 0,0 4 0, 12

0,06 0,18 0, 12 0,08 0,0 4 0,06

0,0 1 0,04 0, 10 0,08 0, 13 0,0.'5 0,06 0,00 0,3.'5 0, 18 0,02 0, 12 -0,01 0,10 0,0 1 0,0 1 0,06

0,0'1

1.00

0,0'1

0, 13 0,0 1 0, 12

1.00 0,09 0,08 0,0.'5 0,06 0, 14

0, 1'1

0,0'1

0, 10 0,22 0,08 0,20 0,28 0,02 0, 1.'5 0,04 -0,01

0,0 1 0, 1.'5 0, 15 0,08 0, 13 0,09 0, 10 0,10 0,04

0, 1'1 0, 12 0,0.'5 0,2 4 0, 1.'5 0,08 0, 1.'5 0,14 0, 10 0,06 0,0.'5

0,08

0,1'1 0,0 4 0,2 4 0,2 4 0, 18 0,02 0,22 0, 1.'5 0,0.'5 0,06 0,03 0,0 4 0, 1.'5 0,02 0,2.'5 1.00 0,06 0,13 0,2 1 0, 12 0,0 4 0,04 0,01

0,22 0,08 0,30 0,02 0, 1.'5 0,02 0,2 4 0,08 0, 1.'5 0,09 0, 1.'5 0,08 0,08 0,18 0,22 0,03 0,06 1.00 0, 10 0, 12 0, 12 0, 1.'5 0, 10 0,06

0, 12

0,0'1

0,0'1

0,09

0, 1.'5 0, 11 0, 13

0,0'1

0, 1'1 0, 11 0,06 0,02 0,24 0,1 4 0,05 0, 1.'5 0,0.'5 0,0.'5 0,02

0, 11 0, 15 0, 1.'5 0, 14 0,08 0, 12 0, 14 0,20 0,08 0,09 0,09 0, 12 0,0 1 0,20 0, 14 0, 10 0, 13 0, 10 1.00 0, 18

0,0'1 0, 16 0,09 -0,03

0,2 1 0,08

0,2'1 0,04 0, 1.'5 -0,0 1 0,0.'5 0,28 0, 13 0,09 0, 11 0,04 0,0 4 0,06 0,24 0,02 0,21 0, 12 0, 18 1.00 0,02 0,2.'5 0,30 -0,04

0,09 0,03 0, 12 0,0.'5 0, 14 0, 10 0, 1.'5 0,02 0,09 0, 16 0,03 0,08 0,09 0, 16 0,0.'5 0,03 0, 12 0, 12

0,0'1 0,02 1.00 0,06 0,03 0, 10

0, 11 0,0 1 0, 10 0,06 0, 12 0, 10

0,0'1 0,0 9 0,09 1.00 0,03 0, 10 0,0.'5 0, 1.'5 0,08 0,0 1 0,0.'5 0,09 0,09 0,09 0, 16 0,08 0,03 0,0.'5

0, 1'1 0,03 0,34 0,0 1 0, 10 -0,0 1 0,0.'5 0, 15 0, 10 0,08 0,0.'5 0,02 0,02 0, 14 0,2.'5 0,01 0,0 4 0, 1.'5 0,16 0,2.'5 0,06 1.00 -0,0 1 0,0 1

0,02

0,02

0, 1'1

0, 13

0,08 0,03 0,08 0,08 0, 1.'5

0,02 0, 12 0,04 0, 13 0, 11

0,0'1

0, 13

0,08 0,03 1.00 0, 14 0, 1.'5 0,2 1 0,06 0,09 0,06 0, 15 0,09 0, 11 0,03 0,0.'5 0,02 0,0 1

0,0.'5 0, 10 0,14 1.00 0,09 0, 13

0,0'1 0, 16 0,03 0,08 0, 12 0,04 0,08 0,02 -0,01 0,08

0,08 0,08

0,06 0,0.'5

0, 1'1

-0,0 1

0,03 0,06 0,0 1 0,0.'5 0,04 0, 10 0,03 0,02 -0,0 1 0,0.'5 0,04 0,08 0,0 1 0,04 0, 10 0,09 0,30 0,03 0,0 1 1.00

0,0 1 0,0.'5 0,06 0,02 -0,01 0,0 4 0,0.'5 0,0 1 0,08 0, 1.'5 0,02 0,0.'5 0,08 0,0 1 0,06 0,03 -0,04 0, 10 0,0 1

-0,0'1

-0,0'1 1.00

Table 6: This table gives the correlations between: (having a cover with group 1 with positive betti number, having a cover with group 2 with positive betti number). The average off-diagonal correlation is 0.09.

WILLIAM P. THURSTON

239

Inventiones mathematicae

Finite covers of random 3-manifolds* Nathan M. Dunfield l , William P. Thurston2 1 2

Mathematics 253-37, Caltech, Pasadena, CA 91125, USA (e-mail: dunf ield@caltech. edu) Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA (e-mail: [email protected])

Oblatum 1O-11I-2005 & 24-IV-2006 Published online: 7 July 2006 - © Springer-Verlag 2006

Abstract. A 3-manifold is Haken if it contains a topologically essen-

tial surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is o. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface E on the set of quotients Jl'l (17) ---+ Q. If Q is a simple group, we show that if the genus of E is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman's theorem that the action of the mapping class group on the SU (2) character variety is ergodic. * Authors were partially supported by the US NSF and the Sloan Foundation. Mathematics Subject Classification (2000): 57M50, 57NlO

WILLIAM P. THURSTON

241

458

N.M. Dunfield, W.P. Thurston

Contents I 2 3 4 5 6 7 8 9

Introduction............. . . . . . . . . . . . . . . . Models of random 3-manifolds . . . . . . . . . . . . . . . . . . Random balanced presentations. . . • . . . . . . • . . . . . . . The profinite point of view . . . . . . . . . . . . . . . . . . . . Quotients of 3-manifold groups . . . . . . . . . . . • . . . . . . Covers of random Heegaard splittings . . . . . . • . . . . . . . Covers where the quotient is simple . . . . . . . . . . . . . . . Homology of random Heegaard splittings . . . . . . . . . . . . Homology of finite-sheeted covers . . . . . . . . • . . . . . . .

......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . ........ . ........ • . . . . . • . .

458 463 470 480 484 491 503 508 512

1 Introduction Here, we study various notions of a random 3-manifold, and try to understand the distribution of topological and group-theoretic properties for such manifolds. Our primary motivation is to try to determine the underlying issues behind the Virtual Haken Conjecture and related problems about properties of finite covers of 3-manifolds. While any hyperbolic 3-manifold has many finite covers - its fundamental group is residually finite - we do not know what most of the covering groups are, much less the properties that we may reasonably expect these covers to have. The reason that the fundamental group of a hyperbolic 3-manifold M is residually finite is that it is a finitely generated group of matrices, and this gives many quotients of 11"1 (M) of the form PS~lF, where IF is a finite field. Lubotzky has shown [Lub2] that such quotients have measure zero among all quotients of 11"1 (M), but his proof provides little insight into what those other quotients might be. For instance, is it reasonable to expect 11"1 (M) to have a quotient which is an alternating group An? Should there be many such quotients? Also, for a particular kind of finite quotient, how likely is it that the associated cover be Haken or have positive betti number? Although there have been many partial results on the Virtual Haken Conjecture, these questions have been hard to address in general by direct deductive reasoning. Another way of thinking about these questions is from a probabilistic point of view. Since the set of homeomorphism types of compact 3-manifolds is countably infinite, there is no uniform, countably-additive, probability measure on this set. Thus the first issue is to define a plausible context in which we can discuss probability. The density of3-manifolds with any given property will depend on the order in which we consider them, unless the property is either true or false for all but a finite set of 3-manifolds. It seems best to us to analyze orders of enumeration that are plausible and tractable, while acknowledging that there may be other equally plausible and tractable orders that give different answers. In Sect. 2, we outline several reasonable models for a random 3-manifold, and then in most of the rest of this paper concentrate on a model of random 3-manifolds which comes from looking at Heegaard splittings generated by random walks in mapping class groups. 242

COLLECTED WORKS WITH COMMENTARY

459

Finite covers of random 3-manifolds

1.1 Random Heegaard splittings

Every closed orientable 3-manifold has a Heegaard splitting, that is, it can be constructed by gluing two handlebodies of genus g together using a homeomorphism between their boundaries. We will look at 3-manifolds of a fixed Heegaard genus g, and consider gluings obtained by a random word in a finite set of generators for the mapping class group of the surface of genus g. In principle, the density of a particular property of the manifolds obtained in this way could depend on the choice of generators for the mapping class group. Indeed, this is plausible since in a non-amenable group such as this one, the correlation between random words of large length in different sets of generators usually tends to O. However, the properties we analyze have limiting densities independent of this choice of generators. In particular, for a finite group Q the probability that the manifold obtained from a random genus g Heegaard splitting has a cover with covering group Q is well-defined (Prop. 6.1), and we denote this probability as p(Q, g). When Q is a simple group, we are able to calculate the limit of these probabilities as Ihe genus g goes to infinity: 7.1 Theorem Let Q be a non-abelian finite simple group. Then as the genus g goes to infinity, the probability of a Q-cover converges:

p(Q, g) ---+ I - e-P.

where

JL = IH2(Q; Z)I/IOut(Q)I.

Moreover, the limiting distribution of the number of Q-covers converges to the Poisson distribution with mean JL.

For example, if Q = PS~F p where p is an odd prime then JL = 1. Thus forlarge genus the probability of aPS~F p coveris about l-e- 1 Rj 0.6321. Hyperbolic 3-manifolds must have infinitely many covers of this form, namely Ihe congruence quotients. However, at least naively, one expects far fewer congruence quotients than given by Theorem 7.1. Another interesting example is Ihe case where Q = An is an alternating group; here again JL = 1 and the probability of an An cover is greater than 63%. Moreover, we show that covers with different groups Qi do not correlate with one another, at least for large genus. As a consequence, we can prove results such as the following, which is a special case of Theorem 7.8:

7.7 Theorem Let E > O. For all sufficiently large g, the probability that the 3-manifold obtained from a random genus-g Heegaard splitting has a An -cover with n 2: 5 is at least 1 - E. Moreover, the same is true if we require some fixed number k of such covers. These results should be contrasted wiIh the analogous results for finitely presented groups with an equal number of generators and relators, where the relators are chosen at random. There, the probability of a An -quotient goes to zero like 1/n! as n goes to infinity, rather than remaining constant WILLIAM P. THURSTON

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(Theorem 3.10). Thus one way of interpreting our results is that they affinn the belief that 3-manifolds have many finite quotients. In Sect. 5, we give some heuristic ways to understand why this should be true, working from a more naive point of view. These stem from the fact that the group relators given by a Heegaard splitting come from disjoint embedded curves on a surface. In particular, we highlight special features of attaching the last two 2-handles in forming a 3-manifold that suggest many extra finite quotients.

1.2 Mapping class group The proof of Theorem 7.1 boils down to understanding the action of the mapping class group of a surface on a certain finite set. Let E g be a closed surface of genus g, and let oMg be its mapping class group. Consider the set .Ag of epimorphisms of 11'1 (Eg) onto our fixed simple group Q, modulo automorphisms of Q. Then oMg acts on.A g via the induced automorphisms of 11'1 (Eg), and we show:

1.3 Theorem Let Q be a non-abelian finite simple group. Then for all sufficiently large g, the orbits of .A g under the action of oMg correspond bijectively to H2 (Q, 7/.,)/ Out(Q). Moreover, the action of oMg on each orbit is by the full alternating group of that orbit. For a finite group Q which is not necessarily simple the orbits can be classified in the same way (Theorem 6.20); that the action on an orbit is by the full alternating group is special to simple groups Q (Theorem 7.4). You can view Theorem 1.3 as saying that when the genus is large the action of oMg on.A g is nearly as mixing as possible. As such, it is directly analogous to Goldman's theorem that the action of oMg on the SU(2)-character variety is ergodic for any genus 2: 2 [GoIJ. Perhaps surprisingly, the proof of Theorem 1.3 uses the Classification of Simple Groups even for concrete cases such as Q = An. However, Theorem 7.1, which is a corollary of Theorem 1.3, also follows from a weaker version which does not use the Classification. What about other types of finite groups? For abelian groups, we give a complete picture of the distribution of HI (M) for a 3-manifold coming from a random Heegaard splitting (Sect. 8). For a general finite group Q, we do not know how to show the existence of a limiting distribution as the genus goes to infinity, but we can show that the expected number of Q-quotients does converge (Theorem 6.21).

1.4 Virtual positive betti number These results give a good picture about the number of different types of covers in many cases, so we now turn to the main question at hand:

1.5 Virtual Haken Conjecture Let M be a closed irreducible 3-manifold with 11'1 (M) infinite. Then M has a finite cover which is Haken, i.e. contains an incompressible surface. 244

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This conjecture was first proposed by Waldhausen in the 1960s [Wal]. It is often motivated as a way of reducing questions to the case of Haken manifolds, where one has the most topological tools available. However, we prefer to view it as an intrinsic question about the topology M itself: does M contain an immersed incompressible surface? If so, can we lift it to an embedded surface in some finite cover? From a more algebraic point of view, one of the fundamental tasks of 3-manifold topology is to understand the special properties of their fundamental groups, as compared to finitely presented groups in general; thus it is natural to ask: does 1l'1 (M) always contain the fundamental group of a closed surface? (Having 1l'1 (M) contain a surface group is equivalent to M having an immersed incompressible surface, but is a priori weaker than having a finite cover which is Haken. The difference is another subtle and interesting question about 1l'1 (M), namely subgroup separability.) Perelman has announced a proof of Thurston's Geometrization Conjecture using Hamilton's Ricci flow [Perl,Per2]; this should reduce the Vrrtual Haken Conjecture to the (generic) case when M is hyperbolic. For hyperbolic M, the Virtual Haken Conjecture fits nicely into a more general question of Gromov: must a I-ended word-hyperbolic group contain a surface group? In any event, it seems to us that it would be very hard to prove the Virtual Haken Conjecture without first establishing Geometrization; for instance, the only known way to show that an "easy to understand" atoroidal Haken manifold M has a non-trivial finite cover is to hyperbolize it to see that 1l'1 (M) is in fact a group of matrices, hence residually finite! Deciding whether a 3-manifold is Haken is difficult, so in the rest of this paper we focus on the stronger version of the conjecture which asks that the finite cover has positive first betti number. (In the case of arithmetic 3-manifolds, this is also the version that relates directly to the theory of automorphic forms.) In our prior work [DT2], we found that this conjecture holds for all RJ 11,000 of the small volume hyperbolic 3-manifolds in the Hodgson-Weeks census. One of our goals here is to determine whether the pattems we observed there are in some sense generic, or are a consequence of special properties of that sample. For simple quotients, the results above give even larger probabilities for such covers than those we observed in [DTZ] (see Sect. 6.5 for a quantitative comparison). However, for the crucial question of whether simple covers have positive betti number, a different picture emerges for our random manifolds here than we saw in [DT2]. In [DT2, §5], we found that covers with a particular fixed finite simple group Q had positive betti number with probabilities between 52-98% depending on Q. However, for our Heegaard splitting notion of random our experimental evidence strongly suggests that these probabilities are O. Moreover, we can show

9.1 Theorem Let Q be a finite abelian group. The probability that the 3-manifold obtained from a random Heegaard splitting of genus 2 has a Q-cover with fh > 0 is O. WILLIAM P. THURSTON

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1.6 Potential uses of random 3-manifolds In combinatorics, studying random objects is done not just for the intrinsic interest and beauty of the subject but also for the applications. For instance, constructing explicit infinite families of expander graphs is quite difficult; the first such construction was based on the congruence quotients of PSLzZ and uses Selberg's 3/16 Theorem (see e.g. [Lubl]). On the other hand, proofs of existence and practical construction can be done by looking at certain classes of random graphs and showing that the desired property occurs with non-zero probability. Closer to the study of 3-manifolds, Gromov initiated the study of groups coming from certain types of random group presentations. These have been used to produce many examples of word-hyperbolic groups with additional properties, such as having Property T [Zuk,Gro2J. Very recently, Belolipetsky and Lubotzky have used random techniques to show that given n and a finite group G there exists a hyperbolic n-manifold whose full isometry group is exactly G [BLJ. Perhaps similar techniques could be applied to questions about 3-manifolds. In particular, to construct 3-manifolds with a certain list of properties, one could try to show that these properties occur with positive probability for a suitable model of random 3-manifold. For such applications, the fact that a random 3-manifold is an ill-defined concept becomes a strength rather than a weakness, since by varying the model one can change the characteristics of the resulting manifolds. Finally, another point of view on random 3-manifolds is that they provide a quantitative context in which to understand one of the central questions in 3-dimensional topology: how do 3-manifold groups differ from finite presented groups in general? As we mentioned, our results show that from the point of view of random Heegaard splittings, 3-manifold groups have many more finite quotients than finitely presented groups in general. Recent work of the first author and Dylan Thurston shows a similar sharp divergence behavior with respect to fibering over the circle, where here a group "fibers" if is an algebraic mapping torus [DT IJ. Surprisingly, the 3-manifolds studied there were much less likely to fiber than similar finitely presented groups.

1.7 Outline of contents In Sect. 2, we discuss several different models of random 3-manifolds and some of their basic properties. In Sect. 3, we discuss groups coming from random balanced presentations, both as a warm up for the 3-manifold case and to provide a point of comparison. We compute the probabilities that such random groups have a particular abelian or simple quotient. Our results about random balanced presentation fit most naturally into the context of profinite groups as we discuss in Sect. 4. Also in Sect. 4, we define a profinite generalization of random Heegaard splittings. In Sect. 5, we discuss some reasons why 3-manifolds should have many finite quotients, working from a more naive heuristic point of view than in later sections. 246

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The remainder of the paper, Sects_ 6-9, focuses on the specific model of random Heegaard splittings and on the finite covers of the corresponding 3-manifolds. Sect. 6 contains as much of the picture as we could develop for an arbitrary finite covering group Q. In Sect. 7, we give much more detailed results in the case when Q is simple. Similarly, Sect. 8 is devoted to the case when Q is abelian. Finally, Sect. 9 discusses the homology of a cover of a random 3-manifold. Acknawledgements. Part of the work for this paper was done while the authors were at Harvard and UC Davis, respectively. The first author was partially supported by an NSF Postdoctoral Fellowship, NSF grant DMS-040549l, and a Sloan Fellowship. The second author was partially supported by NSF grants DMS-9704135 and DMS-0072540. The first author would like to thank Micbael Ascbbacber for helpful conversations. We also thank the referee for a very careful reading of this paper and numerous helpful comments thereon.

2 Models of random 3-manifolds In this section we give several different models of random 3-manifolds, and outline some elementary properties about them. In each case, the idea is to filter 3-manifolds in such a way so that number of 3-manifolds with bounded complexity is finite.

2.1 Random triangulations

Arguably the most natural notion of a random 3-manifold comes from filtering by the number of tetrahedra in a minimal triangulation. To sidestep the difficult problem of determining minimal triangulations, we can make the triangulations themselves the basic objects. Let 13 (n) be the set of oriented triangulations of closed 3-manifolds with n tetrahedra. Here, a triangulation is just an assemblage of 3-simplices with their faces glued in pairs, and need not be a simplicial complex in the classical sense. In the probabilistic setting' we are interested in the properties of the manifolds in 13 (n) as n tends to infinity. For instance, does the probability that M E 13 (n) is hyperbolic go to 1 as n -+ oo? Unfortunately, it seems very difficult to prove anything about the manifolds in 13(n), or even generate random elements of 13(n) for large n. As we will explain (Proposition 2.8), the problem is that if we start with n tetrahedra and glue their faces in pairs we almost never get a 3-manifold. 2.2 Random sUrfaces

We will start with the 2-dimensional case, since one gets a good picture there and it helps explain the problem in dimension 3. Let 12 (n) be the set of oriented triangulations of (not necessarily counected) surfaces; as with studying random graphs, it is convenient to make these labeled triangulations where each triangle is assigned a number in 1, 2, . .. ,n and also WILLIAM P. THURSTON

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has an identification with the standard 2-simplex. Since you can not build a surface from an odd number of triangles, we will always assume that n is even. Another point of view on a triangulated surface is to look at the dual I-skeleton. This is a trivalent graph with labeled vertices where the incoruing edges at each vertex are labeled by 1, 2, 3 according to the label of the edge that they are dual to. Conversely, such a labeled trivalent graph gives a triangulation in :7i (n). This triangulation is uuique since there is only one way to glue a pair of sides on two oriented triangles compatible with the orientations. We can generate elements of:7i (n) with the uniform distribution easily, indeed in time linear in n: simply start with n triangles and pick pairs of sides at random and glue. Note that the dual I-skeleton of a random element of :7i (n) is a uniformly distributed random trivalent graph with n vertices. Thus we can directly apply results about random regular graphs to study properties of :7i(n) (see [Wor] for a survey of regular random graphs). For instance, it follows that the probability that 1J E T2 (n) is connected goes to 1 as n ~ 00. 2.3 Euler characteristic

We will discuss further consequences of the structure of the dual graph later, but first we will explain why the expected genus of a random surface is close to the maximum possible. With a slightly different model, that of gluing sides of an n-gon, the genus distribution is needed to compute the Euler characteristic of the moduli space of Riemann surfaces of a fixed genus. For this reason, it was studied in detail by Harer, Zagier, and Penner [HZ,Pen,Zag]. To compute the Euler characteristic of 1J E :7i(n), we just need to know the number of vertices v as X(1J) = -n/2 + v. So what is the expected number of vertices? Take the point of view of randomly gluing triangles together, and think of how the links of the final vertices are built up by the gluing process. We start with 3n link segments in the corners of the n triangles. These link segments have an orientation induced from the orientation on the triangles. At each stage, we have some number of arcs and circles built up out of these segments. At each gluing of triangles, two pairs of endpoints of arcs are glued together, respecting the orientations. If we were not gluing the link arcs two at a time, we would have exactly the same situation as counting the number of cycles of a random permutation (see e.g. [Fel, §X.6(b)]). We will first describe what would happen with this simplification and discuss the full picture below. With this simplification, at the kth arc gluing we have (3n - k + 1) choices of where to glue the positive end of the given link arc, and exactly one of these choices creates a closed link circle. Thus we expect 1I (3n - k + 1) final vertices to be created per gluing, and the expected total number of vertices is L~:l 11k Rj log(3n) Rj log(n). Note that this says that the expected genus is about nl4 -log(n) where the maximum genus possible with n triangles 248

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is Ln / 4 J. Thus the expecIed genus is quite close to the maximal one. In particular, the probability that E E T2 (n) has any fixed genus g goes to zero as n --* 00. Unfortunately, we do not know how to prove that the expecIed number of vertices is log(n); as this is somewhat tangential, we content ourselves with the following upper bound, which gives many of the same qualitative results: 2.4 Theorem The expected number of vertices for a random surface E T2(n) is at most (3/2) log(n) + 6.

E

Proof First, we will explain where the problem is with the argument we gave above. Call an unglued edge of a triangle bad if the two link arcs which intersect it are actually the same arc. Gluing a bad edge creates a link circle if and only if it is glued to another bad edge. Thus, the expecIed number of circles creaIed by such a gluing depends on the number of preexisting bad edges, which could conceivably be large. We can deal with this problem as follows. At each stage, we always pick a good edge as the first edge in the pair to glue, if there is one. Because we allow the first chosen edge to be glued to any edge, good or bad, every E is generated by this process and we have not changed the distribution on T2(n). However, we have made counting easier. Let Gk be the random variable which is the number of link circles creaIed at the kth stage by a gluing which contains at least one good edge. Let Bk be the number of bad edges creaIed at the kth gluing. (Both of these variable are set to 0 once we have exhausted the good edges.) The number of vertices in the final surface is equal to I: G k plus half the number of bad edges left at the end of the good gluings. The number of bad edges at the end is at most the number created during the whole process. Thus as expectations always add, the expected number of vertices is bounded by 3./2

1 3./2

k=!

k=!

L E(Gk) + 2: L E(Bk). We claim that if there is a good edge left, then E(G k ) = 2/(3n - 2k + 1), where the denominator is the number of choices for an edge to glue to. There are two cases to consider, depending on whether the link arcs of our chosen good edge have other endpoints on the same edge, but the probability is the same in both cases. Similarly, you can see E(Bk) ~ 2/(3n - 2k + 1). Combining, we get that the expecIed number of vertices is less than 3 I:i:!!2 1/(2k - 1) ~ 3/2Iog(n) + 6. 0 We conclude with an outline of how to turn the problem of precisely computing the expecIed Euler characteristic into a problem about the charWILLIAM P. THURSTON

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acter theory of the symmetric group. Suppose we are to create a surface from n triangles. Label the oriented sides of the triangles by 1,2, ... ,3n. A pairing of the sides can be thought of as a fixed-point free involution 7f in the symmetric group S3n. Label the oriented vertex links of the triangles by saying that such a vertex link has the same number as the edge which contains its positive endpoint. We label each triangle so that the 3-cycle (3k - 2, 3k - 1, 3k) rotates the edges of the kth triangle by 1/3 of a turn in the direction of the vertex links. Let a = (123) (456) (789) . .. E S3n be the element which rotates the sides of each triangle in this way. Then the vertex link k is glued to the vertex link (arr)(k). Thus the number of vertices of the surface corresponding to the gluing permutation 7f is just the number of cycles of arr. Hence if C is the conjugacy class of fixed-point free involutions in S3n, then the average number of vertices is:

1

.

ICI ~)num cycles ill arr). >rEC

There are different ways of attacking problems of this kind, and an elementary approach is to use the character theory of the symmetric group, see [Jac,Zag]. For other approaches, based on random matrices, see [HZ,Pen,lZ].

2.5 Local structure An interesting property of random regular graphs is that most vertices have neighborhoods which are embedded trees. More precisely, fix a radius r and let the number of vertices n get large. Then with probability approaching I, the proportion of vertices which have neighborhoods which are embedded trees of radius r is very near 1. The distribution of short cycles in a random regular graph is also understood, and as the following theorem shows, the distribution is essentially independent of the size of the graphs [Wor,

Thm2.5]: Theorem 2.6 (BolloMs) Consider regular graphs where the vertices have valence d. Let Xi,n be the random variable which is the number of cycles of length i in a random such graph with n vertices. Then for i less that some fixed k, the X i,n limit as n --* 00 to independent Poisson variables with ' . _ (d-I)' means I\,~ 2i . One consequence is that if we fix r and pick 1J E T2 (n) with n large, there is a non-zero (albeit small) probability that the shortest cycle in the dual I-skeleton has length :;>: r. That is, the "combinatorial injectivity radius" of a random triangulated surface is large a positive proportion of the time. 250

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2.7 Triangulations of 3-manifolds Now we return to trying to understand random triangulations in :73 (n). Unlike the surface case, we can not study this question by studying random gluings of tetrahedra:

2.8 Proposition Let X be the cell complex resulting from gluing pairs of faces ofn tetrahedra at random. Then the probability that X is a 3-manifold goes to 0 as n --* 00. Proof The link of a vertex in X is always a surface. Moreover, X is a 3-manifold if and only if the combinatorial link of every vertex is a sphere (the "only if' direction follows for Euler characteristic reasons). Intuitively, since we are gluing the tetrahedra at random, a link surface should be a random surface in the above sense. If this were the case, then the probability that the link is a sphere goes to 0 as n --* 00, and so X would almost never be a manifold. However, every time we glue a pair of tetrahedra, we are gluing 3 pairs of link surface pieces at once in a correlated way. We will finesse this issue by using the fact that if X is a manifold then the average valence of an edge is uniformly bounded, and contrast this with the fact that the dual I-skeleton of X is a random 4-valent graph. For the first point, Enler's formula implies that the average valence of a vertex in a triangulation of S2 is less than 6. The average valence of an edge in X is equal to the average valence of a vertex in the vertex links; thus when X is a manifold the average edge valence is less than 6. In particular, since every edge has positive valence, this implies that at least 1/6 of the edges have valence::: 6. Let r be the dual I-skeleton of X. An edge of valence k in X gives a cycle in X of length k. Thus if X is a 3-manifold, the number of distinct cycles in r of length ::: 6 is a definite multiple of the number of vertices. But r is a random 4-valent graph, and by Theorem 2.6, the distribution of the number of cycles oflength ::: 6 is essentially independent of n. Thus as n --* 00, the probability that X is a manifold goes to O. 0 2.9 Remark The proof just given also shows that the probability that a 4-valent graph is the I-skeleton of some triangulation of a 3-manifold goes to 0 as the number of vertices goes to infinity.

All the properties of random surfaces that we described were consequences of the fact that the uniform distribution on T2 (n) was generated by randomly gluing triangles. In the 3-manifold case, we are deprived of this tool, and it seems difficult to say anything at all about a random element of :73 (n). For instance, we do not even know the expected number of vertices for an ME :73 (n), much less whether we should expect M to be irreducible or hyperbolic. Even if one conld not say much theoretically, it would be very useful to be able to generate elements of, say, :73 (100) with the uniform distribution, even approximately or heuristically. It would also be interesting to understand WILLIAM P. THURSTON

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the complexity of uniformly generating elements of 1j(n); perhaps there is simply no polynomial-time algorithm to do so. It is interesting to note that while spheres make up a vanishingly small proportion of 'I2 (n) as n ~ 00, it is actually possible to generate triangulations of S2 in time linear in the number of triangles [PSj.

2.10 Random Heegaard splittings Every closed orientable 3-manifold has a Heegaard splitting, that is, can be obtained by gluing together the boundaries of two handlebodies. Considering such descriptions gives us another notion of a random 3-manifold, and this is the one we focus on in this paper. The version that we will use here takes the following point of view, based on the mapping class group. Fix a genus-g handlebody Hg, and denote BHg by E. Let.Mg be the mapping class group of E. Given cp E .Mg, let Nq, be the closed 3-manifold obtained by gluing together two copies of Hg via cpo Fix generators T for .Mg. A random element cp of .Mg of complexity L is defined to be the result of a random walk in the generators T oflength L. Then we define the manifold of a random Heegaard splitting of genus g and complexity L to be Nq" where cp is a random element of .Mg of complexity L. We are then interested in the properties of such random Nq, as L ~ 00. A priori, this might depend on the choice of generators for .Mg. We will show, however, that certain properties do have well-defined limits independent of this choice (Sects. 6-9). Random Heegaard splittings are much more tractable than random triangulations, in part because every random walk in .Mg actually gives a 3-manifold. Also, for many problems we can reduce the question to a 2-dimensional one, that is, a question about the mapping class group. The disadvantage is that the need to fix the Heegaard genus feels artificial from some points of view. For instance, it means that the injectivity radius of a hyperbolic structure on Nq, is uniformly bounded above [Whij. A very natural question is how often does the same 3-manifold appear as we increase L? For instance, there are arbitrarily long walks cp in .Mg for which Nq, = S3. Thus you might worry that some small number of manifolds dominate the distribution, and so our notion of random is not very meaningful. However, we will show later that if :F is any finite set of 3-manifolds, then the probability that Nq, E :F goes to 0 as L ~ 00. This follows from Corollary 8.5, which shows that HI (Nq" Z) is almost always finite, but the expected size grows with L. The next obvious question is: what is the probability that Nq, is hyperbolic? We believe

2.11 Conjecture As L ~ 00 the probability that Nq, is hyperbolic goes to 1. Moreover, the expected volume of of Nq, grows linearly in L. One expects that the hyperbolic geometry of Nq, away from the cores of the handlebodies should be close to that of a "model manifold" of the 252

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type used in the proof of the Ending Lamination Conjecture. Despite this heuristic picture, a proof of Conjecture 2.11 is likely to be quite difficult. One approach would be to try to show that the expected distance of the Heegaard splitting defining N", (in the sense of Hempel [Hem2]) is greater than 2.1 Namazi's results connecting Heegaard splittings to hyperbolic geometry are also relevant here [Namj. Finally, there are other notions of random Heegaard splittings you could consider. For instance you could think of specifying a random Heegaard diagram of complexity L by choosing one uniformly from among the finite number of such where the number of intersections of pairs of defining curves is ~ L. 2.12 Universal link related notions There are links L in S3 such that every closed orientable 3-manifold is a cover of S3 branched over L. One such link is the figure-8 knot [HLMj. Let K be this knot and M be its exterior. There are only finitely many branched covers of (S3, K) of degree ~ L, since such a cover corresponds to a finite-index subgroup of 1r1 (M). Thus another notion of random 3-manifold is to choose uniformly among all conjugacy classes of subgroups of 1r1 (M) of index ~ L and build the corresponding manifold. As with random triangulations, it is unclear if there are even efficient ways to generate such covers experimentally. While enumerating all subgroups of index ~ L is certaiuly algorithmic [Simj, the number of such subgroups grows superexponentially in our case. This is because 1r1 (M) virtually sUIjects onto a free group on two-generators. So if we wanted to sample branched covers of index ~ L for large L, we would need some way of picking out the subgroups without enumerating all of them. With current technology, it would be difficult to enumerate all subgroups of 1r1 (M) for indices beyond the low 20s. For more about 3-manifolds from the point of view of branched covers of the figure-8 knot, see [Hemlj. 2.13 Random knots based notions

Another notion of random 3-manifold would be to take a Dehn surgery point of view. That is, one could take some notion of a random knot or link in S3 and do Dehn surgery on it, where the Dehn surgery parameters are confined to some finite range at each stage. For this, one would need a good notion of a random knot or link. One could use models based on choosing a random braid and either taking the closed braid or making a bridge diagram. Or you could look at all planar diagrams with a fixed number of crossings. These can be efficiently generated [pSj. Another reasonable notion is to 1 Josepb Malter bas recently announced [Malt] a proof tliat the probability of a Heegaard splitting having distance less than a fixed C goes to 0 as L --> 00; this would establish the first part of Conjecture 2.11, assuming geometrization. WILLIAM P. THURSTON

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build a knot out of a unifonnly distributed collection of unit length sticks stuck end to end (see e.g. [DPS]). These can be generated efficiently, but have the disadvantage that they are typically satellite knots [Jun].

3 Random balanced presentations For a finite presentation of a group, the deficiency is the difference g - r between the number of generators and the number of relators. In the case of a closed 3-manifold group, the natural presentations coming from cell divisions or Heegaard splitting have deficiency O. Deficiency 0 presentations are also called balanced. If a group has a presentation with positive deficiency, then it already has positive first betti number, so deficiency 0 is the borderline case for the virtual positive betti number property of a finitely presented group. In this section, we study groups defined by random presentations of deficiency 0, and otherwise ignore the constraints coming from the topology of 3-manifolds. In particular, we compute the probabilities that they admit epimorphisms to certain finite groups. In later sections, we will contrast these results with those specific to 3-manifold groups. First let us choose a suitable meaning for a "random presentation" by giving a definition of a random relator. Consider the free group Fg on g generators al, ... , ago Given an integer n > 0, consider all unreduced words of length n where each letter is either a generator atl or the identity; there are (2g + l)n such words. A random relator oflength n is such a word selected at random, with each word equally likely. If we fix a number g of generators and number r of relations, a random presentation of complexity n is the group G = (Fg I RI, ... ,Rr ), where each Ri is a random relator of complexity n. Such random presentations have been studied extensively by Gromov and others. In particular, Gromov showed that the probability that G is word-hyperbolic goes to 1 as n --. 00 [Grol,OI]. In the rest of this section, we consider the probabilities that such random groups have different kinds of finite quotients, focusing on the case of deficiency O. What we do here fits well into the context of profinite groups, as we describe later in Sect. 4, and that point of view provides additional motivation for this section. 3.1 Quotients of a fixed type

Let Q be a finite group. We want to consider the probability that a random g-generator r-relator group G has a epimorphism onto Q. We begin by showing that this probability makes sense, and, in later subsections, calculate it for certain classes of Q. First, consider a fixed epimorphism f: Fg --. Q; what is the probability that f extends to G, equivalently that f(RJ = 1 for all i? One way to think of Ri is as the result of a random walk of length n in the Cayley graph of Fg. In this random walk, each edge is equally likely as the next step, and there is a 1/ (2g + 1) probability of not moving at each 254

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stage. The image f(R;) is thus the result of the analogous random walk in the Caley graph of Q with respect to the generators {f(aj)}. The next lemma says that as n ~ 00, the result of such a random walk on a finite graph is nearly uniformly distributed; thus the probability that f(R j ) = 1 converges to l/IQI as n ~ 00.

3.2 Lemma Let r be a connected finite graph. Consider random walks on r with fixed transition probabilities. Suppose that at each vertex the probability of taking any given edge is positive, as is the probability that the walk stays at the vertex. Then the distribution of the position of the walk after n steps converges to the uniform distribution as n ~ 00. The reason for reqniring a positive probability for pausing at each stage is to avoid parity issues, as happens when r is a cycle of even length; therefore, we will always include the identity in the set of generators when constructing a Caley graph. The lemma is completely standard, but as we use it repeatedly we include a proof.

Proof Consider the vector space F of functions from the vertices of r to C. Let L be the linear transformation of F which averages a function over the radius one neighborhood of a vertex, weighted according to the transition probabilities. That is, if f : r ~ C then L(f)(v) =

L

(Probability of transition from v to w)f(w).

WEB,(v)

Let Il E F be the characteristic function for the initial point of the walk. Then the probability distribution for the position of the walk after n steps is L nil. Note that L is a non-negative linear matrix, and that if n is greater than the diameter of r then L n has strictly positive entries. Constant functions are eigenvectors of L, and by the Perron-Frobenius theorem, the successive images by L of any non-negative and non-zero function converge to this eigenspace. In particular, L n Il converges to the uniform measure. 0 Now let us use the same idea to show that the probability that a random G has a Q-quotient is well-defined. More precisely, let p(Q, g, r, n) be the probability that the group of a g-generator, r-relator presentation of complexity n has a Q-quotient (note there are only finitely many such presentations). Then

3.3 Proposition The probabilities p(Q, g, r, n) converge as the complexity n of the presentation goes to infinity. Moreover, the distribution of the number of quotients also converges. In discussing the distribution of the quotients, it is natural to consider two epimorphisms to Q as the same if they differ by an automorphism of Q, and so we will adopt this convention in our counts. This is equivalent to counting normal subgroups with quotient Q. WILLIAM P. THURSTON

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Proof Let 8 be the set of epimorphisms of the free group Fg onto Q, modulo automorphisms of Q. We will fix one representative in each equivalence class in 8, and so regard the elements as actual epimorphisms from Fg to Q. Consider the group Q8, and let P: Fg -+ Q8 be the induced homomorphism where the f-coordinate of p(w) is f(w). Let S ~ Q8 be the image of Fg under P, and let r be the Caley graph of S with respect to the generators {p(aj)}. Suppose R is a word in Fg. Then the f E 8 which kill R are exactly those where the f-coordinate of P(R) is 1. If R is a random relator with high complexity, then by Lemma 3.2 the element P(R) in S is nearly uniformly distributed in S. Thus the probability that some f E 8 kills R is approximately the ratio I{s 01=

E

Sis! = I for some f}1

lSI

.

As the relators are chosen independently, the probabilities p(Q, g, r, n) converge to O/! as n -+ 00. Similarly, the probability that we have a fixed 0 number k of Q-quotients converges for each k. We will call the limiting probability p(Q, g, r). As we saw, it only depends on the finite sets 8 and S, so we turn now to understanding them. First, homomorphisms from Fg to Q which are not necessarily onto are parameterized by Q8. The set 8 is the quotient of the subset of Q8 consisting of g-tuples which generate Q, under the diagonal action of Aut(Q). As Aut(Q) acts freely on this proper subset, we get that 181 < IQI 8/IAut(Q)I. This over-estimate will actually be close to 181 if g is large; as g -+ 00 the proportion of g-tuples in Q8 which do not generate goes to O. Understanding S in general is complicated as it is typically not all of Q8. However. it is easy to compute the expected (average) number of Q-quotients of such random G. Note that for fixed f E 8 the probability that the i th relator is in the kernel of f is l/IQI. As the relators are chosen independently, the probability that f extends to our random group with r relators is 1I IQ I' . Thus the expected number of such quotients coming from f is 1I IQ I'; as expectations add, the expected number of Q-quotients for G is 181 I IQ I'. For any non-negative integer-valued random variable. the chance it is positive is less than or equal to its expectation. Thus p(Q. g. r) ~ 181 II Q I' < IQI8-' IIAut(Q)I. Now, we are most interested in balanced presentations. and this gives: 3.4 Theorem LetQ be afinite group. Theprobability p(Q. g. g) that a random g-generatorbalanced group has a epimorphism to Q is < I/IAut(Q)I.

Now the number of finite groups with IAut(Q) I bounded is finite [LN1. and so the theorem implies that p(Q. g. g) -+ 0 as IQI -+ 00. Thus the larger Q is. the less likely a random balanced G is to have Q as a quotient. In the rest of this section we refine our pictore for the classes of abelian and simple groups. 256

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3.5 Non-abelian quotients We start with the case of a non-abelian simple group Q, where we develop a complete piclure. As in Sect. 3.1, consider the set 8 of epimorphisms from Fg ~ Q, modulo Aut(Q). Most collections of g > 1 elements of a finite simple group Q generate it, especially if g > 2 or if Q is not too small. To get a rough idea of the probability that a random collection of g elements of Q generates Q, consider the contrary hypothesis. If the elements fail to generate, then there is some maximal subgroup H of Q that contains them all. For a particular H, the chance that g elements lie in H is II [Q : H]8. The sum over all maximal subgroups gives an upper bound for the proportion that do not generate, substantially less than 1 in all but a few small cases. These upper bounds with g = 2 in a few of the small cases are (A5, .53), (A6, .57), (A7' .35), (As, .34), (A9, .1S), (PSLzlF7 , AI) (PSLzlFs, .17), (PSLzlF9, .57), (PSLzlFu, .2S) and (PSLzlF13, .11). As the size of the simple group gets larger, the probability of 2 elements generating goes to 1; see the references in [Pak, § 1.1]. The automorphism group of a non-abelian finite simple group contains Q itself as the group of inner automorphisms; the quotient group is the outer automorphism group, which is generally rather small. The upper bound we gave earlier is thus 181::: IQI 8/IAut(Q)1 = IQ8- I I/IOut(Q)I. The preceding paragraph indicates that this bound is actually quite accurate except for small Q and g. As in Sect. 3.1, we now have that the expected number of Q quotients ofa g-generator balanced group is 181/1QI 8 and that this is a bound on the probability p(Q, g, g) for having a Q-quotient. Thus we have 181 1 1 p( Q, g, g) ::: IQ 18 < -:-1Q::-:I-:-::Io'-ut---C(-=-Q:-C)I ::: -QII.

(3.6)

In order to compute p( Q, g, g) exactly, we need to understand the image S of the induced product map Fg ~ Q8 used in the proof of Proposition 3.3. In this case S is actually all of Q8: ~ Qi' where each Qi is a non-abelian finite simple group. Suppose no pair (fi, fj) are equivalent under an isomorphism of Qi to Qj. Then the product map Fg ~ Qi is surjective. It is important in this lemma that Qi be non-abelian. For instance, if we take Q = Z/2, then 181 = 28 - 1 and so Q8 has 22'-1 elements. In contrast, the image of Fg ~ (Z/2)8 is generated by g elements and thus has size at most 28.

3.7 Lemma ([Hal]) Consider epimorphisms fi: Fg

n

Proof This lemma was first proved by P. Hall [Hal]. As it is crucial for us, and unfamiliar to most topologists, we include a proof. We begin with case n = 2. Suppose Fg ~ QI X Q2 is not swjective. Let S be the image; we will show that S is the graph of an isomorphism between WILLIAM P. THURSTON

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QI and~z compatible with the f;. Consider the projection rr: S ~ QIo and let Qz denote the subgroup {I} x Qz in QI x Qz. Let K be the -kernel of rr, that is K = S n Qz. Note that the conjugation action of a q E Qz on Qz can also be induced by conjugating by some S E S, as the projection S ~ Qz is onto; therefore, as K is normal in S it must be normal in Qz. As Qz is simple, K must either be 1 or Qz. In the latter case, S contains Qz which implies S = QI X Q2 as rr is onto. Thus K = 1 and rr is an isomorphism. Similarly, the projection rr': S ~ Qz is an isomorphism. Thus II and fz are equivalent under the isomorphism rr' 0 rr-I. The n = 2 case did not use that the Qi are non-abelian. That hypothesis is used in the form of 3.8 Claim Let N ~ QI X .•• X Qk be a normal subgroup, where the Qi are non-abelian simple groups. Then N is a direct product of a subset of the factors. As before, let Qi denote the copy of Qi in the- product. To see the claim, first observe that N n Q i is either 1 or all of Q i' If the latter case, mod out by Qi to get a case with smaller k. So we can assume N n Qi = 1 for each i. But then [N, QiJ ::: N n Qi as both subgroups are normal, and so [N, QiJ = 1. But then N is central, and thus trivial, proving the claim. To conclude the proof of the lemma, choose the smallest n such that Fg ~ QI X ••• x Q. is not swjective, and let S be the image. Then as in the n = 2 case, the projection rr: S ~ QI X ••• x Q._I is an isomorphism. Let a: QI X .•• x Q._I ~ Q. be the composition of rr- I with projection onto Q •. By the claim, the kernel N of a is a direct product of some of the factors. After reordering, we can assume N = 1 X Qz x ... x Q._I. But then the map Fg ~ QI X Q. is not swjective, and we are back in the n = 2 0 case. We saw above that the limiting probability of getting exactly k quotients with group Q is simply the density of S E S with exactly k of the coordinates equal to 1. Thus as S = Qe the limiting distribution is the binomial distribution: {IQ-quotientsl = k} =

(~)]f (1

- p).-k

(3.9)

where p = l/1QI8 and n = 181. This binomial distribution is wellapproximated by the Poisson distribution. Recall that the Poisson distribution with mean JL > 0 is a probability distribution on :£':>0 where k has probability 'fi-e-I'. Roughly, the Poisson distribution describes the number k of occurrences of a preferred outcome in a large ensemble of events where, individually, the outcome is rare and independent, but in aggregate the expected number of occurrences is JL > O. For instance, it is the limit of the binomial distribution we have here, if JL = 181/1 QI8is kept constant

.

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and n = 181 --+ 00. The difference between (3.9) and the Poisson distribution is usually negligible even in small cases. For instance, p(As, 2, 2) is 0.0052646 ... whereas the Poisson approximation is 1- e-!' R; 0.0052638. Summarizing, we have:

3.10 Theorem Let Q be a non-abelian finite simple group. Let n be the number of epimorphisms from the free group Fg to Q (modulo Aut(Q)), and let /L = nil Q 18 • The probability that a random g-generator balanced group has a Q -quotient is

p(Q, g, g) = 1- (1-IQI-8)IQI'!'

R;

1- e-!'

and the distribution of the number of quotients is nearly Poisson with mean /L. Moreover, as g goes to infinity /L --+ 1/1Aut QI, and the distributions limit to the Poisson distribution with mean 11 IAut QI. We end this subsection with Table 3.11 which summarizes the situation for the first few finite simple groups. As you can see, all the probabilities are very low; we will see that this is not the case for 3-manifold groups.

3.12 Probability of some simple quotient Now, let us consider the more global question: What are the chances that a finitely presented group of deficiency 0 admits an epimorphism to some non-abelian finite simple group? First consider a finite collection e of simple groups. Let G be a random g-generator balanced group with complexity n. For a fixed Q E e, Theorem 3.4 implies that the probability of G having a Q-quotient is < 1/IAut(Q)1 :::: 1/1QI, as long as n is large enough. As there are finitely many Q, we get that for large n the probability that G has a Q-quotient for some Q in e is less than

L QEC

1/IAut(Q)1 ::::

L

1/1QI.

(3.13)

QEC

In fact, quotients for different Q are independent (this follows from Lemma 3.7, just as in the proof that S = Q8 in the context of Theorem 3.10). Therefore, we could replace (3.13) by 1 (1 - 1/IAut(Q)I), but the former will do for us here. If we were to formally carry out this calculation for the collection of all finite simple groups, we would get that the probability of having a nonabelian simple quotient is less than L 1/IAut(Q)1 :::: L l/IQI, where the sum is over all such groups. By the Classification of Finite Simple Groups, eventually nearly all non-abelian simple groups up to a given size are of the form PS~lFq. As PS~lFq has size (q - l)q(q + 1)/2 for odd q and twice that for even q, the sum of III Aut(Q) I over all non-abelian finite simple groups Q is finite. It is not even very large: approximately 0.015.

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Table 3.11 This table gives values and bounds for the expected number of epimorphisms from a random deficiency 0 group to a finite simple group Q. The simple group Q is listed in the tirst column. The secnnd column is the order of Q, and the third column Gen pairs is the number of pairs of elements that generate Q, up to automorphisms of Q. Colunm Ont gives the order of the outer automorphism group of Q. Column Exp 2-gen is the expected number of epimorphisms to Q among groups with 2 generators and 2 relators. Column Exp n-gen is 1/1 Aut(Q)I, which is an upper bound for the expectation independent of the number of generators, the limit of the expectation as the number n of generators goes to infinity, and a good approximation to the expectation when n > 2. Quotient

Order

As PSLzlF7 A6 PSLzlFs PSLzlF l1 PSLzIF 13 PSLzIF 17 A7 PSLzlF 19 PSLzlF 16

60 168 360 504 660 1092 2448 2520 3420 4080 5616 6048 6072 7800 7920 9828 12180 14880 20160 20160 25308 25820 29120 32736

PS~1F3

U3(lF3)

PSLzIF23 PSLzIF25 Ml1

PSLzlF27 PSLzlF29 PSLzlF31 As PSL31F4 PSLzlF37 U4(lF2)

Sz(lFs) PSLzIF32

Genpairs 19 57 53 142 254 495 1132 916 1570 939 2424 2784 2881 1822 6478 1572 5825 7135 7448 1452 12291 11505 9534 6330

Ont 2 2 4 3 2 2 2 2 2 4 2 2 2 4 1 6 2 2 2 12 2 2 3 5

Exp2-gen .005278 .002020 .000409 .000559 .000583 .000415 .000189 .000144 .000134 .000056 .000077 .000076 .000078 .000030 .000103 .000016 .000039 .000032 .000018 000004 000019 .000017 .000011 .000006

Expn-gen .008333 .002976 .000694 .000661 .000758 .000458 .000204 .000198 .000146 .000061 .000089 .000083 .000082 .000032 .000126 .000017 .000041 .000034 .000024 .000004 .000020 .000019 .000011 .000006

However, this does not give a proof that many random balanced groups have no non-abelian simple quotients; for a fixed group G, the relators certainly do not map to uniformly distributed random elements of Q as IQI ---+ 00. For one thing, a relator of length R is confined to the ball of radius R in the Cayley graph of Q, and this ball has fewer than «2g)R - 1)/(2g - 1) elements. IT the relators were uniformly distributed in these balls, then the probability of an epimorphism to Q would be bounded below (although very small), so one would expect there to eventually be an epimorphism of G to some phenomenally large simple group Q. But this argument is also invalid, since among all finite simple groups with a choice of a sequence of g generators, there are only finitely many isomorphism classes of balls of radius R, so we have only finitely many chances to find an epimorphism.

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The real question, which appears to be a difficult issue, is how many different isomorphism classes of balls of radius R exist among all nonabelian finite simple groups. It seems reasonable to us that most of these balls fall into patterns with relatively few new variations when log IQI is large compared to R. A good estimate of this sort could imply that most groups of deficiency 0 have ouly finitely many epimorphisms to finite simple groups, often no such homomorphisms. Since a random group in our sense is word hyperbolic, this would imply that there are word hyperbolic groups that are not residually finite. In any case, for a typical deficiency 0 group that has no quotients among the first few non-abelian simple groups, it is clear that if it has any such quotient, the index must be so astronomically large as to be far beyond brute force computation. From the calculation above, a random balanced presentation with 3 or more generators has about 1.5% probability to admit an epimorphism to a non-abelian simple group of manageable size, and a 2-generator group has about 1.3% probability. To test our thinking, we made 1000 random 2-generator presentations with statistics similar to the census manifolds used in our paper [DTI], and computed all epimorphisms to the first few non-abelian simple groups. Only 15 of these groups had any such quotients, and only 4 had more than one such quotient. This fits reasonably well with the estimate of 1.3% above.

3.14 Abelianization Let us begin by looking at the abelianization of a random balanced group from a different, more global, perspective. The abelianization is the quotient of zg by the subgroup generated by the abelianization of each relator. In other words, we can make a matrix MR whose columns correspond to the relators so that the (i, JJ entry is the exponent sum of the occurrences of generator gi in relator Rj • If the abelianization is infinite, the determinant of MR is 0, otherwise the determinant of MR is the order of the abelianization. For a random relator Rj of length n, the corresponding column is just the result of a snitable random walk in the integer lattice zg. Individual entries can also be thought of as generated by random walks, and like all I-dimensional walks their absolute value is proportional to y'n. Thus the typical determinant grows large as r grows large - indeed, it grows as n g / 2 (to see this rigorously, note that the distribution of (1/ y'n)MR converges to that of matrices with independent Gaussian entries). However, for our purposes it is more important to determine the probability that a random presentation admits a finite-sheeted covering of a given type, as we did in the previous subsection. Any finite abelian group is the product of its p-Sylow subgroups, so we focus in on just one prime. We Will think about this from the point of view of the p-adic integers Zp. As rational integers which are coprime to p have inverses in Zp, the p-Sylow subgroup of the cokemel of our matrix MR is the same as the quotient of Z~ WILLIAM P. THURSTON

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by the Zp-submodule generated by the columns of MR. We are interested in asymptotics as the complexity n of our presentation goes to infinity, and so we want to understand the limiting distribution of the MR. More precisely, let mn be the probability measure on elements of Fg coming from random walks of length n. This gives us a measure on the set of balanced presentations with g-generators, with finite support. Then one has

3.15 Lemma The pushlorward of the measure mn to the space of p-adic g x g matrices converges weakly to the uniform distribution, i. e. in the limit the entries are elements of Zp chosen uniformly and independently with respect to Haar measure. Proof The Haar measure on Zp can be understood by thinking of Zp as the inverse limit of Z/P'Z. Showing that the weak limit is Haar measure is tantamount to checking that, for each k, the distribution of the entries modulo I' converges to the uniform distribution as n ~ 00. The mod I' abelianization of a random relator is the same as going for a random walk in the Caley graph of (Z/ P'Z)g. As always, that distribution becomes the uniform one as n ~ 00, proving the lemma. 0 This p-adic point of view is eqnivalent to considering (Z/P'Z)g modulo the subgroup generated by a random sequence of g elements, and looking at the limiting distribution of quotient groups as k goes to infinity; however, it has the advantage of giving us a concrete limiting object to calculate with. First, let us compute the distribution for the orders of the p-Sylow subgroups. This is just the largest power of p which divides det(MR); if I . Ip denotes the p-adic norm, this is the same as 1/1 det(MR)l p. Thus, we need to understand the distribution of det(MR), where MR is a g x g matrix with entries in Zp, chosen uniformly. The easy case is when g = 1, for then det(MR) is just uniformly distributed. As an element in Z/P'Z has a p-k chance of being 0, an element in Zp has a p-k chance of being in P'Zp. Thus the chance thatz E Zphas Izlp = p-kisck = p-k(l-l/p).Auseful way to encode the sequence {Ck} is to use a generating function:

~

k p-l L...Ck t = - - . k=O p-t

In the general case we will show:

3.16 Proposition Let d k be the asymptotic probability that the order of the p-Sylow subgroup of the abelian group defined by a random g-generator balanced presentation is 1'. The generating function for the sequence {dd is (p - 1)(p2 - 1) ... (pH - 1)

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Except for small primes p, this is close to the distribution for the case g = 1. Thus, except for small p, the probability that the p-Sylow subgroup is non-trivial is close to 1/p. This is the same as asking that the group sUIject onto Z/p, and by Theorem 3.4 we already knew this probability was < 1/ IAut QI = 1/(p - 1). Thus in this case the general estimate is close to correct. We now prove the proposition.

Proof A vector in Z~ has probability 1/pkg to be in pkz~. This tells us the distribution of the maximal p-adic norms of an element in the first column of M R : the probability that the maximal p-adic norm equals 1/pk is a geometric progression, with generating function (p8 - 1) / (p8 - t). If the first column equals pkw where k is as large as possible, then Z~/ (W) is isomorphic to Z~-I. Moreover, the remaining columns map to independent random elements of this module. Thus I det(MR) Ip is the product of p-k with I det(N) Ip where N is a random (g - 1) x (g - 1) matrix. Therefore we can get the generating function for I det(MR ) Ip by multiplying the generating 0 functions for these two things. Inducting on g completes the proof.

3.17 Remark It is worth noting that the proof shows that the Sylow subgroups for distinct primes p and q are independent, essentially because the quotient maps from Z to Z/ p and Z/ q induces a sUIjection Z ~ Z/ p x Z/ q; thus a random walk in Z pushes forward to the (nearly) uniform distribution onZ/p x Z/q. Now, we will delve further and determine the typical isomorphism type for the p-part of the homology. One way to describe the isomorphism class of an abelian p-group A is to specify the sequence of ranks of Pi (A) of pi A/(pi+l A). For example, the group (Z/ p)2 E9 Z/ E9 Z/ corresponds to 4, 2, I, I, I, 0, with all subsequent terms also O. Introducing a variable tk to denote an instance of (Z/ p)k, then an isomorphism class corresponds to a monomial in the tk; the example corresponds to t4t2tr. With this notation, there is a fairly nice and straightforward computation for the power series in tl, ... , tg whose coefficients give the asymptotic probability that a g-generator, g-relator group has the particular isomorphism type of p-Sylow subgroup of its abelianization; this series is a rational function AFPg • This is a bit of a digression for studying 3-manifolds, so we will content ourselves with stating the formulae for 1, 2, and 3-generator groups:

r

r

AFP 3 = (-1

+ p)\1 + 2p + 2p2 + p3)(p8 + rtl + p6 t1 + rt2 + p3 t2 + tlt2) (p - t[)(p4 - t2)(p9 - t3) WILLIAM P. THURSTON

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For instance, the 2-Sylow subgroup of the abelianization of a random 2-generator 2-relator group has probability 3/8 to be trivial, 9/32 to be 7/.,/2,9/64 to be 7/.,/4, 3/128 to be (7/.,/2)2, etc. Independently, the 3-Sylow subgroup has probability 16/27 to be trivial, 64/243 to be 7/.,/3, 64n29 to be 7/.,/9, 16/2187 to be (7/.,/3)2, and so on. To see how to compute AFPg , note that in our case, where A is the quotient of 7/.,~ by the subgroup generated by a random sequence of g elements, the probability that Po(A) = k is the probability that g elements of (7/.,/ p)g generate a subgroup of rank g - k. Similarly, when PiCA) = h, the conditional probability that PHI = k is the probability that a random sequence of h elements of (7/.,/p)h generate a subgroup of rank h - k.

4 The profinite point of view In the last section, when we studied the finite quotients of a "typical" balanced group, we worked with asymptotic probabilities p(Q, g, g), which were limits of finite probabilities as the size of the presentation increases. In the case of abelian groups, we saw that these probabilities could be thought of as probabilities on a certain p-adic object, where the notion of probability came from the natural Haarmeasure (Sect. 3.14). In this section, we explain how this picture holds true in general by considering random quotients of profinite free-groups; this helps clarify why we got well-defined probabilities such as p(Q, g, g). At the end, we discuss a natural analog of a Heegaard splitting in the profinite context. 4.1 Profinite completions

We begin with a brief sketch of the theory of profinite groups and completions (for more, see e.g. [Wil,RZ], and [DdSMS]). Let G be a finitely generated group. The profinite completion G of G is a compact topological group defined as the inverse limit of the system of all finite quotients of G. (Note that whenever QI and Q2 are any two finite quotients, both quotients factor through the image of G in the product map to QI x Q2, so the set of finite quotients does form an inverse system.) If G has only finitely many finite quotients, then G is a finite group (possibly trivial). Otherwise, G has the topology of a Cantor set, whose stages of refinement give particular finite quotients. The natural map G ~ G is injective if and only if G is residually finite. To reconstruct the finite quotients of G, take small open and closed neighborhoods V of the identity in G, form the subgroup W generated by V, and then pass to the intersection of the finitely many conjugates of W to obtain an open and closed neighborhood X which is a normal subgroup. The quotient G / X is a finite group, and the finite quotients obtained in this way from any neighborhood basis of 1 are cofinal among all finite quotients of G. In general, a profinite group is any compact -

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topological group that has a neighborhood basis of the identity consisting of open and closed subgroups. Equivalently, a profinite group is a group that is the inverse limit of finite groups. Since a profinite group G is a compact topological group, it has a unique bi-invariant probability measure, its Haar measure. This measure is the inverse limit of the counting measures on its finite quotients. Thus, any property of elements of G has a well-defined probability (provided the set of such elements is measurable). 4.2 Profinite presentations In the profinite context, a finitely presented group is the following. Consider the profinite completion Fg of the free group on g generators. Given a finite set of elements {Rl, ... , R r } of F g , let K be the topological closure of the normal subgroup they generate. The quotient topological group G = FgI K is the group of the profinite presentation with g generators and relations {R j }. Now focus on the set:Bg of all g-generator balanced profinite presentations, which is just the product of g copies of F g , one for each relator. As such, it has a natural probability measure coming from the product of Haar measures on each factor; equivalently, we are thinking of each relator as being chosen independently at random. Thus we can talk about the probability that G E :Bg has some particular property. In the case of the property of having a epimorphism to a finite group Q, this is really the same question we encountered before:

4.3 Theorem Let Q be a finite group. Let G be the group defined by a randomly chosen g-generator balanced profinite presentation. Then the probability that G has a epimorphism to Q is p(Q, g, g). The quickest way to see this would be to repeat the proof of Proposition 3.3 in this context, and see that one gets the same answer. We will phrase it a little differently to make clear why we get the same answer - after all, the set of regular (non-profinite) presentations has measure 0 in :Bg , and so it is hardly given that asymptotic probabilities of regular presentations are the same as the corresponding probabilities for profinite presentations. Consider random walks on Fg of length n, and let mn be the probability measure on Fg given by the endpoints of such walks. We can also think of mn as a measure on F g • Then we have:

4.4 Lemma The measures mn converge weakly to Haar measure on F g • Proof On a totally disconnected space such as F g , locally constant functions are uniformly dense among continuous functions. So it suffices to check that for a locally constant function f: Fg --+ JR, the integrals of f with respect to mn converge to the integral of f with respect to Haar measure. Since every locally constant function on Fg is the pullback from a function on some finite quotient, this lemma follows from Lemma 3.2. 0 WILLIAM P. THURSTON

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If S c /B g is both open and closed, then its characteristic function is continuous. Hence, if we look at regular (non-profinite) presentations Lemma 4.4 implies lim P{ G

n-+oo

E

S I G a random balanced group of complexity n} = 1t(S),

where It is the natural measure on /Bg. For instance, the property of having an epimorphism to a fixed finite group Q is both open and closed; thus Theorem 4.3 follows from Lemma 4.4. Passing to random profinite presentations makes it possible to estimate the probability that a group has no non-abelian simple quotients at all. For regular presentations, we weren't able to show this probability was positive, but the formal calculation in Sect. 3.12 actually applies in the profinite context. In particular, the subset of /B g consisting of groups which surject onto Q has measure less than 11 IAut QI. As L l/lAut QI is finite and indeed about 0.015 we have: 4.5 Theorem Let G be the group defined by a random g-generator profinite balanced presentation. Then with probability 1, the group G has only finitely many non-abelian finite simple quotients. If g 2': 3, the probability that G has no such quotients is about 98.5%; if g = 2, about 98.7%. The abelian quotients of a random balanced G can be understood directly from Sect. 3.14. Usually, but not almost always, the abelianization A of a random balanced profinitely presented group is the inverse limit of cyclic groups. This is equivalent to the condition that there is no prime p such that A admits a continuous epimorphism to Zip x Zip. Most of the exceptions are for p = 2, with most of the remaining exceptions for p = 3; the probability for the existence of such a homomorphism is only about 1I p4 for larger p. Among balanced profinitely presented groups with 1 through 5 generators, the probabilities that all finite abelian quotients are cyclic are about 1.0, 0.924, 0.885, 0.865, 0.856. The limiting value for a large number of generators is about 0.847.

4.6 Profinite generalizations of 3-manifold groups In this subsection, we define a class of profinite groups that includes the profinite completions of all 3-manifold groups; this class comes with a natural probability measure. While we will not make direct reference to these ideas elsewhere in this paper, they provide a natural context for the results of Sects. 6-8, just as groups with balanced profinite presentations do for the results of Sect. 3. Consider a Heegaard diagram of a 3-manifold, and let Sg be the fundamental group of the Heegaard surface. Looking at the fundamental groups of the two handlebodies gives us a diagram of groups

Fg +- Sg 266

"""* Fg.

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There is a corresponding diagram of profinite completions:

Fg +- Sg -+ Fg. The profinite completion of the fundamental group of the 3-manifold is the quotient of Sg by the topological closure K of the normal closure of the kernel of the two homomorphisms. Since Sg is finiteg generated and residually finite, there is a neighborhood basis for 1 in Sg that consists of invariant subgroups of finite index, that is, subgroups invariant under all automorphisms. For any invariant subgroup, the mapping class group .Mg acts as an automorphism of the quotient group. Therefore .Mg is also residually finite, and furthermore, the action of .Mg on Sg extends to a continuous action of .Mg. 4.7 Remark These actions are not necessarily faithful, in particular it is not for g = 1. The torus case is the first case of the congruence subgroup problem: does every finite index subgroup of SLkZ contain a principal congruence subgroup? (A principal congruence subgroup is the kernel of a reduction mod n to SLkZ/nZ.) For SLzZ the answer is no, basically since SLzZ is virtually a free group and thus it is easy to find quotients which are simple groups not isomorphic to PSLzlFpo (Tangentially, the answer to the congruence subgroup problem is yes for k ~ 3.) It is unknown if the action of .Mg is faithful in genus greater than 1.

This picture gives us some justification in considering profinite Heegaard diagrams Fg +- Sg -+ Fg which are limits of diagrams of actual 3-manifolds; in other words, they are obtained by gluing two copies of the standard map Sg -+ Fg by an element of .Mg. Associated with such a diagram is a locally compact, totally disconnected topological group, which we will refer to as a profinitefold group: the quotient of Sg by the smallest normal, closed subgroup K containing the kernels of the two homomorphisms to Fg • (This construction is special to dimension 3, so we will not bother with a dimension indicator such as "3-profinitefold group".) Let Tg be the subgroup of .Mg consisting of homeomorphisms of the surface that extend to homeomorphisms of the handlebody. TWo elements /!, h E .Mg define eqnivalent Heegaard diagrams if Tg \JdTg = Tg \h/Tg. Similarly, it makes sense to define two Heegaard profinite diagrams to be eqnivalent if the gluing automorphisms are in the same double coset in 'Tg\.Mg/'T g. Haar measure on .M g pushes forward to a measure on this double coset space. This gives a probability measure on the set of profinitefold groups, which we will use to make sense of statements about random profinitefold groups. The first homology of any finite sheeted cover of any irreducible 3-manifold M can be reconstructed from 7rJ (M): if r is the fundamental group of this finite sheeted cover, the profinite completion of the abelianization of G is the same as the abelianization of r, which is the corresponding subgroup WILLIAM P. THURSTON

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of finite index in ]1'1 (M). In particular, the abelianization of r is infinite if and only if the abelianization of r admits a continuous epimorphism to Z. In the profinite context, first consider a group G defined by a random balanced profinite presentation. We claim that with probability I, G has no continuous homomorphism to Z. In this context, Proposition 3.16 says that the probability that G has a continuous homomorphism to 7l,j p is about 1 j p. As these probabilities are independent as we vary p (see Remark 3.17), the probability that we have one to all 7l,j p is zero. Turning now to the case of profinitefold groups, the natural analog of the virtually positive betti conjecture is 4.8 Question Does the profinitefold group G defined by almost every profinite Heegaard diagram have a subgroup of finite index with a continuous epimorphism to Z?

It is too weak a condition merely to reqnire that G have an infinite abelianization. In fact, the abelianization of almost every profinitefold group is indeed infinite, because the first homology group of a 3-manifold is typically a finite group that is large if the manifold is complicated. There are uncountably many isomorphism classes of Heegaard profinite diagrams up to isomorphism, so the countable set coming from profinite completions of actual Heegaard diagrams forms a set of measure O. Thus, Question 4.8 and the question of whether all 3-manifolds with infinite fundamental group have virtually positive betti number does not appear to have any easy logical implication one way or the other - the divergence between them involves different orders of taking limits. Nevertheless, they are intuitively and heuristically connected, and so it would be quite interesting to settle Question 4.8. 5 Quotients of 3-manifold groups Group presentations coming from Heegaard splittings of 3-manifolds differ substantially from random deficiency-O presentations because the relators, rather than being generic elements in the free group, are given by a g-tuple of disjoint simple closed curves on a genus g handlebody. Indeed, 3-manifold presentations are a vanishingly small proportion of all deficiency-O presentations since the number of simple closed curves with word length R grows polynomially in R rather than exponentially. Geometrically, the curves' embeddedness forces the words to be far from independent, and typically there are many repeating syllables at varying scales (for a graphical illustration of this, see [DT1, Fig. 1.5]). In this section, we try to explain why these geometric properties force there to be more finite quotients than for a general deficiency-O group. Later we will examine this question from the point of view of random Heegaard splittings (Sects. 6-9), but in this section we take a more naive heuristic point 268

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of view. In particular, we try to explain why a given quotient f: Fg --+ Q is much more likely to extend over the last 2 relators.

5.1 Last relator One reason to expect 3-manifold groups to have more finite quotients than random deficiency-O presentations has to do with the last relator. To describe this topologically, if we attach 2-handles to the handlebody along g -1 of the curves, we obtain a 3-manifold M whose boundary is a torus; the remaining curve is a simple closed curve on the torus. Thus, if an epimorphism Fg --+ Q satisfies the first g - 1 relators, the remaining relator is restricted to an abelian subgroup A of S that can be generated by at most 2 elements. Assuming that the distribution in A is nearly uniform, this suggests that there is approximately a 1/ IA I chance that the last relator is satisfied, as compared to a 1/1 Q I chance for a general relator. Actually, the situation is more complicated because the last relator is a simple closed curve on the torus aM. Consider a torus T and a finite quotient f: 1l'1 (T) --+ A. Simple closed curves on T correspond to primitive elements of 1l'1 (T) = 7l}, and so we are interested in the probability that a primitive element lies in the kernel of f. If A is cyclic of order a, then one can change basis so that f is the factor-preserving map Z Ell Z --+ Z/a Ell O. Now look at all primitive lattice points in Z2 in some large ball; we want to know the proportion of them which lie in the kernel of f. It turns out that this is not quite l/a, but rather 1/ f3(a) where f3 is a function of the prime decomposition of a given by f3(p~l p~2 ... p~)

= p~'-1(pl + l)p~2-1(p2 + 1) ... p':;-l(Pn + 1).

On the other hand, if A is non-cyclic then you can change basis for 1l'1 (T) = Z2 so that f is the factor-preserving map Z Ell Z --+ Z/a Ell Z/ab; thus every element of the kernel is divisible by a, and so there are rw primitive elements in the kernel. Returning to our original situation, suppose we are attaching the last of g relators and want to know if a given epimorphism Fg --+ Q extends over this final handle. This leads us to ask: what is the distribution of possible subgroups A of Q which are the image of the fundamental group of the remaining torus T? Not all 2-generator abelian subgroups can occur. For instance, the image H 2 (A) --+ H 2 (Q) must be trivial, since the torus T is the boundary of a 3-manifold and H 2 (T) --+ H 2 (A) is surjective. This condition reduces the number of non-cyclic A we need to consider (though it need not eliminate them completely), which is good since those never extend to the resulting manifold. For example, in As the subgroups isomorphic to Z/2E1lZ/2 are eliminated by this criterion, and so the relevant abelian subgroups are just the cyclic subgroups, which have orders 1, 2, 3 or 5. If each type of cyclic subgroup occurs equally often, this would lead to the guess that the last relator would WILLIAM P. THURSTON

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be satisfied 35% of the time. But even if the cyclic group of order 5 occurs much more frequently than the others, this would still give an estimate that the last relator would be satisfied 16.7% of the time, far more than the 1.7% predicted for a random relator. If one looks at a random map from Z2 --+ As, then the cyclic group of order 5 is indeed the most common image, occurring about half the time. 5.2 Genus 2

There is also a special argument that sometimes applies to the next-to-Iast relator, which works in particular for epimorphisms to PS~Fq' The surface E of genus 2 has a special homeomorphism of order 2, the hyperelliptic symmetry r, that is centralized by the entire mapping class group of E. The quotient E Ir orbifold is a sphere with 6 elliptic points of order 2. Any simple closed curve on E can be isotoped to be set-wise invariant under r. If the curve is non-separating, then it is mapped to itself with reversed orientation. If we fix a hyperbolic metric on E which is invariant under r, then the geodesic representative of a non-separating curve passes through exactly 2 of the 6 fixed points of r. The consequences of this are easiest to describe for the boundary of a genus 2 handlebody H: Any non-separating simple closed curve on aH describes a circular word in the free group F2 = 1rI (H) that is the same read backward or forward. This is because the hyperelliptic symmetry r of aH extends over H; the induced action r.: 1rI (H) --+ 1rI (H) sends standard generators {a, b} of 1r1 (H) to their inverses {a-I, b- I }. As mentioned, r sends a non-separating curve on aH to itself with reversed orientation. Thus if w is a word in 1r1 (H) which is the image of a non-separating curve, we have that r. (w) is conjugate to w- I ; if w is regarded as a circular word this is the same as saying that it is the same read backward or forward. It may or may not be possible to conjugate the linear word w to read the same backward and forward. This depends on which pair of the 6 fixed points of r the curve passes through, as we now explain. Pick dual discs for our chosen basis of 1r1 (H) which are invariant under r; these contain 4 of the fixed points of r. Running around the geodesic representing w looking at intersections with the discs reads off the word w. If the geodesic goes through one of the middle 2 fixed points of r which are not near the dual disks, then reading off starting at one of those points results in a linearly palindromic w. If instead we start reading from a fixed point of r on one of the discs, then we get a w so that r(w) = SW-IS- I where s is one of the generators. Thus, it is always possible to conjugate w so that r.(w) = sw- I s-I where s E {I, a±l, b±l}. In this case, we will say that w is in standard form. In the case of an epimorphism 1r1(H) --+ PS~Fq, we will show that the involution r. on 1r1 (H) pushes forward to one on the image group. We will use this fact to greatly restrict the possibilities for the image of a non-separating curve in PS~Fq. In particular, we will show: 270

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5.3 Theorem

Let H be a handlebody of genus 2, and p: 7f1 (H) --+ be an epimorphism where q is an odd prime power. If w is a word in standard form coming from a non-separating embedded curve in fJH, then the image of w under p lies in a subset of PS~lFq of size at most (I/2)(q2 + q + 2). PS~lF q

For comparison, the order ofPS~lFq is (I/2)q(q + I)(q -1). Nole also there is always a non-separating curve on 1J in the kernel of p, as there is one in the kernel of 7fI (1J) --+ 7fI (H). Thus from this theorem one would naively expect that a non-separating curve is about q times more likely to be in the kernel of p than a random word in 7f1 (H). In the theorem, the subset of PS~lFq mentioned depends on the standard form of w, more precisely on the s such that r.(w) = sw-Is- I . If you prefer a stalement which is independent of s, just multiply the size of the subset by 5 (the number of possibilities for s). Before proving Theorem 5.3, let us further contrast the pictore it gives with that of random words in 7f1 (H). Consider 3-manifolds M obtained by attaching a single 2-handle to H along a non-separating curve (these are examples of tunnel-number one 3-manifolds). For comparison, look at twogenerator, one-relator groups where the generator is chosen at random. For such a random group, we can work out the probability of a Q = PS~lFq quotient just as we did before; there are Rj IQ I/IOut Q I epimorphisms from F2 onto Q, and each factors over the relator with probability II IQ I. Thus the number of Q-quotients should be roughly Poisson distribUled with mean I/IOut QI. If we specialize to the case that q is prime, then lOut Q I = 2 and so the probability of a Q-quotient for the random group is I 2 Rj 39%. In particular, this probability is essentially independent Rj 1- e- / of Q. In contrast, Theorem 5.3 suggests that the number of quotients of a 3-manifold group should be Poisson distribuled with mean Rj qllOut QI. In the case where q is prime, this leads to the probability of a Q cover being 1 - e- q / 2 , which goes to 1 as Out Q --+ 00. The last column of Table 6.6 gives some data on this, using random curves coming from our notion of a random Heegaard splitting. It suggests that, at least qualitatively, this last prediction of Theorem 5.3 really does hold. To prove Theorem 5.3, we first show that r. pushes forward to the image group PS~lFq.

5.4 Lemma Let A and B be elements of

PS~lF q

where q is an odd prime power. Suppose that A and B do not have have a common fixed point when acting on pi (IFq). Then there exists an element T in PG~lFq of order 2 such that

Proof In general, the trace of an element in PG~lFq depends on the lift to G~lF q. However, the elements T in PG~lFq which have order 2 are exactly those where the trace of any lift is O. Now lift A and B to elements of S~lFq' WILLIAM P. THURSTON

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and consider the equations tr(T)

= 0, tr(TA) = 0,

and

tr(TB)

=0

where T is a 2 x 2 matrix over IFq (possibly singular). As these are homogeneous linear equations, there is a non-zero solution, call it T. We claim that T must be non-singular. If not, change to a basis where the first vector spans the kernel of T, and so

T -- (00

ot)

where t

¥- O.

But then tr(TA) = 0 forces A to be upper-triangular. As the same is true for B, we have a contradiction in that A and B have a common fixed point in p1(IFq). In PG~lFq, the elements T, TA, and TB have order 2, which proves the lemma. 0

Now consider an epimorphism p: 7fl(H) -+ PS~lFq. By the lemma, there exists aTE PG~IFq so that the involution of PS~IFq induced by T is the push-forward of the involution T. on 7fl (H). So if w is a word in standard form coming from a non-separating curve, then there is a U of order 2 in PG~lFq such that Up(w)U- 1 = p(W)-I, where U is one of T, A±IT, or B±IT. Thus just knowing that w comes from a non-separating curve implies that p( w) is sent to its inverse by an involution of PS~IFq which is completely determined by p and the symmetry points of w. The next lemma shows that this restricts p( w) to a proper subset of PS~IFq' and completes the proof of Theorem 5.3.

5.5 Lemma Let q be an odd prime power, and U an element of PG~IFq of order 2. Then the number of W E PS~IFq such that UWU- 1 = W- 1 is

21 (q2 + (2E_ -

l)q + 2,,+),

where ,,± is 1 if ± det U is a square in IFq and 0 otherwise. Proof Requiring that UWU- 1 = W- 1 is the same as saying that UW has order 1 or 2. In the former case, W = U and this contributes to our count only when U E PS~lFq, that is, when = 1. The latter case is the same as counting solutions to the equations,

"+

tr(UW) = 0

and

det(W) = 1 ,

where here we have lifted everything to G~lFq. One can write out these equations explicitly (one is linear and the other quadratic), and it is not to hard to see that the number of solutions is equal to q2 + (2,,- - l)q. Passing to PS~ from S~ reduces the number of such W by half. Combining, we get the count claimed for W. 0 272

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5.6 Second to last relator Having worked out the simpler handlebody case, we return to our original question about attaching the second to last relator. Again, let lJ be a surface of genus 2, and set

G

= 1f1 (lJ) = (a, b, c, d I [a, b] = [c, d]),

where our convention is that [a, b] = aba- l b- l . If we choose the base point for 1f1 (lJ) to be the second leftmost fixed point of -r, then -r acts on G in the following way -r(a) = a-I, -r(b) = b- l , -r(c) = xc- l X-I, and -r(d) = xd-lx- l , (5.7) where x = a-lb-ldc. By taking arcs from the base point to the other fixed points of -r, we see that every non-separating simple closed curve on lJ can be represenied by a element w E G such that -r.(w) = sws- I where s is in S = {I, a, b, x, xd}. Such a w is said to be in standard form. Now consider a homomorphism G --+ PS~IFq' The next theorem gives a criterion for when -r pushes forward to an automorphism of PS~IFq. 5.S Theorem Let G be the fundamental group of a surface of genus 2. Let -r be the automorphism of G coming from the hyperelliptic irwolution. Consider an irreducible homomorphism f : G --+ PS~IFq' where q is odd. If f lifts to a homomorphism into S~IFq' then there is an element T E PG~IFq such that Tfr- I = f o-r. Now let M be a 3-manifold with boundary lJ, and suppose f is the restriction of a homomorphism 1f1(M) --+ PS~q. The obstruction to lifting f to S~IFq is an element of 1f2(G, Z/2Z), which vanishes as f extends over M. So if f is irreducible, then it has the above symmetry, and !his restricts the image under f of a non-separating simple closed curve similar to as before. If w EGis a standard form representative for such a curve, then f(w) must satisfy Tf(w)T- 1 = f(s)f(w)-l f(S)-I, where s is one of the five elements of S and T E PG~IFq is the element inducing the symmetry. If instead f is reducible, its image lies in a proper subgroup of PS~IFq.

Thus, in either case, the image of f( w) is restricied. So as long as there is some non-separating simple closed curve in the kernel of f, one would expect that the probability that f extends over the second to last relator should be much higher than for a random word in a free group. However, as in the torus case, there could be situations where there are no such curves in ker f. Unlike the torus case, we do not know any examples where !his actually occurs. Now let us prove the theorem.

Proof Before begimIing the proof itself, let us rephrase the question in order to make the algebra that follows seem more natural. Let X be the hyperbolic orbifold lJI -r. We have 1 --+ G --+

1f1 (X)

--+ Z/2 --+ 1, WILLIAM P. THURSTON

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where 1l'1 (X) can be obtained by adding an element t to G subject to the requirement that conjugating by t induces the action of r given in (5.7). Thus the problem at hand is given f: G -+ PS~lFq' does f extend to a homomorphism 1l'1(X) -+ PG~lFq? The separating curve in 1J representing [a, b] maps down to a curve y which separates X into two regions containing 3 cone points each; in 1l'1(X) we have y2 = [a, b]. Roughly, Lemma 5.4 says that we can extend f over each half of X, and the issue is whether these agree along y. It toms out that f lifting to S~lFq guarantees this. Let denote the lift of f to SL2lFq , and set A = l(a), B = l(b), etc. To prove the theorem, it suffices to find aTE G~lFq such that

1

tr(T) = tr(TA) = tr(TB) = tr(X- 1T) = tr(X-ITC) = tr(X- 1 TD) = 0

(5.9) as then all of the above elements have order 2 in PG~lFq and so TfT- 1 = for. The main case is when f is irreducible when restricted to both of the subgroups (a, b) and (c, d), and we begin there. Think of (5.9) as homogeneous linear equations in the entries of a 2 x 2 matrix T. By the proof of Lemma 5.4, any non-zero T which satisfies the first 3 trace conditions is necessarily non-singular. Thus we just need to prove that the dimension of the solution space of (5.9) is positive dimensional. To check this, we are free to enlarge our base field from IFq to its algebraic closure k. Now over k, the equations (5.9) have a non-zero solution if and only if there is one with det(T) = 1, since any non-zero determinant has a square root in k. So henceforth, we try to solve equations (5.9) for T E S~k. By Lemma 5.4, we can choose T, U E S~k so that tr(T) = tr(TA) = tr(TB) = 0

and

tr(U) = tr(UC) = tr(UD) =

o.

Our goal is show U = X-IT. As we are working in S~ rather than PS~ we have T2 = -1 not 1, and the same for TA, U, etc. Thus we have (BAT)2 = - [B, A] = - [D, C] = (DCU)2.

So BAT and DCU have the same square, which is not -1 because if A and B commuted we would be in the reducible case. Any S f. -1 in S~k has at most 2 square roots. Thus, after replacing U with -U if necessary, we have BAT = DCU which implies U = X-I T, as required. This completes the proof of the theorem in the case when f is irreducible when restricted to both (a, b) and (c, d). Now suppose instead f is reducible when restricted to (a, b). This is equivalent to tr(ABA- I B- 1) = 2, and the same reasoning implies that f must be reducible when restricted to (c, d) as well. Now ABA- 1 B- 1 cannot be parabolic, as then f itself would be reducible. Therefore, we are down to the case where ABA-1 B- 1 = I, i.e. both (A, B) and (C, D) are abelian. Now by changing generators of G we can assume that (A, C) do not have 274

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a common fixed point in pl. By Lemma 5.4, there is aTE G~(lFq) such that tr(T) = tr(TA) = tr(TCBA) = O.

Now as B commutes with A, the above also forces tr(TB) = tr(TBA) = tr(BAT) = O. Rewriting tr(TCBA) = 0 as tr(BAT· C) = 0, the commuting of C and D gives tr(BAT· Y) = 0 for all Y E (C, D). Expanding X in (5.9), it follows that all those equations hold, completing the proof. 0 6 Covers of random Heegaard splittings In this section, we address the following question: Fix a finite group Q and genus g. Consider a manifold M obtained from a random genus-g Heegaard splitting. What is the probability that M has a cover with covering group Q? What is the distribution of the number of covers? In this section, we will begin looking at these by showing that the answers to both these question are well-defined (Prop. 6.1). We will then give three examples which illustrate some of the key issues in computing these probabilities. In later sections, we will compute these probabilities exactly for abelian groups (Sect. 8), and we will give a complete characterization of these probabilities for non-abelian simple groups in the limit when g is large (Sect. 7). Fix a genus-g handlebody H g, and denote aHg by E. Let .Mg be the mapping class group of E. Given t/J E .Mg , let Nt/> be the closed 3-manifold obtained by gluing together two copies of Hg via t/J. Our notion of a random Heegaard splitting of genus g is as follows. Fix generators T for .Mg. A random element t/J of .Mg of complexity L is defined to be the result of a random walk in the genemtors T of length L. Then we define the manifold of a random Heegaard splitting of genus g and complexity L to be Nt/>, where t/J is a random element of .Mg of complexity L. We begin with the following proposition which shows that the questions we are interested in make sense in the context of random Heegaard splittings:

6.1 Proposition Fix a Heegaard genus g and ajinite group Q. Let N be the manifold ofa random Heegaard splitting ofcomplexity L, and let p(L) be the probability that 17:!(N) has an epimorphism onto Q. Then p(L) converges to a limit p(Q, g) as L goes to infinity. Moreover; p(Q, g) is independent of the choice of generators for .Mg , and the probability distribution of the number of epimorphisms also converges. Proof Consider the collection .A. of all epimorphisms from 17:! (E) to Q, up to automorphisms of Q. Let t/J be in .Mg, and consider the associated 3-manifold Nt/>. Identify E with the boundary of the first copy of Hg in Nt/>, so that a homomorphism f in .A. factors through to one of 17:\ (Nt/» if and only if f and f 0 t/J:;! both extend over H g • Let 8 C .A. consist of those homomorphisms which do extend over H g . Before proving the full proposition, let us consider the simpler question: WILLIAM P. THURSTON

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Given an f in 8, what is the probability that it extends over the copy of Hg in Nq,? That is, how often does f 0 cfJ;1 also lie in 8? Consider the (left) action of .M.g on A by precomposition: cfJ • f = f 0 cfJ;l. Thus we are interested in the probability that cfJ • f lies in 8. Now look at the Caley graph of this action of .M.g on A. That is, consider the graph whose vertices are A and whose edges correspond to the action of the chosen generators T of .M.g. If we do a random walk in .M.g, the corresponding sequence of cfJi • f moves around this graph according to the labels on the edges. As in Lemma 3.2, the distribution of the cfJi • f converges to the unifonn measure on the orbit of f in A. Let e be the orbit of f in A. Since the cfJi • f are nearly uniformly distributed in e for large i, we see that the probability that f extends to N q, converges to Ie n 81/1C1. Note that this limit depends only on the orbit e and not on the choice of generators for .M.g. Returning to the question of the probability of Nq, having an epimorphism to Q, consider the action of .M.g on subsets of A. Again, if we look at the images of 8 under cfJi, these are nearly uniformly distributed in the orbit of 8 in the power set of A; thus peL) converges to the proportion of subsets in the orbit of 8 which intersect 8. Finally, the distribution of the number of epimorphisms also converges, to the corresponding finite averages over 0 the orbit of 8 under .M.g. Next, we will give three detailed examples in genus 2 where we calculate these probabilities exactly. These illustrate the some of the main issues and techniques that arise later in Sects. 7-8.

6.2 Example: Z/2 For our first genus 2 example, let us begin with Q = Z/2. In this case, A is isomorphic to the non-zero elements of HI(lJ; Z/2). Thus IAI = IS, and similarly 181 = 3. In order to compute the probabilities, we need to understand the image of .M.2 in the symmetric group of .A.. While the action is transitive, its image is much smaller than all of Sym(A): it is the 4-dimensional symplectic group over ]F2, which we will call G. As we do a random walk in .M.2 , the images under.M.2 --* Sym(Ad converge to the unifonn distribution on G. Thus p(Z/2, 2) is the same as the probability that (g. 8) n 8 f. 0 for a random g E G. First, let us compute the expected number of Z/2-quotients. To begin, focus on whether a fixed f E 8 extends to 7fJ(Nq,). For g E G, we have f E g. 8 if and only if g-J . f E 8. Since the action of G is transitive, {g-I . f}geG is uniformly distributed in A. Therefore, the probability that f E g·8is 181/1AI = liS. Thus the expected number ofZ/2 quotients from f is 115. As expectations add, the overall expected number of quotients is 181 2 II A I = 3/5. Computing p(Z/2, 2) is more complicated because there may be correlations between different f in 8 extending (see the discussion in the next example). In this case, it turns out that averaging over all 720 elements of G gives that p(Z/2, 2) = 7/15. 276

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Of course, p(7L./2,2) is the same as the probability that HI(N4); 7L./2) In Sect. 8, we will work out the distribution of the homology for general 7L./ p. Readers uninterested in non-abelian groups can skip ahead to that section now.

f= O.

6.3 Example: As Next, let us consider the smallest non-abelian simple group Q = As. In this case, I.>'\, I = 2016 and 1&1 = 19. The action of .M2 on .>'\, has two orbits .>'\,1 and .>'\,2 of size 1440 and 576 respectively, and 8 is completely contained in .>'\,1. In order to compute peAs, 2), we only need to understand the action on .>'\, I. It turns out that this is the full alternating group AIt(.>'\, I). As before, we get that peAs, 2) is the probability that (g . 8) n 8 f= 0 for a random g E AIt(.>'\,I). Now Alt(.>'\,I) acts transitively on subsets of .>'\,1 of size 19, as 19 ::: I.>'\, II - 2. Thus g . 8 is uniformly distributed over all subsets of .>'\,1 of size 19. Hence the probability that (g . 8) n 8 f= 0 is just the probability that a randomly chosen subset of 19 elements of .>'\,1 intersects 8. Thus

p(As,2) = 1-

C~~I)/C:O) ~ 22.43%.

As in the previous example, the expected number of As quotients is 181 2/ I.>'\, tI ~0.2507.

The case where .Mg acts as the alternating group of an orbit is one that we will find in general for non-abelian simple groups, so it is worth discussing here the connection to the Poisson distribution. Recall that the Poisson distribution with mean JL > 0 is a probability distribution on 7L.+ where k has probability i~ e- fL • Roughly, the Poisson distribution describes the number k of occurrences of a preferred outcome in a large ensemble of events where, individually, the outcome is rare and independent, but in aggregate the expected number of occurrences is JL > O. In our context, we have a set.>'\, of size n which contains a marked subset & of size a; we then pick another subset 8' of size a and want to know the size of 8 n 8'. If n is large, the distribution of 18 n 8'1 is essentially Poisson with mean JL = a2 /n. In particular, the probability of at least one intersection is l-e- fL • In the case of As, this approximation gives the probability of an As cover at 22.17%. It is worth mentioning that the alternating group action here makes it very easy to compute p(Q, g) from just the sizes of the orbits of.>'\, and &, and that this is not true in general. For instance, returning to the previous example, p(7L./2,2) = 7/15 ~ 0.4667 but the probability that 181 = 3 items chosen from I.>'\, I = 15 things intersects a fixed set of size 3 is larger: 47 /91 ~ 0.5165. The reason for the difference is that the action of.M2 is not 3-transitive, and there are positive correlations between different elements of 8 factoring through to 7f I (N4»; in particular, because the number of WILLIAM P. THURSTON

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Z/2-quotients is IH1(Nt/>, Z/2)1- 1 = 2n Z/2-quotient then we have 3 of them.

6.4 Example:

-

I, if we have more that one

PS~lF 13

Let us look at a more complicated example. Consider Q = PS~lF 13, a group of order 1092. In this case, I.AI = 623520 and 181 = 495. In this case there are four orbits of sizes 235680, 94080, 278400, and 15360. Only the first two orbits intersect 8 in subsets ofsize 307 and 188 respectively; set 8 i = 8 n.Ai. The action on the first two orbits .A 1 and .A2 are again by the full alternating groups Alt(.Ai). Since the two alternating groups have different orders, by Lemma 3.7 the map .M.2 --+ A1t(.Ad x Alt(.A2) is surjective. Therefore g. 8 1 and g . 8 2 are independent of each other, and 2

p

(PS'

_11>'

2) = 1 -

'-"2"'13,

8 IT (l.Ad - 1 d)/(I.A d ) i=1 18d 18d

~ 54.02%.

Further, the expected number of quotients is L~=l 18d 2/ l.Ad ~ 0.7756. Although 8 is contained in two orbits, the overall distribution of quotients is still nearly Poisson since the sum of two independent Poisson variables is also Poisson.

6.5 Example: Small simple groups As we will see in Lemma 6.10, for simple groups the size of .A g grows like IQ12g - 2. Thus even for genus 2, it is difficult to compute the action of .M.g on .A g when IQI is large. However, the proof of Proposition 6.1 suggests a way to approximate p(Q, g) by looking only at IQlg- 1 epimorphisms. Namely, we first compute 8g and consider cp E .M.g which is evolving by a random walk in our fixed generators. The time average of (cp. 8g ) n 8g will then converge to p(Q, g). For genus 2, we did this for the simple groups of order less than 7000. The resnlts are shown in Table 6.6, and we compare them to the results of our earlier experiment [DT21, as well as the limit of p(Q, g) as g --+ 00.

6.7 The general picture The rest of this section is devoted to what we can say in general about p(Q, g) for an arbitrary group, with particular emphasis on the limiting picture as the genus g goes to infinity. We would like to say that the probability distributions on the number of covers for each g converge to a limiting probability distribution as g --+ 00; in particular, this would suggest some robustness in our notion of random Heegaard splitting and that, perhaps, we 278

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Table 6.6 This table gives the percentage of genus 2 manifolds which have particular finite shnple quotients, and compares this data to other samples. The first 4 columns list the quotient group Q and some of its basic properties. The number p(Q, 2) is the probability that a manifold coming from a random Heegaard splitting of genus 2 will have a cover with covering group Q; as discussed in Sect. 6.5, these numbers are actually an approximation coming from looking at a random walk in oM2 of length 106. The Census column is the corresponding probability for those genus 2 manifolds in the Hodgson-Weeks census that we studied in [DTI, §5]. The number p(Q, (0) is the limit of p(Q, g) as g --> 00; by Theorem 7.1 we have p(Q, (0) = I - e-IH21/10utl . Finally, we looked at attaching a single 2-handle to a genus 2 handlebody along a non-separating curve to give a tonnel number one manifold with torus boundary; as before, the attaching curve is chosen by a random walk in oM2. The last column records the probability that the resulting manifold has a Q cover. This data is interesting to compare with the discussion in Sect. 5.2. 100ti

IH21

p(Q,2)

Census

p(Q, (0)

Sing. Quo

A6

60 168 360

2 2 4

2 2 6

22.4 30.8 28.9

26.9 28.2 31.4

63.2 63.2 77.7

72.09 88.95 85.04

PSL,lFg PSL,lFu PSL,IF13

504 660 1092

3 2 2

1 2 2

22.1 41.7 54.0

21.7 32.8 41.1

28.3 63.2 63.2

89.37 98.63 99.85

PSL,IF17 PSL,1F19

2448 2520 3420

2 2 2

2 6 2

56.3 60.1 55.9

43.1 45.8 44.4

63.2 95.0 63.2

99.97 99.86 99.99

PSL,1F16 PSL31F3 U301'3)

4080 5616 6048

4 2 2

1 1 I

19.3 40.5 31.5

18.3 28.0 18.0

22.0 39.3 39.3

97.36 99.76 99.57

PSL,IF23

6072

2

2

58.9

47.6

63.2

99.99

Q

Order

As PSL,1F7

A7

should expect to find this same limiting distribution with other notions of random 3-manifold. While we will build up quite a bit of information about .;\, g' «g and .;\, g / .M.g for large g, we do not know how to prove the existence of a limiting distribution in general. Instead, we will show here that the expected number of covers does have a limit as g --* 00 (Theorem 6.21). In the special cases of abelian and simple groups, we are able to obtain a complete asymptotic picture (see Sects. 7 and 8). The case of abelian groups is essentially independent of the rest of this section, so if that is your primary interest you can skip ahead to Sect. 8. Examples 6.2--6.4 illustrate the key issues that we encounter here. As usual, let Hg be a handlebody of genus g, I:g = aHg , and .M.g be the mapping class group of I:g • We want to compute the probability p(Q, g) that a 3-manifold associated to a random genus-g Heegaard splitting has a cover with group Q. As we saw in the examples, what we need to know is how many epimorphisms 11"1 (I:g ) --* Q there are, and how .M.g acts on them. As we will see, the answer to the first question is easy (Sect. 6.8), and it is the second question that requires more work. For the latter question, for a general Q we will only be able to classify the orbit set .;\,g / .M.g (Sects. 6.12--6.19). The basic idea is this: given a homomorphism f: 11"1 (I:g ) WILLIAM P. THURSTON

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-+ Q in Ag we get a map from Ig to the classifying space BQ, and thus an element of Hz(Q, Z). This homology class is invariant under the action of .M.g on A g • The key is to show that for large g this class is the only invariant of the action of .M.g on Ag (Theorem 6.20). Finally, this section concludes with a more group-theoretic point of view on some aspects of this section as they relate to 8g (Sect. 6.24). 6.8 Counting A and 8

In this subsection, we determine the number of elements of A and 8 for a fixed group Q as the genus gets large. For resnlts where the genus is fixed but the (typically simple) group Q gets large see [LSI] and [LS2]. If Q is a group, Q' will denote the commutator subgroup. The following lemma computes for us the probability that a random homomorphism Fzg -+ Q factors through JT 1 (Ig).

6.9Lemma LetQbeafinitegroup.lf(a\,b\, ... ,ag,bg) E QZgischosen uniformly at random, the probability that O[ai, b i ] = 1 converges to III Q'I as g -+ 00. Proof The set T = {[a, b]}a,bEQ generates Q'. Thus choosing (ai, bi) E QZg at random is the same as choosing a string of g elements of T. That is, [ai, bi ] is the resnlt of a random walk in Q' with respect to T. As such, it

o

converges to the uniform distribution on Q'. Therefore, the probability of it being 1 converges to l/IQ'1 as g -+ 00. 0 For non-abelian simple groups, the set T above is very large, conjectorally all of Q = Q'. In this case, the random walk is on a (weighted) complete graph with vertices Q. Thus it will converge to the uniform distribution very quickly. Let Ag be the set of all epimorphisms of JTl(Ig) to Q, modnlo automorphisms of Q. 6.10 Lemma Let Q be afinite group. Then as g -+ 00.

IAgl

~

IQI Zg1(1 Q'I IAut(Q) I)

Proof Consider JT\ (Ig) in its standard presentation with 2g generators. A 2g-tuple (ai, b;) E QZg gives a possible homomorphism JT\(I g) -+ Q. If the 2g-tuple is randomly chosen, the probability that the entries generate Q goes to 1 as g -+ 00. By the preceding lemma, the probability that it induces a homomorphism of JTl (Ig) is III Q'I. Combining the above, this says that the number of elements of QZg which give epimorphisms is asymptotic to IQI Zg II Q'I· Since the action of Aut(Q) on epimorphisms is free, we get the

claimed formula for

IA I.

0

We also record here the corresponding result about 8. 6.11 Lemma Let Q be afinite group. Then 280

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6.12 Associated homology classes

For us, an important invariant of an epimorphism f: 71'(17g) -->- Q is the associated homology class cf in H2(Q, Z) coming from the induced map on classifying spaces 17g -->- BQ. In the context of finite groups, H 2(Q, Z) is usually called the Schur multiplier, and it is an important invariant there, especially in the study of simple groups (see e.g. [WieD. In this section all homology groups will have Z coefficients, so we will stop including this in the notation. It is not quite true that this homology class is well-defined for elements of .A,g, but we can associate an element of H2(Q)/Out(Q). The issue here is that Aut(Q) may act non-trivially on H2(Q); after all, elements in .A, g are equivalence classes of epimorphisms modulo the action of Aut(Q). Now inner automorphisms of Q induce self-maps of BQ which are homotopic to the identity, and so such automorphisms act trivially on H2(Q). Thus associated to an [fl E .A,g we get a well-defined homology class c[f] in H 2(Q)/ Out(Q). For simple groups, it is often the case that H2(Q) is trivial or Z/2; the action ofOut(Q) must be trivial in this case. The first simple group where the action is non-trivial is A 6 , where H2 (Q) = Z/6 and Out(Q) -->- Aut(H2(Q)) = Z/2 is surjective. In general, the map Out(Q) -->- Aut(H2(Q)) need not be surjective, as the example of PSylF4 shows. Now the map .A,g -->- H 2(Q)/Out(Q) is invariant under the action of .Mg , since acting by a mapping class just re-marks the surface and so does not change the image of the map 17g -->- BQ. Thus we get a well-defined map .A,g/.Mg -->- H2(Q)/Out(Q). Later, we will show that this map is a bijection for large g. For this reason, we are interested in the number of epimorphisms in each homology class: 6.13 Lemma Let Q be afinite group. Fix [cJ E H 2 (Q, Z)/Out(Q). Let k be the number of elements of H2(Q, Z) which are in the equivalence class [cJ. Then the ratio

IU E

.A,g

I cf = [c]}l/I.A,gl

converges to k/IH2(Q, Z)I as g -->-

00.

Proof First, we explain how to compute cf directly using Hopf's description of H2 of a group (see e.g. [Bro, §II.SD. Let G be a finitely generated group, and express it as a quotient of a free group G = F/ R. Then H 2 (G) can be naturally identified with (F' n R)/[F, RJ. From this point of view, H 2 (G) is a subgroup ofleftmost term of the exact sequence

1 -->- R/[F, RJ -->- F/[F, RJ -->- G -->- 1;

(6.14)

note that the leftmost term is central in the middle term. When G is finite, R/[F, RJ is the direct sum of (F' n R)/[F, RJ and a free abelian group A (see e.g. [WieD. Taking the quotient by A we get a exact sequence of finite groups

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where S is called a Schur cover of G. Here, H 2 (G) is central in Sand contained in Sf. The Schur cover need not be unique unless G is perfect; in that case S ~ Ff I[ F, R] and S is the universal central extension of G. Now, let I: 1l'1(17g) ~ Q be a homomorphism. Consider standard generators ai, b i of 1l'1 (17g); in the Hopf picture, H2 (.L') is generated by the standard relator n[ai' bil. Fix a Schur cover S of Q. Consider an induced map of the sequences (6.14) for 1l'1(17g) and Q. Thus cf is given by the following proceedure: Pick lifts Si, ti E S of the images I(ai), I(bi ), and then Cf = n[Si, til. Now we determine the proportion of elements of ""'g which have a fixed homology class. It is easier here to work directly with epimorphisms I: 1l'1(17g) ~ Q before quotienting out by Aut(Q). As in Lemma 6.10, consider a random 2g-tuple (Si' ti) of elements of S and let C = fUSi, til. The images of (Si, ti) in Q are also uniformly distributed, and therefore the image of C in Q is nearly uniformly distributed in Qf if g is large. Recall that H2(Q) lies in Sf. As we know C is nearly ur!iformly distributed in Sf, if we restrict to those (Si' ti) which induce a homomorphism 1l'1 (17g) ~ Q, the probability that C is some particular element of H2(Q) is essentially I/IH2(Q)I. Modding out by Aut Q gives the probability claimed in the original statement. 0 6.15 Stabilization

Now, we want to discuss our main topological tool for understanding ""'g when g is large. Let I: 1l'1 (17g) ~ Q be in ""'g. A stabilization of I is an element If of ""'g+h obtained by viewing 17g+h as 17g#17h and setting If to be I on 17g and the trivial homomorphism on 17h; that is, If is the composition of I with the map on 1l'1 induced by the quotient map 17g#17h ~ 17g. Note here we are not fixing a particular identification of 17g+h with 17g#17h, so each I E ""'g usually gives rise to many elements in ""'Hh. So that stabilization respects the associated classes in H2(Q)1 Out Q, we do require the identification of 17Hh with 17g#17h to be orientation preserving. Looking at it another way, an If in ""'g+h is the result of an h-fold stabilization if there is an essential subsurface S of 17g+h which is a once-punctured 17h, and where If is the trivial homomorphism on 1l'1 (S). First, we will show that for large enough genus every element of ""'g is a stabilization: 6.16 Proposition Let Q be afinite group.1lg > IQI then every I: 1l'1 (17g) ~ Q is a stabilization.

Before giving the proof, let us point out an important consequence. If E Ag are in the same orbit under .Mg, then so are their stabilizations in ""'g+1 under the action of .Mg+1. Thus we get a map of orbit sets ""'g l.M g ~ ""'g+1I .Mg+!' The proposition says thatfor g > IQ I these maps are swjective; thus once some threshold is crossed no new orbits appear,

Ii, fz

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though existing orbits can merge. But since there are only finitely many orbits, this merging process must eventually stop. Thus we have

6.17 Corollary Let Q be a finite group. Then for large enough g, the number of.M.g orbits in .ll. g is constant. Itis worth noting that this argument gives no indication of when l.II.g/ .M.gl stabilizes, even though Proposition 6.16 gives an explicit bound. Later, we will show the stable number of orbits is just IH2(Q, Z)/ Out(Q)I. We now prove the proposition. Proof View E g as g punctured-torus spokes attached to a central hub which is a g-times punctured sphere. Consider standard 2g generators {£I;, bi } of lb 11"1 (Eg), where each (ai, bi ) pair are generators of the i spoke. lfwe orient everything symmetrically, any element Wi = al • a2 •.. £I; of 11"1 (Eg), with 1 ::: i ::: g, can be represented by an embedded non-separating simple closed curve in E g. Now fix f E A g • We need to find an essential punctured torus on which f is the trivial homomorphism. By the pigeon hole principle, there exists i < j such that f(Wi) = f(wj). Therefore, f(ai+! ... aj) = I, and so we can find one non-separating simple closed curve in the kernel of f. To turn this into an entire handle where f is trivial, consider a maximal disjoint collection CI, .•. ,Ck of such non-separating curves in the kernel of f. By changing the basis of 11"1 (E), we can make these Ci be the curves b l , ••• ,bk in our preferred basis. Then as before, there exists some W = £1;+1 ••• aj in the kernel of f. By rnaximality, the curve W must intersect one of our Ci, say CI. As W and CI intersect in a single point, they have a regular neighborhood which is a punctured torus whose fundamental group is mapped trivially under f. Thus f is a stabilization. 0 To complete our understanding of stabilization, we characterize the stable equivalence classes of epimorphisms. Two epimorphisms a: 11"1 (Eg) -+ Q and fJ: 11"1 (Eh) -+ Q are called stably equivalent if they have a common stabilization. In particnlar, Ca = Cp in H2(Q). The following theorem of Livingston [LivJ shows that this homological condition is sufficient as well as necessary.

6.18 Theorem ([Liv]) Let Q be a finite group, and consider two epimorphisms a: 11"1 (Eg) -+ Q and fJ: 11"1 (Eh) -+ Q. Then a and fJ are stably equivalent ifand only ifca = cp in H2 (Q, Z). Since this result is crucial to our characterization of .ll.g/.M.g and is quick to prove, we include a complete proof, following [LivJ. Unfortunately, the proof gives no control over the amount of stabilization required. One example where stabilization is needed is PSLzlF 13 where there are mnltiple orbits of .11.2 which correspond to 0 in H2 (see Example 6.4). For some meta-cyclic groups, Edmonds showed that no stabilization is required [EdmJ. Zimmermann [ZimJ gave a quite different, purely algebraic, proof of this theorem wbich might be useful for an analysis of the degree

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of stabilization required for particular Q. We now give the proof of the theorem.

Proof Suppose a and f3 are as above, with Ca = cll in H2(Q, Z). Now bordism is the same as homology in dimension 2, so, in particular, the two induced maps I:g ~ BQ and I:h ~ BQ can be extended over some cobordism between I:g and I:h [Con,CFj. Thus there exists a3-manifold Mwith two boundary components IhM = I:g and /hM = I:h, and an epimorphism f: :n: I (M) ~ Q which restricts to :n: I (a j M) as a and f3 respectively. Take a relative handle decomposition of M with all the 2-handles added after the I-handles; that is M is (aiM) x I, plus some I-handles, plus some 2-handles, plus (/hM) x I. Let MJ be (aiM) x I and the I-handles, and let M2 be the 2-handles and (a2M) x I. Thus M is the union of MI and M2 along a surface I:. To conclude, we will show that the restriction of f to :n: I (IJ) is a stabilization of both a and f3. First consider a and MI. Because a is sUIjective, we can slide the attaching maps of the I-handles of MI around so that all their cores map trivially under f. This shows (I:, f) is a stabilization of a. If we flip the handles over, the same reasoning applies to M2 and f3. Thus a and f3 have a common 0 stabilization. 6.19 Characterization of .A g/ .M.gfor large g

Recall from Sect. 6.12 that we have a natural map .Ag/.M.g Out(Q). We will now show:

~

H2(Q)/

6.20 Theorem Let Q be a finite group. For all large g, the map

.Ag/.M.g ~ H2(Q, Z)/Out(Q) is a bijection. Proof First, by Corollary 6.17 the size of .Ag/.M.g is constant for large g, and the stabilization maps .Ag/.M.g ~ .Ag+I!.M.g+1 are bijections. Moreover, these stabilization maps are compatible with the maps to H2 (Q). Theorem 6.18 implies that any two elements of .A g which represent the same class in H2(Q)/ Out(Q) become the same in some .Ag+h/ .M.g+h; thus .Ag/.M.g ~ H2(Q)/ Out(Q) is injective for all large g. Lemma 6.13 shows that every class is realized for large enough g, so the map .Ag/.M.g ~ H2(Q)/Out(Q) is sUIjective as well (alternatively, this follows because bordism is the same as homology in dimension 2). Thus .Ag/.M.g ~ H2(Q)/ Out(Q) is a bijection for all large g, as claimed. 0 Unfortunately, this theorem and its constituent parts do not seem to be enough to show the existence of a limiting distribution of the number of quotients as the genus g goes to infinity. However, it is easy to show that the number of expected quotients does converge: 284

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6.21 Theorem Let Q be afinite group. Let E(Q, g) be the expected number of covers with covering group Q of a 3-manifold associated to a random Heegaard spliUing of genus g. Then as the genus g goes to infinity

E(Q

) --+ IQ'II H2(Q,Z)1 ,g IAut(Q) I •

Proof Let g be large enough so that .Io g/.M g --+ H 2(Q)/ Out(Q) is a bijection. Let A~ be those elements f of .log which have cf = 0 in H2(Q)/ Out(Q); by choice of g, .Io~ is a single orbit of .Mg. By Lemmas 6.10 and 6.13, we have

As usual, let 8g be those epimorphisms which extend over the handlebody. By Lemma 6.11, we have 18g 1 ~ IQlg/ Aut(Q). Since 8g C .Io~ and the action of .Mg on .Io~ is transitive, as in Example 6.2 we have that E(Q

2 ) = 18g 1 ~ IQ'II H2(Q)1 ,g l.Iogl IAut(Q) I '

o

as desired. 6.22 Characterization of .Io~/ .Mg for large g

In Sect. 7, we will need a slight variant of Theorem 6.20 and its precursors in Sect. 6.15. For a finite group Q and genus g, define .Io~ to be the set of epimorphisms 11"1 (.E) onto Q; unlike .log we do not mod out by the action of Aut(Q). As with .log, elements in .Io~ have associated homology classes, but now these are well-defined in H2(Q, Z). In this context, we have the following analog of Theorem 6.20:

6.23 Theorem Let Q beafinite group. For all large g, the map .Io~/.Mg --+ H2(Q, Z) is a bijection. The proof of this is the same as for Theorem 6.20, after modifying Corollary 6.17 and Lemma 6.13 by replacing .log by.lo~ and H2(Q)/ Out(Q) by H2 (Q). In case you are wondering why we do not work with .Io~ throughout, the reason is that the action of .Mg on .Io~ can not be highly transitive in general, e.g. the full alternating group of each component; this is because 18; n (cfJ • 8;) I must always be an integer multiple of Aut( Q). WILLIAM P. THURSTON

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6.24 Orbit of 8 g under .M.g In the case ofPS~lF 13 (Example 6.4) we saw that the action of .M.2 on 82 was not transitive, and there were two distinct orbits. However, one consequence of Theorem 6.20 is that for any Q the action of .M.g on 8g is always transitive when g is large enough. In fact, something more is true: the action of the stabilizer of 8 g is transitive for large g. Moreover, when Q is a non-abelian simple group, the image of the stabilizer of 8g in Sym(8g ) contains Alt(8g ). Both of these facts follow immediately from the well-developed theory of the action of Out(Fg) on epimorphisms Fg --+ Q. We outline the needed results below, following the survey [Pak], to which we refer the reader for further results along these lines. The material in this subsection is not strictly necessary for the rest of this paper, but has two merits. The first is that it gives explicit bounds, in terms of Q, of when these phenomena occur. The second is that the strong result in the case of simple groups allows us to avoid the use of the Classification of Simple Groups for some of the results in Sect. 7. First, note that the stabilizer of 8g in .M.g contains the mapping class group .M.(Hg) of the handlebody Hg. Let Fg = ]1'\ (Hg) be the free group on ggenerators. The mapping class group.M.(Hg) acts on 8g = {f: Fg ---» Q}/ Aut(Q) via a homomorphism .M.(Hg) --+ Out(Fg); based on the generators of Out(Fg) that we list below, it is easy to see that .M.(Hg) --+ Out(Fg) is surjective. Thus, understanding the action of .M.(Hg) is equivalent to understanding the action of Out(Fg) on 8g . In the language of [pak, §2.4], elements of 8g are called T-systems, and 8g is denoted by Eg(Q). Before continuing, recall that Out( Fg) is generated by three kinds of elements: replacing a generator by its product with another generator (on either the left or the right), permuting two generators, and inverting a generator. If we think of elements of 8g as g-tuples of elements of Q (up to Aut(Q», then these generators carry over to natural operations on such tuples. Now let us state the main fact of this subsection:

6.25 Proposition Let Q be a finite group. Then for all large g, the action

ofOut(Fg) on 8g is transitive. Hence the action of.M.g on 8g C .;\,g is also transitive for large g. How big does g need to be for the conclusion to hold? For simple groups, a conjecture of Wiegold posits that g ~ 3 suffices, and this is known for some classes of such groups [pak, Thm. 2.5.6]. In general, the bound in the proof we will give is proportional to log 1Q I. Now, let us prove the proposition, following Prop. 2.2.2 of [Pak], which in turn follows [DSC].

Proof Let d be the mjnjmal number of generators of Q. Let

d

be the maximal size of a minimal generating set for Q; that is, a generating set for which no proper subset generates. We will show that the action of Out(Fg) is transitive provided thatg ~ d+d, which we now assume. Letqlo'" ,qd be aminimalgeneratingsetforQ.Consider fo = (q\, ... ,qd, 1, ... ,1)in8g • 286

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Let (rt, ... rg) be any element in 8g, which we will move to fo by elements in Out(Fg). Since there is some subset of the rj of size d which generate Q, permute the rj so that rd+t, ... , rg generates Q. Now by using elements of Out(Fg) which multiply (rt, ... , rd) by elements of (rd+t, .. .Tg), we can turn (rt, ... , rg) into (qt, ... , qd, rd+t, ... , rg). Repeating !his using the first d places instead lets us get to fo by further elements of Out(Q). Thus 0 the action of Out(Fg) on 8g is transitive. Before moving on, let us note that d is no more than the length of a properly increasing sequence of subgroups

{I} = Qo < Qt < ... < Qk = Q. Since the index of one Q j in the next is at least 2, we get that d ~ logz IQ I. Thus in Proposition 6.25, we can take g ?: 210g z IQ I as d + d ~ 2d. Again, see [Pak] for sharper results about particular groups. Since we are particularly interested in non-abelian simple groups, the following theorem is of interest. 6.26 Theorem Let Q be a non-abelian finite simple group and g ?: 3. Then there is some orbit 8' of 8 g where Out(Fg) acts on 8' as either the full symmetric or alternating group. In particular, when Out(Fg) acts transitively, the image of the homomorphism Out(Fg) -+ Sym(8g ) is either the whole thing or has index two. Theorem 6.26 was proved by Gilman [Gil] for g ?: 4 and improved to g = 3 by Evans [Eva]. The statements they give seem to put restrictions on the simple group, but their conditions actually hold for all finite simple groups; see the discussion in [Pak, ThIn. 2.4.3]. The proof of Theorem 6.26 is fairly short and elementary but we will not reproduce it here; see [Gil] for more. We should mention, though, that as we stated it the result is more complex because it uses that every finite simple group is generated by two elements, one of which has order two; as such it uses part of the Classification of Finite Simple Groups. However, for the way we will use Theorem 6.26, we only need that the conclusion holds for all large g, and !his is exactly what Gilman's elementary argument shows.

7 Covers where the quotient is simple This section is devoted to the proof of the following theorem, which gives a complete asymptotic picture in the case of a non-abelian simple group 7.1 Theorem Let Q be a finite non-abelian simple group. Then as the genus g goes to infinity,

p(Q, g) -+ 1 - e-P,

where

JL = IHz(Q, Z)I/IOut(Q)I.

Moreover, the limiting distribution on the number of Q-covers converges to the Poisson distribution with mean JL. WILLIAM P. THURSTON

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As an example, if Q = PSLzlF p where p is an odd prime then /.t = 1. Thus for large genus the probability of a PSLzlFp cover is about 1 - e- 1 ~ 0.6321. Unfortunately, the proof does not give information about the rate of convergence, and we will discuss this issue in detail later. It is interesting to compare this result with the corresponding result for a general random group coming from a balanced presentation. Theorem 3.10 shows that as the number of generators goes to infinity, the limiting distribution is also Poisson. However, the mean is much smaller, namely 1/IAut(Q)1 ::::: 1/IQI. In particular, the mean goes to 0 as the size of the groups increases. In contrast, we just saw that for 3-manifold groups the limiting mean of the number of PSLzlFp covers is independent of the size of the group. To prove the theorem, the thing that we need beyond Sect. 6 is a better understanding of the action of the mapping class group .M.g on the set of epimorphisms .A.g (notion is as in Sect. 6). In particular, we will show that for a fixed non-abelian simple group Q the action is eventually by the full alternating group of each component (Theorem 7.4). Before stating the first lemma, we need some definitions. For a finite group Q and an element C E Hz(Q)/Out(Q) set .A.~ = {f E .A. g I cf = c}.

Recall that the action of a group on a set Q is k-transitive if the action on k-tuples of distinct elements of Q is transitive. The following lemma is the heart of the proof of Theorem 7.1: 7.2 Lemma Let Q be a non-abelian finite simple group, with C E Hz(Q)/ Out(Q). Let k be a positive integer. Then for all large g, the action of .M.g on .A.~ is k-transitive. Proof Consider the direct product Qk of k copies of Q. We will use .A.g(Q) and .A.g(Qk) to denote epimorphisms to Q and Qk respectively. As in Sect. 6.19, we will also consider the set .A.~(Qk) of epimorphisms to Qk where we do not mod out by Aut(Qk). By Theorem 6.23, for all large g, the map .A.~ (Qk) / .M.g --+ Hz (Qk) is a bijection; henceforth we will assume this holds. Now consider k-tuples ([fd, ... , [fkl) and ([hd, ... , [hkl) of distinct elements of .A.~(Q). Let e be a lift of C to Hz(Q, Z), and choose our representatives fi' hi: 11'1 (17g) --+ Q so that all the Cfi and Chi are equal toe in Hz(Q). Now consider the induced product maps f, h: 11'1 (17g) --+ Qk. Since Q is simple and the Ii represent distinct classes in .A. g, the homomorphism f is also surjective (Lemma 3.7). Similarly h is also an epimorphism, and so both f and h are elements of .A.~(Q). 288

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As Q is non-abelian, HI(Q, Z) = 0, and so H2(Qk) is the direct sum (H2(Q)l. Thus cf = Ch = 7!' in H2(Qk). By our choice of genus, there must be a t/J E .Mg such that h = t/J. f. Then t/J carries ([ftl, ... , [AD to ([htJ, ... , [hkD and so the action of .Mg is k-transitive. 0 We will also need the following variant of the preceding leruma, whose proof is essentially identical.

7.3 Lemma Consider a finite set Qi of non-abelian finite simple groups, andjix Ci E H 2(Qi)1 Out(Qi). Suppose there are no isomorphisms from Qi to Qj which take Ci to Cj. Then the action of.Mg on .A.~' (QI) x· .. x .A.~n(Qn) is transitive.

Now we use Lemma 7.2 to prove:

7.4 Theorem Let Q be a non-abelian finite simple group, with ajixed class C in H2(Q)1 Out(Q). Thenfor all large g, the action of.M g on .A.~ is by the full alternating group Alt(.A.~). Moreover, the actions on different .A.~ are independent, in the sense that the map .Mg --+ Alt(.A.~) is surjective.

n

You can view this theorem as saying that the action of .Mg on .A. g is nearly as mixing as possible, at least when the genus is large. As such, it is analogous to Goldman's theorem that the action of .Mg on the SU(2)character variety is ergodic for any genus ~ 2 [GoIJ. While the proof here does not give an explicit bound on when this mixing behavior occurs, we suspect that it typically occurs as soon as g ~ 3. Proof One consequence of the Classification of Simple Groups is that a group which acts 6-transitively on a finite set il must contain Alt(il) (see e.g. [DM, Thm. 7.3AD. By Theorem 7.2, the action of .Mg on .A.~ is 6-transitive for all large g; hence the action contains Alt(.A.~). Since .Mg is a perfect group for genus at least 2, the image of .Mg --+ Sym(.A.~) must be Alt(.A.~) and not Sym(.A.~), proving the first part of the theorem. Alt(.A.~) is not smjective then there For the independence, if .Mg --+ are distinct C and d such that.Mg --+ Alt.A.~ x Alt.A.: is not smjective. Then there is a bijection a: .A.~ --+ .A.: which is compatible with the.Mg action on both of these sets; in particular, the action of .Mg on pairs (f, h) E .A.~ x .A.: is not transitive. But that contradicts Lemma 7.3, finishing the proof. 0

n

You might wonder if we really need the Classification of Simple Groups to prove this theorem. In the case of the orbit .A.~, which is what we need for Theorem 7.1, we can replace the Classification by some much less difficult results. First choose g large enough so that the action on .A.~ is 2-transitive. Let G be the image of .Mg --+ Sym(.A.~). Gilman showed (see Theorem 6.26) that the action of the stabilizer of 8 g in G is all of Alt(8g ) for large g. Thus G has subgroups K O. Thenfor all sufficiently large g, the probability that the 3-manifold obtainedfrom a random genus-g Heegaard splitting has an An-cover for some n 2: 5 is at least 1 - E. Moreover; the same is true if we require that there be at least some fixed number k of such covers. What about similar questions where we consider a collection {Qi }~o of non-abelian simple groups? For this we will show:

7.8 Theorem Let {Qi}~ be a sequence of distinct non-abelian finite simple groups; set ILi = IHz(Qi)I/IOut(Qi)l. Suppose further that I: ILi = 00. Fix E > O. Thenfor all sufficiently large g, the probability that the 3-manifold obtained from a random genus-g Heegaard splitting has an Q;-cover for some i is at least 1 - E. Moreover; the same is true ifwe require that there be at least some fixed number k of such covers. Since IHz(An) II IOut(An) I = 1 for all large n, this theorem immediately implies Theorem 7.7. Another class where it applies is Qi = PSLzlFPi where Pi is prime; an example of where it does not is Qi = PSLzlF pi2 where p is a fixed prime. In fact, it is not hard to work out exactly which sequences {Q i} satisfy the hypothesis from the Classification of Finite Simple Groups (see [CCNPW, §3]). In particular, any sequence {Qi} must either contain an infinite number of An or an infinite number of Chevalley groups. In WILLIAM P. THURSTON

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the former case, Theorem 7.8 applies. For the Chevalley case, these are groups of Lie type, and let qi = pf be the field of definition of Qi. Then f.Li = cd fi where Ci is a non-zero rational number which is universally bounded above and below (see [CCNPW, §31, especially Table 5). Thus Theorem 7.8 applies in this case if and only if L 1/Ii diverges. Thus Theorem 7.8 applies to PSL;zFpi but not PSL;zFpi2. Now we will give the proof of the theorem. Proof of Theorem 7.8 The key lemma is the following, which says that covers with different Qi are essentially independent. 7.9 Lemma Let QI, ... , Qn be non-abelianfinite simple groups. Thenfor all large g, the map .Mg -+ Alt(.A,~(Qi)) is surjective.

n

First,let us see why this lemma implies the theorem. For a fixed genus g, let Xi denote the random variable corresponding to the number of Qi covers. Fix a positive integer n. Choose g large enough so that the lemma holds for Xl, ... ,Xn, and so that the expectations E(X i ) are very nearly f.Li for 1 :'0 i :'0 n. From the lemma, the Xi are independent random variables which are nearly Poisson with mean f.Li. Thus their sum XI + ... + Xn is nearly Poisson with mean Mn = f.LI + ... + f.Ln. In particular, the probability of having a cover with group one of QI, ... , Qn is about 1 - e- M•• Since L f.Li diverges, by increasing n we can make this probability as close to 1 as we like (of course, g may well have to increase as we change n). Similarly, we can require some fixed number k of Qi covers. This proves the theorem modulo the lemma. As for the lemma, by Theorem 7.4 gives a g so that.Mg -+ Alt(.A,~(Qi)) is surjective for i :'0 n. If the product .Mg -+ Alt(.A,~(Qi)) is not surjective, then because the factors are simple, there must be a pair i, j such that 1.A,~(Qi)1 = 1.A,~(Qj)1 and a bijection .A,~(Qi) -+ .A,~(Qj) which is compatible with the .Mg action on both of these sets. In particular, the action of .Mg on pairs (j, h) E .A,~(Qi) X .A,~(Qj) is not transitive. But that would contradict Lemma 7.3, thus proving Lemma 7.9 and hence the theorem. 0

n

8 Homology of random Heegaard splittings In this section, we work out the distribution of the homology of a 3-manifold coming from a random Heegaard splitting of genus g. First, let us set up the point of view we will take in this section. As always, let Hg be a genus-g handlebody, Eg its boundary, and.Mg the mapping class group of E g . Given tjJ E .Mg let Mtf> be the resulting 3-manifold. Let J be the kernel of the map HI (E g ; Z) -+ HI (Hg ; Z). Then HI (Mtf>; Z) is the quotient of HI (E g ; Z) by the subgroup (J, tjJ;I(J). Note that counting algebraic intersections between two cycles gives HI (8Hg ; Z) a natural symplectic form. With respect to this symplectic structure J is a Lagrangian subspace. The natural map .Mg -+ SP2gZ coming from the action of mapping classes 292

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on HI (17g; Z) ~ Z2g is swjective. Thus understanding the distribution of the homology of M", boils down to considering the distribution of Lagrangians under the action of SP2gZ. 8.1 Symplectic groups over IFp

As with the case of random finitely presented groups, we will work one prime at time. We will work in the following framework, where p is a fixed prime (or prime power). Let J be a vector space over IF p of dimension g, and let K be its dual. A nice way of thinking about the symplectic vector space over IF p of dimension 2g is to consider V = J Ell K with symplectic form given by (h, k l ), (h. k2)) = kl(h) - k2(h). Notice that both J and K are Lagrangian subspaces of V. Now set G = SpY = Sp2gIFp. Identify V with HI(17g; IFp) so that the kernel of the map to HI (Hg ; IF p) is the Lagrangian J. The action of .M.g on homology gives a swjection .M.g -+ G. If tjJ E .M.g is the result of a long random walk then its image in G is nearly uniformly distributed. Now by the analog of the Gram-Schmidt process for a symplectic vector space, it is easy to show that the action of G on the set of Lagrangians is transitive. Thus tjJ;;I(J) is nearly uniformly distributed among all Lagrangians. Hence the asymptotic probability that dim(HI (M",; IFp)) = k is simply the ratio of the number of Lagrangians L in V such that dim(J n L) = k to the total number of Lagrangians.

8.2 Counting Lagrangians First, let us count the total number of Lagrangians in V; we will do this by computing the size of the stabilizer S ~ G of the fixed Lagrangian J. Notice we have a natoral homomorphism S -+ GL( J). This map is swjective, for if A E GL(J) then recalling that K = J*, we have that the map A Ell (A*)-I of V respects the symplectic form and hence is in S. An element s of the kernel of S -+ GL(J) is determined by its restriction to K. Write the restriction as BI Ell B2 where BI maps K into J and B2 maps K into K. For each j E J andkE Kwehave(k,j) = (s(k),s(J)) = (BI(k) + B 2(k) , j) = (B2(k),j) and hence B2 = Id on K. Moreover, the requirement that s preserve the symplectic form is then equivalent to (BI (kd, k2) = (BI (k2) , k l ) for each kl' k2 E K. Such a BI is called symmetric. If we write BI with respect to a pair of dual basis for J and K then this is equivalent to the resulting matrix being symmetric. Thus g

lSI = I{g x gsymmetricmatrices }I·IGL(J)I = pK(g+I)/2n(pK _ pH) k=1

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To complete the count of Lagrangians, we also need to know the order of G itself. Fix a basis for V consisting of a basis {ei} for J and the dual basis {e i } for K. To build an element of G, there are g - 1 choices for the image v of ej, and as the image w of e l must satisfy (w, v) = 1, there are p2g-1 choices for w. The other basis vectors must be sent into span(v, w).L. Thus inductively one has

r

g

IGI =

g

n(p2k _1)p2k-1 = pK2 n(p2k -1). k=1

k=1

Thus, the number of Lagrangians in V is

IGI

lSI =

p2k - 1 g k -1 = n(p + 1).

n pk g

k=1

k=1

8.3 Counting transverse Lagrangians

Now we will calculate the probability that HI (M¢; IFp) = O. To do this, we need to count the number of Lagrangians L which are transverse to our base Lagrangian J. Given such an L, projection of L onto K is swjective; thus we can view it as the graph of a map B: K --+ J. The requirement that the graph L is Lagrangian is equivalent to (k l

+ B(kj} , k2 + B(k2»

= 0 {==} (B(k l ), k 2) = (B(k2), k l )

for all kl' k2

E

K.

That is, B is symmetric in the same sense as before. Thus the number of Lagrangians transverse to J is equal to the number of symmetric g x g matrices over IF P' that is pK(g+I)/2. Combining this with our count of Lagrangians gives the following: 8.4 Theorem Fix a Heegaard genus g. Then the asymptotic probability that g

1

n l+p-k k=1

as we let the length of the random walk generating r/J E .M.g go to infinity.

Notice that for a fixed genus g the larger p is the closer this probability is to 1. Thus the asymptotic probability that HI (M¢; IQI) vanishes is 1, as you would expect. On the other hand, if we consider a pair of primes p and q then Lemma 3.7 implies that the induced map

.M.g --+ SP2gIFp x SP2gIFq 294

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is swjective. Thus homology with IF p versus IFq coefficients are asymptotically independent variables. Moreover, the sum of the probabilities that HI (M; IFp) ¥- 0 diverges (albeit slowly, like the harmonic series). Thus the probability that some HI (M; IFp) ¥- 0, where p is larger than some fixed C, goes to I as the complexity of t/J goes to 00. Hence the expected size of HI (M; Z) goes to infinity as the complexity of t/J increases. Summarizing: 8.5 Corollary Fix a Heegaard genus g. Then with asymptotic probability I the rational homology HI (M; Q) vanishes. However, for a fixed C the probability that IHI (M; Z) I > C goes to 1 as the complexity of t/J -->- 00.

Notice that the second conclusion implies that while a fixed manifold N occurs with many different t/J, as the complexity of t/J goes to infinity the probability that N = M goes to zero. Thus asymptotic probabilities are, in particular, not highly disgnised statements about some finite set of manifolds. It is also interesting to compare Theorem 8.4 to the corresponding results for the group r of random g-generator balanced presentations. In that case, by Proposition 3.16, the probability that HI (r; IFp) = 0 is g

n ( l - p-k). k=1

Comparing term by term shows that this is less than the same probability for oM; thus the homology of the random group is more likely to be non-zero than for the random 3-manifold. Note also that in both cases the probabilities have the same limit (namely 0) as p goes to infinity. Both of these facts are the reverse of what we saw for non-abelian simple groups.

8.6 General distribution To complete the picture of the IF p homology, we need to find the probability that dim(HI (M; IF p» = d for general d. To do this, we start by parameterizing those Lagrangians L where dim(L n J) = d. Let us fix A = L n J. Any Lagrangian L intersecting J in A is contained in A1- = J EEl Ann(A), where Ann(A) is the subset of K = J* which annihilates A. Note that A 1- 1A = (11 A) EElAnn(A) inherits a natural symplecticform, and L projects to a Lagrangian in A1- 1A which is transverse to JI A. Indeed, Lagrangians in A1- which are transverse to JI A exactly parameterize Lagrangians in V which intersect J in A. Thus the latter are parameterized by (g - d) x (g - d) symmetric matrices. Hence the number of Lagrangians intersecting J in a d dimensional subspace is:

I{(g - d)

x (g - d) symmetric matrices} I . I{dim d subspaces of J} I =

n d

p(g-d+I)(g-d)/2

k=1

1

.,g-k+1 y-

-.

pk -1

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If we set Cd to be the probability that HI (M; IF p) has dimension d, then a concise way of summarizing this distribution is

n n cog

Co

=

(1

+ p-k)-I

and

k=1

-Cd -

d

1

-g+k-I

-p ---0-------:-

pk-l

k=1

for 1 ::::. d ::::. g.

8.7 Large genus limit As with the case of simple groups, the distributions of dim HI (M; IFp) have a well-defined limit distribution as the genus g goes to infinity. Namely, it is the probability distribution given by

n+ n(~ 00

Co = _

:d = Co

(1

p-k)-I

and

k=1 d

1)-1

for 1 ::::. d.

k=1

The reader can check that this is really a probability distribution, i.e. that it has unit mass. One way to do this is to use that the finite approximates have unit mass, and then observe that Cd/CO is an increasing function of g.

8.8 p-adic point of view It is possible to work out the p-adic distribution of homology, analogous to Sect. 3.14. However, we will not go into this here. The key technical tool needed is the ability to count non-singular g x g matrices over IF p' which is done in [Car] and [Mac]; see also [Sta, §4].

9 Homology of finite-sheeted covers In this section, we try to determine the probability that a cover has Ih > 0, where the covering group Q is fixed. As before we work in the context of random Heegaard splittings. Our original goal was to find groups where this probability is positive, and so demonstrate that the Virtual Haken Conjecture is true in many cases, perhaps with probability 1. Unfortunately, our results are in the other direction; in particular the main result in this section is:

9.1 Theorem Let Q be a finite abelian group. The probability that the 3-manifold obtained from a random Heegaard splitting of genus 2 has a Q-cover with /31 > 0 is O. 296

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The restriction to genus 2 is almost certainly artificial and simply due to a technical difficulty in the general case that we were unable to overcome. What about when Q is non-abelian? We did some computer experiments, and could not find any groups Q for which Q-covers seemed to have fh > 0 with positive probability. However, these experiments are not completely convincing, even for the groups examined; as you will see in the proof of Theorem 9.1, one needs to consider certain subgroups of the mapping class group of truly staggering index, even for fairly small Q. Thus one can not rule out experimentally some very small probability that a Q-cover has fh > o. We still suspect that the probability of fh > 0 is 0 for all Q, and would be surprised if there was a Q for which the probability of fh > 0 was greater than, say, 10-4 . Let us now contrast these results with our earlier experimental results in [DT21. There, we examined about 11,000 small volume hyperbolic 3-manifolds. Of these, more than 8, 000 had 2-generator fundamental groups, and therefore almost certainly have Heegaard genus 2. About 1.7% of these genus 2 manifolds had a 7£,/2 cover with fh > 0, and at least 1.7% had a 7£,/3 cover with fh > O. Overall, 7.3% of the genus 2 manifolds have an abelian cover with fh > 0 (of the full sample of 11,000 manifolds, this rises to 9.6%). In comparing this with Theorem 9.1, however, it is important to keep in mind that the covering group Q is fixed in the theorem. In particular, we do not know the answer to the following question, even experimentally:

9.2 Question Let M be obtained from a random Heegaard splitting of genus g. What is the probability that the maximal abelian cover of M has fh > O? Compared to the experiments in §5 of [DT21 which dealt with nonabelian simple covers, the contrast becomes much more marked. In [DT2, §51, we found for a fixed such group Q that a very large proportion of the covers (> 50%) had fh > 0; indeed, the expectation for fh grew linearly with 1Q I. However, in our experiments for random genus 2 Heegaard splittings we found that the probability seems to be 0 for the first few group (As, PS~lF7' A6, and PS~lF8), and we expect that this pattern continues for all simple groups.

9.3 Homology and subgroups of the mapping class group The goal of this section is to prove Theorem 9.1. While Theorem 9.1 restricts to genus 2 and abelian covering groups, with the exception of Lemma 9.6 this section will be done without these restrictions. So from now on, we fix a Heegaard genus g ?: 2 and an arbitrary finite group Q. Let H be a handlebody of that genus, and 1J be aH. Let .M. denote the mapping class group of 1J, with a fixed generating set T. We focus on a single epimorphism f: ll"\(1J) --+ Q which extends over H. Proposition 6.1 shows that for r/J the result of a random walk in .M., the probability that f extends to the WILLIAM P. THURSTON

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3-manifold Nt/> converges to a positive number as the length of the walk goes to infinity. We want to show that the probability that the corresponding cover of Nt/> has fh > 0 is 0; as there are only finitely many choices for f this will suffice to prove Theorem 9.1. Recall from the proof of Proposition 6.1 that f extends over Nt/> if and only if cp . f = f 0 cp:;1 extends over H. We will consider each of the possibilities for cp . f individually. We start with the case that cp . f = f in the set .A of all such epimorphisms, up to Aut(Q). This case is a little simpler, and it forms the core for the argument in general. Let K be the kernel of f: 1l'1 (.E) -+ Q. We are focusing on the subgroup

.M.f =

{cp E.M. I cp. f

= fin.A,orequivalentlycp.(K) = K},

which has finite index in .M.. The mapping classes in .M. f are exactly those which lift to the cover 15 -+ 1J corresponding to K, and let .M.f be the group of all sucJ.Llifts, modulo isotopy within this restricted class of hom,.!?omorphisms of 1J . Regarding Q as the group of covering translations of 1J, we have the exact sequence 1 -+ Q -+

.Mf

-+ .M.f -+ 1.

H,Sre, we are using that g ::: 2, as in the torus case Q need not inject into .M.f. Note that Q is not central unless .M. f acts trivially on 1l'1 (1J)1 K. If cp E .M.f then the Q-cover corresponding to f will be denoted Nt/>. If 4J E .Mf is a lift of cp, then Nt/> = Ni' The covering group Q acts on the homology HI (Nt/>; 1Ql), and we can consider the decomposition of this action into irreducible (over ~) representations. This is the same decomposition as considering L = HI (Nt/>; Z)/(torsion), which can be regarded as a lattice in HI (Nt/>; 1Ql), and looking at the sublattices on which Q acts irreducibly. (The direct sum of these sublattices has finite index in L, but is not necessarily all of L.) We will examine L by looking at the submodules individually. Equivalently, for each irreducible representation p: Q -+ GL(Vi) where Vi is a lattice, we consider the homology HI (Nt/>, Vi) with coefficients twisted by p. To understand the homology of Nt/>, we first consider the action of the covering group Q on V = H I (15; Z). Again, the action decomposes into a sum of (rationally) irreducible representations. We group these representations by isomorphism type, and so express a finite index sublattice of V as Vo Ell VI Ell ... Ell V., where the action on Vi is a direct sum of copies of aJlingle representation, which differs for distinct indices i. The action of .M. f on V preserves the decomposition into Vi because Q is a normal subgroup of .Mf . Now consider the cover ii of the handlebody H corresponding to f, and set W to be the kernel of V -+ HI(ii; Z). Then we have HI(Nt/» = -I - VI (W, CP. (W». As the action of Q on 1J extends over H, the kernel W is 298

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a Q-invariant subset of V. Hence it also decomposes into WoE9 WI E9 ... E9 Wn where Wi = W n V;. 9.4 Lemma Each Wi is half-dimensional in Vi. Proof Basically, we can compute both dimensions explicitly using the Euler characteristic. Consider an irreducible rational Q-module T ~ qt

corresponding to some V;. Suppose X is a finite CW complex with a epimorphism 11'1 (X) ---+ Q. Then the Euler characteristic of the twisted homology H.(X, T) is just dim(T)x(X), as is clear from the chain complex used to compute the former. Moreover, HO(X, T) and Ho(X, T) both vanish because T is irreducible; in general, ~(X, T) is the submodule of T consisting of invariant vectors, and Ho(X, T) is the module of co-invariants (= V/(qv - v I q E Q, vET» [Bro, §ill.1J. Consider the homology H.(H, T) of the handlebody H. As H is homotopy equivalent to a bouquet of g circles, we have that the only non-zero homology is in dimension 1. Thus dimHI(H, T) = -X(H.(H, T» = (g -l)dimT

and so HI (ii, 1Ql) contains exactly (g - 1) copies of T. Next, let us consider the homology H.(E, T). Poincare duality implies ~

H2(E,T) HO(E, T) = 0 [Bro, Vill.lOJ. Thus again we have dimHI(E, T)

= -X(H.(E, T» = (2g -

2) dim T.

and so V; consists of (2g - 2) copies of T. Counting dimensions now shows that Wi must consist of (g - 1) copies of T, proving the lemma. 0 We now focus attention on one summand V;, looking at the homology piece by piece. As we only care about rational homology, set Ui = (Vi/(Wi';;~ (Wi»)) ® 1Ql.

Consider the homomorphi~m p: .M.f ---+ GL(V;) induced by the projection V ---+ V;. We want to show that Ui almost surely vanishes, or equivalently that the image of p almost surely takes Wi to a complementary subspace. More precisely we need: 9.5 Claim Given E > 0 there exists a Co such that the following holds. If t/J is the result of a random walk in .M oflength C ~ Co then the probability

P{t/J

E

.Mf

and

Ui

¥- O}
E .M I then Uf.O implies that if> is in one of IB I cosets of r in .MI' As always, if if> is the result of a long random walk in .M then the location of if> in the finite coset space .MI r is nearly uniform. Thus for long walks

r

r

r

P{if> E.MI

and


is easy to describe: the preimage of f32 consists of IQ I curves, one on each T2 spoke, while the preimage of PI is a single curve running once round the TI hub; 300

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the lift 4) is simply the Dehn twisLalong this disjoint collection of curves. The important thing for us is that I/J takes the Lagrangian kernel of H1(15; Z) --+ H1(H; Z)

to something transverse with itself; moreover, the images of this Lagrangian under proper powers are all distinct, mutually trans,!erse subspaces. Now consider the particular summand Vi of H1(IJ; Z) in the case at hand. For 4), we know that the orbit of the Lagrangian Wi inside Vi is infinite. Therefore, by choosing p large we can make P as large as we want. Turning now to understanding B, by the Euler characteristic calculation of ~emm.!! 9.4, Wi consists o£a single irreducible Q-module. Therefore, for 1{; EMf the intersection 1{;(Wi ) n Wi is either {OJ or all of Wi since this intersection is Q-invariant. Thus B consists solely of Wi and IB I = 1. Since we can make IP I as large as we want, the Lemma follows. 0

;p

This completes the proof of Theorem 9.1 in the case that I/J. f = f. We now tum to the general case, wherel/J· f = g for an arbitrary g: 1l"1(lJ) --+ Q extending over H. Consider

.M.f,g =

{I/J E .M. II/J· f

= g in .A" or equivalently I/J.(K) = ker g},

which is no longer a subgroup but is a coset of .M.f. Fix rPo such that rPo . g = f· Given I/J E .M.f,g we have rPo 0 I/J E .M.f. Schematically, we have: 15 f

1

~ --'-'---'-+

15 f

1

io- I --'-'---'-+

15 g

1

where we have distinguished the Q-covers of 1J corresponding to

f

and

g by SUbscripts. Let Vf = H 1(15f ; Z) and let Wf be the kernel of Vf --+

H1 (Hf; Z); similarly, let Vg and Wg be the corresponding lattices for 15 f. Then moving from the right hand column of the diagram to the middle we get H 1(N4)) = Vg/ (Wg, 4).(Wf») ~ Vt/ (rPo.(Wg),

(4)0 0 4),(Wf»)·

Now the subspace rPo.(Wg) is Q-invariant, so as before we can break this question into separate questions for each summand Vf,i of H1 (15f; Z) corresponding to an irreducible representation. Again, we are interested in the orbit P of the Lagrangian Wf,i C Vf,i under the elements of .M.f. The only difference is that the set B should now be taken to be those X E P which are not transverse to rPo.(Wg,i) (rather than to Wf,i)' If we now assume that the genus is 2, then, as in the proof of Lemma 9.6 the Q actions on rPo.(Wg,i) and elements of P are irreducible. Thus B has either 0 or 1 element depending on whether rPo.(Wg,i) E P. The general case can now be completed in exactly the same way as before. This finishes the proof of Theorem 9.1. WILLIAM P. THURSTON

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9.7 Possible generalizations It would be nice to remove the restrictions that Q is abelian and that the genus is 2 from the hypotheses of Theorem 9.1. These two conditions are actually quite separate from the point of view of the proof, so we discuss them each in turn. First, the fact that Q is abelian (or really, cyclic) ensured that the cover 15 ~ IJ was concrete enough to exhibit an element rjJ E .M.f showing that each P is infinite. (Note that the rjJ given in Lemma 9.6 easily genera1izes to any genus.) Such rjJ could probably also be found for other groups, especially small cases like 83 • More ambitiously, if one wanted to do a whole class of simple groups, e.g. alternating groups, one would be badly hampered if one could not remove the genus restriction - after all, Sect. 7 only applies in that case. at only one point - to show IB I = 1 The genus 2 hypothesis is used and thus allowing us to make IB II IP I small simply by knowing that P is infinite. For higher genus, we would need a more detailed picture of the image

.M.f

~

Sp(V)

where V = HI (15; Z) in order to compare the relative sizes of B and P. However, a more abstract point of view might also work to circumvent this issue. Consider a finitely generated subgroup r of SP2n (Z), which we think of as sitting inside G = SP2n (JR). Fix a standard integral Lagrangian W in JR2n and set D = {g E G

I g(W) n W

j'qO}} ,

which is a proper real-algebraic subvariety (but not subgroup) of G. Now fix generators of r, and consider the probability that a random walk of length N in r lies in D. It seems very reasonable that, as long as r does not have a finite-index subgroup which is contained in D, then this probability goes toOas N ~ 00. Indeed one can prove this with some additional hypothesis on the Zariski closure of What one needs is the following. Consider the mod p reduction OFp) of which has a natural structure as an algebraic variety over IFpo By hypothesis, DOF p) is a proper subvariety. It follows that

r

r

r.

r,

as each has roughly as many points as the projective space over IF p of the appropriate dimension (this is a weak form of the Wei! Conjecture). Thus if one has that the mod p reduction map

(9.8) 302

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is surjective for arbitrarily large p, this would prove the desired claim. It is known that (9.8) holds when r is a lattice in r (IFp), or if r (IF p) is simple and simply connected. (The latter is the celebrated Nori-Weisfeiler Strong Approximation Theorem [Wei], see also [LS, §Windows] for a discussion.) It is unclear if either of these hypotheses hold in our setting. References [AS]

Alon, N., Spencer, J.R.: The Probabilistic Method, 2nd edo.. WtleyInterscience Series in Discrete Mathematics and Optimization. New York: Wtley-Interscience 2000 [BL] Belolipetsky, M., Lubotzky, A.: Finite groups and hyperbolic manifolds. Invent. Math. 162, 459-472 (2005). arXiv:math.GR/0406607 [Bol] Bollobas, B.: Random Graphs. London: Academic Press 1985 [Bm] Brown, KS.: Cohomology of Groups. Graduate Texts in Mathematics, vol. 87. New York: Springer 1982 [Car] Carlitz, L.: Representations by quadratic forms in a finite field. Duke Math. J. 21, 123-137 (1954) [CF] Conner, P.E., Floyd, E.E.: Differentiable Periodic Maps. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., Band 33. Berlin: Springer 1964 [Con] Conner, P.E.: Differentiable Periodic Maps, 2nd edo. Leet. Notes Math., vol. 738. Berlin: Springer 1979 [CCNPW] Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Eynsham: Oxford University Press 1985 [DSC] Diaconis, P., Saloff-Coste, L.: Walks on generating sets of groups. Invent. Math. 134,251-299 (1998) [DPS] Diao, Y., Pippenger, N., Sumners, D.W.: Onrandomknots. In: Random Knotting and Linking (Vancouver, BC, 1993), of Ser. Knots Everything, vol. 7, pp. 187197. River Edge, NJ: World Sci. Publishing 1994 [DdSMS] Dixon, J.D., du Sautoy, M.P.F., Mann, A., Segal, D.: Analytic pro-p Groups, 2nd edo. Cambridge Studies in Advanced Mathematics, vol. 61. Cambridge: Cambridge University Press 1999 Dixon, J.D., Mortimer, B.: Permutation Groups. Graduate Texts in Mathematics, [OM] vol. 163. New York: Springer 1996 [DTI] Dunfield, N.M., Thurston, D.P.: A random tunnel number one 3-manifold does not fiber over the circle. Preprint, 2005. arXiv:math.GT/0510129 [DT2] Dunfield, N .M., Thurston, W.P.: The virtual haken conjecture: experiments and examples. Geom. Topol. 7, 399-441 (2003). arXiv:math.GT/0209214 [Edm] Edmonds, A.L.: Surface symmetry. n. Michigan Math. J. 30, 143-154 (1983) [Eva] Evans, M.J.: T-systems of certain finite simple groups. Math. Proc. Cambro Philos. Soc. 113, 9-22 (1993) [Eve] Everitt, B.: Alternating quotients of Fuchsian groups. J. Algebra 223, 457-476 (2000) [Fel] Feller, W.: An Introduction to Probability Theory and its Applications. Vol. I, 3rd edn. New York: John Wiley & Sons Inc. 1968 [Gil] Gilman, R.: Finite quotients of the automorphism group of a free group. Can. J.Math.29,541-551 (1977) [GoI] Goldman, W.M. Ergodic theory on moduli spaces. Ann. Math. 146,475--507 (1997) Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group [Grol] Theory, vol. 2 (Sussex, 1991), Lond. Math. Soc. Lecture Note Ser., vol. 182, pp. 1-295. Cambridge: Cambridge University Press 1993 WILLIAM P. THURSTON

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520 [Gro2] [Hal] [HZ] [Heml] [Hem2] [HLM] [IZ]

[Jac]

[Jun] [LN] [LSI]

[LS2] [Liv] [Lubl] [Lub2] [LS] [Mac] [Mah] [Nam] [01]

[pak]

[pen] [perl] [per2] [PS] [RZ]

[Sim] 304

Gromov, M.: Random walk in random groups. Geom. Funct. Anal. 13, 73-146 (2003) Hall, P.: The Eulerian functions of a group. Quart. J. Math. 7, 134-151 (1936) Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math. 85, 457-485 (1986) Hempel, J.: The lattice of branched covers over the figure-eight knot. Topology Appl. 34, 183-201 (1990) Hempel, J.: 3-manifolds as viewed from the curve complex. Topology 40, 631657 (2001) Hilden, H.M., Lozano, M.T., Montesinos, J.M.: On knots that are universal. Topology 24, 499-504 (1985) Itzykson, C., Zuber, J.-B.: Matrix integration and combinatorics of modular groups. Commun. Math. Phys. 134, 197-207 (1990) Jackson, D.M.: Counting cycles in permutations by group characters, with an application to a topological problem. Trans. Am. Math. Soc. 299, 785-801 (1987) Jungreis, D.: Gaussian random polygons are globally knotted. J. Knot Theury Ramifications 3, 455-464 (1994) Ledermann, W., Neumann, B.H.: On the order of the automorphism gronp of a finite group, I. Proc. R. Soc. Lond., Ser. A. 233, 494-506 (1956) Liebeck, M.W., Shalev, A.: Fuchsian groups, coverings of Riemann surfaces, subgroup growth, random quotients and random walks. J. Algebra 276, 552-601 (2004) Liebeck, M.W., Shalev, A.: Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159, 317-367 (2005) Livingston, C.: Stabilizing surface symmetries. Mich. Math. J. 32, 249-255 (1985) Lubotzky, A.: Discrete Groups, Expanding Graphs and Invariant Measures. Progress in Mathematics, vol. 125. Basel: Birkhiiuser 1994 Lubotzky, A.: Subgroup growth and congruence subgroups. Invent. Math. 119, 267-295 (1995) Lubotzky, A., Segal, D.: Subgroup Growth. Progress in Mathematics, vol. 212. Basel: Birkhiiuser 2003 MacWilliams, J.: Orthogonal matrices over finite fields. Am. Math. Monthly 76, 152-164 (1969) Maher, J. Random walks on the mapping class group. Preprint 2006. arXiv:math.GT/0604433 Namazi, H.: Heegaard Splittings and Hyperbolic Geometry. PhD thesis, Yale 2005 Ol'shanskiI, A.Y.: Almost every group is hyperbolic. Int. J. Algebra Comput. 2, 1-17 (1992) Pale, I.: What do we know about the product replacement algorithm? In: Groups and Computation, ill (Columbus, OH, 1999). Ohio State Univ. Math. Res. Inst. Publ., vol. 8, pp. 301-347. Berlin: de Gruyter 2001 Penner, R.C.: Perturbarive series and the moduli space of Riemann surfaces. J. Differ. Geom. 27, 35-53 (1988) Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint 2002. arXiv:math.DG/0211159 Perelman, G.: Ricci flow with surgery on three-manifolds, Preprint 2003. arXiv:math.DG/0303109 Poulalhon, D., Schaeffer, G.: Optimal coding and sampling of triangularions. Preprint, 2003 Ribes, L., Zalesskii, P.: Profinite Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 40. Berlin: Springer 2000 Sims, C.c.: Computation with Finitely Presented Groups. Cambridge: Cambridge University Press 1994 COLLECTED WORKS WITH COMMENTARY

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[Wie]

[W1l]

[Wor]

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Stanley, R.P.: Spanning trees and a conjecture of Kontsevich. Ann. Comb. 2, 351-363 (1998). arXiv:math.C0/9806055 Waldhausen, E: The word problem in fundamental groups of sufficiently large irreducible 3-manifolds. Ann. Math. 88, 272-280 (1968) Weisfeiler, B.: Strong approximation for Zariski-dense subgroups of semisimple algebraic groups. Ann. Math. 120,271-315 (1984) White, M.E.: lojectivity radius and fundamental groups of hyperbolic 3-manifolds. Commun. Anal. Geom. 10, 377-395 (2002) Wiegold, I.: The Schur multiplier: an elementary approach. 10: Groups---St. Andrews 1981 (SI. Andrews, 1981), Lond. Math. Soc. Lecture Note Ser., vol. 71, pp. 137-154. Cambridge: Cambridge University Press 1982 Wilson, I.S.: Profinite Groups. Lond. Math. Soc. Monographs, New Series, vol. 19. New York: The Clarendon Press Oxford University Press 1998 Wormald, N.C.: Models of random regular graphs. 10: Surveys in Combinatorics, 1999 (Canterbury). Lond. Math. Soc. Lecture Note Ser., vol. 267, pp. 239-298. Cambridge: Cambridge University Press 1999 Zagier, D.: On the distribution of the number of cycles of elements in symmetric groups. Nieuw Arch. Wiskd. 13, 489-495 (1995) Zimmermann, B.: Surfaces and the second homology of a group. Monatsh. Math. 104, 247-253 (1987) :lui

j

B2 - - B 3

>

I .I I

H 2 - - H3

I P>

Iq

S~M v

v

>

Remark The restriction of j to S~ takes S~ onto S~ and is a p(G(S»-invariant 2-sphere-filling Peano curve. A more precise description of the topology of j will appear in Section 15. At this pofut we simply highlight the structures that we will use to prove the existence of the extension j. In H2 the important structures will be the leaves of the two foliations A') and A; formed in H2 by lifting the pseudo-Anosov foliations A) and ,1.2 on S to the universal

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cover. The critical property of ,1.'1 and A; is summarized in the following theorem which is a consequence of Theorem 10.2. Theorem 4.1 (Neighborhoods at infinity in the ball) Each point x E s~ has arbitrarily small neighborhoods in B2 = H2 U s~ bounded by the closure in B2 of a single leaf of ,1.'1 U A; . In H3 there are two important structures. They arise from viewing H3 as a topological product: H3 = H2 X (-00,00). The first of the two structures is geometric and consists of those subproducts of H2 x (-00,00) having the form Lx (-00,00) where L is a leaf of either ,1.'1 or A;. The second of our two structures is metric and consists of a Riemannian pseudo-metric-with-singularities ds on H3 defined in Section 5. The metric ds is obtained as the equivariant lift of a pseudo-metric from M. Hence, in particular it is G(M) invariant. It is carefully chosen to have two properties. (I) The metric ds is near enough to the standard hyperbolic metric that it induces the same topological structure at infinity. (2) The metric ds respects the product structure and foliations enough so that the sets Lx (-00,00) are metrically nice. Property (I) is explained in Theorem 5.1. Property (2) is explained in Theorem 5.2. Properties (I) and (2) are exploited together in Theorem 5.3. And finally all of these properties combine to give the proof of the Main Theorem at the end of Section 5.

5 The approximate metric structure of M The universal cover H3 of the fibered manifold M is topologically the product H2 x R of the universal cover H2 of the fiber S and the universal cover R of the base Sl of the fibration. The monodromy map ¢: S --+ S and its associated laminations (AI, dx) and (,1.2, dy) and multiplier k > I supply a natural pseudometric dS6 = dx 2 + dy2 for H2 x {O} (see Section 12, Measured laminations binding a surface). In order to extend this pseudometric in a natural way to all of H2 x R so as to be G (M) invariant, we note that invariance requires

{

kdx

Jyx{t}

{ Jyx{t}

dy

{

kdy

Jif;-l(y)x{t}

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COLLECTED WORKS WITH COMMEN TARY

{ JYX{t+l}

k dy.

1321

Group invariant Peano CUf1!es

where ¢: H2 --+ H2 covers ¢: S --+ S. That is, the measure must stretch by a factor of k in the dx direction and by 1/ k in the dy direction when t E R increases by I. This suggests the G(M) invariant, infinitesimal pseudometric ds on H3 "" H2 X R given by the formula

where A is a positive constant that may be chosen arbitrarily. We choose A = log k because the metrics ds 2 = k2t + (logk)2 dt 2 and ds 2 = k- 2t + (logk)2 dt 2 on the plane R x R are isometric with the upper half-plane model of hyperbolic geometry H2 under the maps (x, t) --+ (x, k- t ) and (y, t) --+ (y, kt) , respectively. We denote the associated G(M)-invariant global pseudometric by (s = fy ds): H3 x H3 --+ [0, 00). (See Sections 11-14 for properties of f ds.) Theorem 5.1 (Quasicomparability of metrics) The hyperbolic metric d H and the G(M)-invariant global pseudometric s = fy ds are quasicomparable. Proof The quasicomparability of d H and s requires the existence of positive numbers K > I and K' > I such that

max(s,dH»K'

=}

(~)dH:':S:':KdH'

The facts we need to know about s are the following: (1) s is G (M) invariant. (2) s is obtained by integrating a continuous infinitesimal pseudometric ds along paths of H3. (The continuity of s is convenient but not esssential; a weak kind of continuity like that of Section 11 , Measured laminations, would suffice.) (3) Each compact subset of H3 has finite s-diameter. (4) Each pseudometric s-neighborhood

N(x, E; s) = {y EH 3 Is(x,y) < E, x E H3,E E (0, oo)}, has compact closure in H3. Condition (1) is easily checked. The continuity of ds follows from Section 13 (PseudoAnosov diffeomorphisms). Conditions (3) and (4) follow easily from the fact proved in Section 12 (Measured laminations binding a surface) that the metrics d H and p = f(dx 2 + dy2)1/2 on H2 are quasicomparable. Qeometry & Topology, Volume 11 (2007)

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Realizing as we do that d H on H3 also satisfies (1), (2), (3) and (4), we need not distinguish between Sj = sand S2 = d H provided we only use properties (1-4), Hence it suffices to prove the single implication: there exist K, K' > I such that

S2(X,y) > K' Pick a > I so large that, if z

E

S2(X,y):,: KSj(x,y).

=}

H3 , then the G(M)-translates of

N(z,a;sj) ={x

3 EH Isj(z,x)

I so large that, if z E H 3 , then N(z,a;sj) C N(z, p; S2) (Property (4». Take K and K' > 4p and suppose that x, y Let y denote a path from x to y satisfying Sj

(x, y):,:

i

E

dS j
K'.

+ 1.

Let B j , B 2, ... B n denote a minimal covering of y by neighborhoods of the form N(z, p; S2)' It is then clearly true that

S2(X, y):,: 2p ·n. But, subdividing y into segments Yo, Yj, ... , Yk, satisfying 0:': { dS j < I,

{ dS j = I

i yo

lYi

it becomes obvious from the choice of a and B j , ••• , Bn that n :': k. Hence

s2(x,y):':2p'n:':2p

i

(i = L.,k),

p and the minimality of the covering

dS j K' > 4p implies that S2(X, y)/2
N, then the Euclidean diameter of Lx (-00,00) in H3 = H2 X (-00,00) C B3 = H3 U S~ is less than E, Proof From the quasicomparability of j(dx 2 + dy2)1/2 and d H on H2 and the quasicomparability of s = j ds and d H on H 3 , it follows that dH(z, L) --+ 00 (as L approaches infinity in H 2 ), From the quasicomparability of ds and d H on H3 it follows (see [3]) that, since Lx (-00,00) is quasi-totally geodesic in H3 with respect to ds, it is also quasi-totally geodesic in H3 with respect to the hyperbolic metric d H in the following sense,

(*) There is a W > 0 such that if L is any leaf of ),'1 U),;, if x and y are any two points of L x (-00,00) and g is the hyperbolic geodesic from x to y, then there is a path y from x to y in Lx (-00,00) within W of the geodesic g, Qeometry [3 Topology, Volume 11 (2007)

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But since hyperbolic geodesics g that miss large compact subsets of H3 must have uniformly small Euclidean diameter in the Poincare disk model H3 C B3 = H3 U s~, it follows that corresponding s-geodesics y must have uniformly small Euclidean diameter, Thus if L x (-00,00) misses a large compact subset of H 3, then L x (-00,00) has small Euclidean diameter in B 3, 0 Proof of Main Theorem There exists a natural continuous, possibly multivalued extension j: B2 --+ B3 of i: H2 --+ H3 defined as follows for x E Sl = 3H 2 , Let C l , C2 , . denote a sequence of compact neighborhoods of x in B2 whose intersection is {x}. Let U l = C l nH2, U2 = C2 nH2, etc. Let j(x)=n~~liUnCB3

To prove the theorem it suffices to show that j is single valued. We can do this simply by showing that lim diam (i Un) = o. n-+oo

Theorem 4.1 (Neighborhoods at infinity in the ball) implies that we lose no generality in assuming that Cn is bounded in B2 by the closure of a single leaf Ln of A'l U A; . By Theorem 5.3, lim diami(Ln x (00,00)) =0.

n-+oo

But i (Ln x (-00,00)) separates i Un from a large compact subset of H3. It follows that lim diam (i Un) = 0, n-+oo

hence that j is a continuous function as desired.

o

6 Fully one-sided degenerate groups A typical discrete faithful action of a surface group in Isom (H3) is a quasi-Fuchsian group, where there are exactly two components of the domain of discontinuity separated by a Jordan curve. The surface groups coming from fibers of fibrations are completely opposite to these typical actions: in the case of fibers, both components of the complement of the curve have shrunk to nothing. As we have seen, the Jordan curve becomes a Peano curve in these cases, filling the sphere. There is also an intermediate case, surface groups for which one of the expected components of the domain of discontinuity is completely absent but the other component

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is healthy. Such a group is called afolly one-sided degenerate group. We will show that at least for certain of these groups. the Jordan curve becomes a continuous map of the circle that traces out a fuzzy tree embedded on S2 . The existence of fully one-sided degenerate surface groups was established by Bers. who analyzed what happens if a quasi- Fuchsian group is varied so that the conformal structure on the quotient surface 5) of one of the domains of discontinuity is held constant. Here in outline is the proof that they exist. According to the deformation theory of quasi-Fuchs ian groups. the conformal structure on the quotient 52 of the other domain can be varied arbitrarily over Teichmiiller space. If a normalization of the groups is chosen in some appropriate way. this gives rise to a family of conformal embeddings of the universal cover of 5) into the sphere S~. It is easy to see that the closure of this family of embeddings is compact in the topology of pointwise convergence, by the theory of normal families. All the limit embeddings give rise to discrete faithful group actions. Those actions for which there is more than one component of the domain of discontinuity are contained in a countable union of complex subvarieties of the deformation space that have less than full dimension. Therefore, most of the limit groups have to be fully one-sided degenerate. Unfortunately, this construction does not give much of a cluse as to the nature of the fully one-sided degenerate groups. In [8] and [7], a good deal of theory for these groups is developed, expressed in terms of geodesic laminations. To each one-sided degenerate group pen) (5)) is associated a certain geodesic lamination on 5, called the ending lamination t (p) for the group. The ending lamination can be defined in terms of the rough location of geodesics in the hyperbolic three-manifold Mp = H3 / pen) (5)). For any compact set K C M p, let 5 CK denote the set of simple closed curves on 5 whose geodesics in Mp do not intersect K. Thurston showed that for a limit of quasi-Fuchs ian groups that is a one-sided degenerate group, this set is nonempty no matter how large the compact set K. As K increases toward all of M, 5 CK decreases. The limit set for 5 CK in the space of measured laminations on 5 consists of all measures supported on a certain geodesic lamination teMp). If a limit of quasi-Fuchsian groups degenerates at both ends, then there are two ending laminations, one for each end of the limit manifold. For more details, see Thurston, [8] and [7]. More recently, Francis Bonahon [2] has proven that the theory of ending laminations extends to the case when a surface group is not known to be the limit of quasi-Fuchsian groups. Our main theorem in the case of fully one-sided degenerate groups is the following:

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Theorem 6.1 (Ending stable gives fuzzy tree) If ¢ is any pseudo-Anosov homeomOlphism of S, then there is a fully one-sided degenerate group pen! S) whose ending lamination is the stable lamination for ¢ such that any n! -equivariant map from H2 = § onto H3 extends to a continuous map ofB 2 = H2 U S~ to B3 = H3 U S~. It would be nice to be able to say what happens for other ending laminations. The problem in general is equivalent to knowing whether or not the limit set for the group is locally connected. No examples are known where the limit set for a surface group is not locally connected. Our method depends on knowing the approximate metric structure of the associated 3-manifold. We do not have a good analysis of this metric structure in the general case, although our method would probably extend before encountering insurmountable obstacles to the case of ending laminations belonging to an uncountable but rare set of laminations having the regularity properties enjoyed by the stable and unstable laminations for pseudo-Anosov homeomorphisms. The success of Bonahon's proof that a general surface group is geometrically tame gives more hope for understanding the topology of the general limit set.

The analysis in this case is inevitably less elementary and less self-contained than in the case of limit sets for surface groups that are fibers of fibrations, since the hypothesis already involves the idea of the ending lamination of an end. Basic references are Thurston, [8] and [7].

7 Approximate metric structure, one-sided degenerate case The three-manifold Mp associated with a fully one-sided degenerate surface group p: n! (S) ---+ Isom(H3) has two ends, one of which flares out exponentially and develops out toward the domain of discontinuity, the other of which has bounded diameter and somehow controls the topological and geometrical structure of the limit set. In order to analyze the topology of the limit set as we did in the case of a surface group coming from a fiber, we have to analyze the approximate metric structure of this latter end. The idea is that when the ending lamination is the stable or unstable manifold of a pseudo-Anosov homeomorphism ¢ of S, we should expect the end to look metrically similar to one of the two ends for the doubly degenerate group that is the fiber of the mapping toms M¢. We will prove this by a trick of passing to a limit of a sequence of representations of the surface group differing from the original by an automorphism of the surface group. There is a compactification T of the Teichmiiller space T that was discovered by Thurston in conjunction with the theory of pseudo-Anosov homeomorphisms, in which

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a sphere of projective classes of measured laminations on S is adjoined to the open ball of hyperbolic structnres on S to form a closed ball. Any homeomorphism of S acts as a homeomorphism of this ball. If ¢ is a pseudoAnosov homeomorphism. then ¢ fixes exactly two points in the ball. both of which are on the boundary. These two points are the stable lamination A1 and unstable lamination A2 of ¢. On S. the transverse measure of A1 is increased by ¢ while the transverse measure of A2 is decreased by ¢. Simple closed curves on S tend to look closer to Al after applications of ¢. On T. the stable lamination A1 is an attracting fixed point for the action of ¢. while the unstable lamination is a repelling fixed point. In fact, under iteration of ¢, the entire ball except for A2 converges toward AI. The ending lamination is unfortunately not defined as a boundary point of T -it is only a topological lamination that admits a transverse measure, not a measured lamination. If a lamination is uniquely ergodic, it defines a unique point on aT. The stable and unstable laminations of ¢ are uniquely ergodic. Let R be the set of conjugacy classes of representations of n 1 (S) up to conjugacy that are quasi-Fuchsian, with conformal structnres on the two domains of discontinuity of the form (¢-n(go), ¢m(go)), where go is a fixed conformal structure and n, m::: O. The double limit theorem of Thurston [8] implies that the closure of R is compact. The added elements of the closure consist of: (I) if m --+ 00 but n stays bounded, fully one-sided degenerate representations such that one ending lamination is AI, while the remaining component of the domain of discontinuity has conformal structure ¢ -n (go); (2) if n --+ 00 but m stays bounded, fully one-sided degenerate representations such that one ending lamination is A2, while the remaining component of the domain of discontinuity has conformal structure ¢m (go); (3=1 +2) if n --+ 00 and m --+ 00, doubly degenerate representations with ending laminations A1 and A2.

A hyperbolic structure for the mapping torus of ¢ was constructed from limits of type (3) by adjoining an additional transformation T¢ that induces the automorphism ¢ of nl (S). Since T¢ acts as a quasiconformal automorphism of each group in R, with uniformly bounded quasiconformal constant, it acts also as a quasiconformal automorphism of any of the limiting groups. But in case 3, there are no quasiconformal automorphisms except isometries, by a theorem of Sullivan or a theorem of Thurston. Thus, any limit of type (3) gives rise to a hyperbolic 3-manifold for Mo. Mostow's rigidity theorem asserts that such a hyperbolic structnre is unique, so there is a unique limit of type (3). Qeometry & Topology, Volume 11 (2007)

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We can also obtain the limit of type (3) by first forming the limits of type (I), then letting n go to infinity, The map 1 acts on the set oflimits of type (I) by incrementing n, This means that we can choose a fixed representation Po of type (I), and then the limit of

r;

Po

0

r 1>- k

must exist and equal the unique limit of type (3), (Every subsequence must have a convergent subsequence, but there is only one possible limit for these subsequences, so the entire sequence must converge,) There is also a geometric form of the statement about algebraic convergence, The quotient manifold Mo = H3 / PO(nl (S» has one geometrically infinite tame end, If we take a sequence of points tending toward this end, this defines a sequence of based hyperbolic manifolds, There is necessarily a convergent subsequence, and the limit of the sequence is isometric to the quotient manifold by the limit of type (3), with some choice of base point This fact can be proven from the existence of the algebraic limit above, by first considering the sequence of base points {Pk} where Pk comes from the point in H3 that minimizes the total translation length of a fixed set of generators for nl (S) The normalization forces the actual sequence of under the representation Po 0 representations to have a convergent subsequence, not just a subsequence convergent up to conjugacy, The geometric limit of the manifolds based at Pk is covered by the quotient of the group coming from the algebraic limit representation, But this manifold is only a covering space in very special ways: anything it covers is either compact, or is covered with finite degree, These kinds of covering spaces cannot arise in a geometric limit, so the geometric limit is simply the quotient of the representation of type (3),

r;k,

A knowledge of all possible geometric limits as the base point moves toward the end gives a description of the approximate geometry of the geometrically infinite tame end: it is exactly the same approximate geometry as the previous case, the infinite cyclic covering of the mapping torus of ¢, In the geometrically finite direction, the approximate geometry is easy to understand: it is just an exponentially flaring collar, We can write down an expression for the approximate pseudometric: if dx and dy are the transverse measures for A1 and A2, then on § xR, the metric ds 2 = exp(2lt l)dx 2 + exp(2t)dy2 + dt 2 is quasicomparable to the hyperbolic metric, where t is a parameter for R, In the ds pseudometric for § x R, any leaf of the lamination AJ sweeps out a surface that is locally isometric to a hyperbolic plane, The retraction of S x R to this hyperbolic plane decreases distances, as before, Therefore, each of these hyperbolic planes is

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isometrically embedded in the pseudometric, hence quasi-isometrically embedded in H3. Its boundary forms a circle in S~, with the leaves of )q at the geometrically infinite end being collapsed to points on the limit set for the group.

8 Hyperbolic surfaces and limit geodesics A hyperbolic surface is a 2-manifold S with a complete Riemaunian metric ds of constant negative curvature -I. The universal cover of S is hyperbolic 2-space H2. The group G(S) of covering transformations is a group of hyperbolic isometries of H2. The covering map p: H2 --+ S is a local isometry.

We consider only the case where S is orientable and closed of genus g::: 2. The metric ds then defines an area element dA on S with respect to which the area dA of S is finite and equal to 2n g.

Is

Theorem 8.1 (Limit geodesics) Every geodesic ray r: [0,00) --+ S has a (possibly nonunique) limit geodesic r': (-00,00) --+ S having the following property. Given E > 0 and any geodesic path R' in r', there are infinitely many geodesic paths R in r pointwise within E of R'. Proof By the compactness of S, some sequence of segments r I [n), n) + I], r I [n2,n2 + I], (n) < n2"') converges to a geodesic segment r': [0,1]--+ S. The segment r' extends to a geodesic r': (-00,00) --+ S. Let E > 0 and any geodesic path R' in r' be given. We may assume R' has the form R' = r' I [-n, n] for some n. Then for r I [ni, ni + I] close enough to r' I [0, I], a requirement that can be realized simply by choosing ni > n very large, r I [ni - n, ni + n] will be pointwise close to R' = r' I [-n, n], as desired. D

9 Geodesic laminations A geodesic lamination A on S is a collection of disjoint, simple (that is, nonsingular) geodesics on S called leaves whose union IAI is closed in S. Theorem 9.1 (Lamination complement) The complement S -IA I of a lamination is nonempty (since no closed surface ofgenus> I admits a foliation without singularities). Each component C of S -IAI is (clearly) a convex hyperbolic polyhedron (that is, C has convex universal cover) that has only ideal vertices and is bounded by leaves of A. The set S -IAI has only finitely many sides. Qeometry & Topology, Volume 11 (2007)

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Proof Augment A by adding to A disjoint, simple closed geodesics J 1, h, . m S -IA I until it is impossible to add more. The process stops after finitely many additions since (i) no closed geodesic is homotopically trivial. (ii) no compact surface contains infinitely many disjoint, nontrivial, non-freely-homotopic simple closed curves, and (iii) freely homotopic closed geodesics coincide. We recall that the area of a hyperbolic triangle with angles a, p, and y is n -00- p-y. Area calculations will allow us to prove that, with A so augmented, the only forms assumed by components C of S \ IAI are the following: (I) C is simply connected and has area 0 < Area(C) = (n - 2)n :': 2n g, where n is the number of sides of C. (The polygon C has its n vertices at 00, hence can be divided into n - 2 ideal triangles, each of area n.) (2) C is an open annulus, one boundary component is a simple closed geodesic J, the other boundary has n ::: I sides formed from non-closed geodesics, and the area of C is 0 < Area (C) = nn :': 2n g. (The polygon C cut along a geodesic joining J to some other boundary component can be divided into n + 2 triangles of angle sum 2n .) (3) C is a pair of pants (an open disk with two holes) and each of the three boundary components consists of a single simple closed geodesic. The area of C is o < Area( C) = 2n :': 2n g. (The polygon C can be divided into two rightangled hexagons. Each hexagon can be divided into 4 triangles with angle sum 6·n/2 = 3n.) Indeed, any component C of X -IAI not having form (I), (2), or (3) contains a simple closed curve J' not freely homotopic into the boundary of C. Since C is convex, the geodesic J freely homotopic to J' must lie in C U ac, hence in C. But then J' could be used to further augment A, a contradiction. From the area calculations of (I), (2) and (3) it follows immediately, as asserted, that S -IA I has only finitely many components, each having only finitely many sides. 0 Theorem 9.2 (Lamination area) Area

ClAI) = o.

Proof Adding leaves in S -IA I, one can complete A to a foliation-with-singularities completely filling S. The index theorem allows one to calculate the Euler characteristic in terms of these singularities. The estimate thus obtained on the singularities alllows one to calculate via (I), (2) and (3) the total area of S -IA I. It equals Area (S). Hence 0 Area ClAI) = o.

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10 Laminations binding a surface A pair of geodesic laminations Al and ,1.2 is said to bind S if they satisfy the four

conditions of the following theorem. Theorem 10.1 (Laminations binding a surface) For geodesic laminations Al and ,1.2 on S the first three of the following conditions are equivalent and imply the fourth.

(1) Each simple geodesic on S crosses a leaf of Al U ,1.2' (2) Each simple geodesic rayon S crosses a leaf of Al U ,1.2' (3) Each geodesic rayon S crosses a leaf of Al U ,1.2' (4) Let A' denote the lift of A to the universal cover H2 of S. Then each component ofH 2 - (1,1.'11 U 1,1.; I) is the interior of a compact, convex hyperbolic polyhedron in H2. Proof That (3)

=}

(2) and (2)

=}

(1) is clear.

=} (2): Let r denote a simple geodesic rayon S and r' a limit geodesic of r (Section 8, Hyperbolic surfaces and limit geodesics). By (1), r' crosses a leaf of Al U ,1.2' Hence, by Section 8, r crosses the same leaf of Al U ,1.2'

(1)

(2) =} (4): Each component C of S - (1,1.11 U 1,1.21) clearly has convex lifts in H2. We claim that C is simply connected. For otherwise C U aC would contain a simple closed geodesic J; by (2) J would cross a leaf of Al U ,1.2 and J would not lie in C U ac, a contradiction. If a lift C' of C did not have compact closure in H2, then C' would contain a geodesic ray r'. Since C is simply connected, the projection C' --+ C is a homeomorphism. Hence the image r of r' would be a simple geodesic ray in S not crossing any leaf of Al U ,1.2' Hence C' has compact closure in H2, and (4) follows. =} (3): Let r denote a geodesic rayon S. Since (2) =} (4), any lift r' of r to H2 intersects a leaf of ,1.'1 U A;. Hence r intersects a leaf of Al U ,1.2' If r does not cross that leaf, then r coincides with that leaf, hence is simple, hence crosses some ~~m. D

(2)

Remark Easy examples, where ,1.'1 and A; share a leaf that does not appear in the boundary of any component of H2 \ (1,1.'11 U 1,1.; I), show that (4) does not imply (1). Theorem 10.2 (Neighborhoods at infinity) Suppose laminations Al and ,1.2 bind S. Then each point x E S~ at infinity in B2 = H2 U S~ has arbitrarily small Euclidean neighborhoods in B2 bounded by the closure in B2 of a single leaf of ,1.'1 U A; .

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Proof Since)'.j and A2 bind S, A) and A2 have no leaf in common, Hence no leaf L) of A') and leaf L2 of A; have a common endpoint at infinity; for otherwise geodesic rays r) and r2 in the projections of L) and L2 to S would have a common limit geodesic L (Section 8, Hyperbolic surfaces and limit geodesics) and L would be a leaf common to A) and A2'

Let r denote a geodesic ray in H2 with infinite endpoint x, If possible, take r to m lie in IA')I or IA;I, Since A) and A2 bind S, there exist leaves L), L 2 , L 3, . A') U A; such that L; crosses r in the Euclidean (1/ i)-neighborhood of x. We claim that L; -+ x as i -+ 00. Suppose not. Then some subsequence converges to a leaf L of A') U A; with infinite endpoint x, say L in IA')I. By the choice of r it follows that r is contained either in IA')I or in IA;I. By the previous paragraph, since Lis contained in IA') I, it is impossible that r be contained in IA; I. Hence r is contained in IA')I. But then each L; is a leaf of A; , hence L is contained in IA;I. But this contradicts the previous paragraph. Hence L; -+ x as asserted, and the L; cut off small neighborhoods of x in B2 as desired. 0

11 Measured laminations A transverse measure on a geodesic lamination A is a positive measure dm defined on local transversals to the leaves of A, invariant under local projection along the leaves of A and positive and finite on nontrivial compact transversals to the leaves of A. Such a measure lifts to a G(S)-invariant transverse measure on the inverse A' of A in H2, the lifted measure also denoted by dm. A geodesic lamination with a transverse measure is called a measured lamination.

If dm is a transverse measure on a lamination A and y: [a, b]-+ H2 is any path, one may define the integral fy dm : if X is a closed subsetofH 2 , define mIX) to be the measure of the projection of xn IA'I into the leaf space of A'; let P = a = 00 < a) < ... < an = b denote a partition of [a, b]; define m (y, P) = L.:7~) m (y [0;_), ail) - L.:7;;; m (y (a;»; take fy dm as the supremum of m(y, P) over all partitions P of [a, b]. The negative term in m(P) avoids doubling the contribution to fy d m of a point lying on a leaf that supports an atom of d m .

i

Given two points x and y of H2 , if y is the geodesic segment joining x and y, and J is any path joining x and y, then fy d m :': fo d m . We present four theorems. The first two describe limitations on the leaves of a measured lamination. The second two describe weak continuity properties of fy d m .

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Theorem 11.1 (Atoms and closed curves) If (y, dm) is a measured lamination on S and L is a leaf of A carrying an atom of d m, then L is a simple closed curve. Proof Otherwise, let R denote a limit geodesic of L in the sense of Section 8 (Hyperbolic surfaces and limit geodesics) and let X denote a compact transversal to R. Then X intersects L infinitely often and d = 00 a contradiction. D

Ix m

Theorem 11.2 (Zero-angle) Suppose that (A, dm) is a measured lamination on S, and suppose that rl and r2 are distinct geodesic rays in H2 each compatible with A' in the sense that either ri n IA'I = ri or ri n IA'I = 0. Then if rl and r2 have the same endpoint x atinilnity, and there is a neighborhood N of x in B2 = H2 U S~ such that either (I) no point of N

n IA'liies in the open angle of zero measure between rl and r2;

or (2) exactly one leaf L of A' intersects the closed angle of zero measure bounded by rl and r2 in N; L has inilnite endpoint x; L is isolated and carries an atom of d m: and the projection of L in S is a simple closed curve. Proof Since rl and r2 are compatible with A', N may be chosen so small that if L is any leaf of A' that intersects the closed angle between rl and r2 in N, then L has x as an infinite endpoint. If no leaf L of A' intersects the open angle between r 1 and r2 in N, (I) is satisfied and we are done. Hence we suppose the existence of a leaf L with infinite endpoint x containing an infinite ray r that lies strictly between r 1 and r2 in N. The image p(r) in S has a limit geodesic r' in the sense of Section 8 (Hyperbolic surfaces and limit geodesics). Let X denote a short compact geodesic arc transverse to some short geodesic subarc Y of r' in S. We may assume X n Y is a single point. Arbitrarily short sub segments of X near X n Y lift to segments in H2 crossing the entire angle from rl to r2 in N. The integral of dm along each of these subsegments is the same strictly positive number. If these segments may be chosen to be disjoint, then dm = 00, a contradiction. Otherwise we conclude that r' = p(r), that r' is isolated, and that r' carries an atom of dm. Hence, by Theorem ILl (Atoms and closed curves), r' = peL) is a simple closed leaf. Any other leaf L' satisfying the properties of L must have projection peL') with the same limit geodesic r'; as above, peL') = r'. Two lifts of the same simple closed curve with a common endpoint at infinity are equal. Hence L = L' and (2) is satisfied. D

Ix

Measured laminations satisfy the following two weak continuity properties.

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Theorem 11.3 (Continuity property 1) If(A,dm) is a measured lamination and A a positive constant, then there is a positive constant B such that each geodesic segment y in H2 oflength :0 A has integral J y d m :0 B. Proof Otherwise, take segments Yi of length :0 A converging to a segment in the interior of a segment Y, Jy, d m --+ 00. Let X denote a segment transverse to y . For all ; sufficiently large, each leaf of A' hitting Yi hits Y or X. Hence 00

= lim

1 1 dm:o

dm
length(y) - I. If Inty intersects IA'I , choose y_ to contain a point of IA'I. Let y + be a geodesic segment containing y in its interior with length (y +) < length(y) + I. Let X denote a geodesic segment transverse to y at a point of y_ with length(X) < I. If y_ intersects IA'I and does not lie in IA'I, take Xc IA'I. Case 1 y C IA'I Note that J y dm :0 Jx dm < B < C. For all ; sufficiently large, each leaf of A' intersecting Yi also intersects X. Hence Jy, d m :0 JX d m < B < C. Hence IJy dm - J y, dml < C. Case 2 (Inty) n IA'I = 0 Then for y and for all Yi, ; sufficiently large, IA'I intersects each of y and Yi in two segments of total length < I. Hence by choice of B,Jydm :o2B = C and J y, dm :o2B = C. Case 3 (Inty) n IA'I oj 0 and Xc IA'I For all ; sufficiently large, each leaf of A' that intersects y_ also intersects Yi. Hence

i

dm:o

(L

dm) +2B:o ( { dm) +C.

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Also, for all i sufficiently large, each leaf of A' that intersects Yi intersects Y+, Hence

1 1 1 dm :':

Yi

dm:':

dm

Y+

+ 2B =

Y

(1

dm)

+c

Y

Cases 1,2 and 3 exhaust the possibilities and complete the proof of Property 2,

0

12 Measured laminations binding a surface Two measured laminations (A 1, dx) and (A2, dy) are said to bind S if the associated geodesic laruinations A1 and A2 bind S in the sense of Section 10 (Laruinations binding a surface), Theorem 12.1 (Laminations give good pseudometric on surface) Suppose (A 1, dx) and (A2, dy) denote measured laminations binding S, and define

d/ = dx 2 +dy 2 Then dp lifted to H2 defines a G (S) -invariant pseudo-metric p on H2 quasicomparable with the hyperbolic metric d H . Proof If x E H2 , define p(x, x) = O. If x and yare distinct points of H2 and y is the geodesic segment joining x to y, define

p(x,y) =

i

dp.

Then p clearly satisfies the conditions for a pseudometric: 0 = p(x, x), 0 :': p(x, y) < 00, pIx, y) = pry, x), and p(x, z) :': p(x, y) + pry, z). The quasicomparability of p and d H entails the existence of positive constants K and K' such that

We prove the existence of K and K' as follows. By Theorem 11.3 (Continuity property I) there is a constant D such that each geodesic segment of dH -length not exceeding I has p-Iength not exceeding D. Thus if dH(x, y) > I, with dH(x, y) = n + k, 0:': k < I, n :': I,

p(x,y):,: (n

+ l)D =

n+1 --kDdH(x,y):,: 2DdH (x,y).

n+

This proves the second inequality of formula (*). The first inequality of (*) follows in an exactly analogous way from the following assertion: there is a positive integer N Qeometry & Topology, Volume 11 (2007)

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and a positive integer i such that each geodesic segment y of dH-length exceeding N has p-Iength exceeding 1/ i . Indeed. if the assertion is correct. let K' > K > 2N i . If p(x,y):,: dH(x,y) > K', then dH(x,y) = (n +k)N, k:': I positive and n:c: I integral. Since d H (x, y) > n . N , our assertion implies that p(x, y) > n . (l / i) so that

dH(x,y)

K

nN+kN


I such that ¢(IAil) = IAil, f¢Cy)dx = I/kfydx and f¢Cy)dy = kfydy for each path y on S. The lamination A1 is called the stable lamination and A2 the unstable lamination for ¢. For the remainder of this section we fix a pseudo-Anosov diffeomorphism ¢ with associated laminations (A 1, dx) and (A2, dy) and multiplier k > I. We define d¢2 = dx 2 + dy2 as in Section 12 (Measured laminations binding a surface). Theorem 13.1 (Stable laminations irreducible) The laminations Al and A2 have no simple closed leaves. The measures dx and dy have no atoms. The pseudometric p = f dp is continuous on H2 x H2 .

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Group invariant Peano CUf1!es

Proof The third assertion clearly follows from the second. The second assertion is. by Theorem II.I (Atoms and closed curves) a consequence of the first. Hence, it suffices to show that neither Al nor ,1.2 has a simple closed leaf. Suppose, on the contrary, that A 1 had a simple closed leaf L. The surface S cannot contain infinitely many disjoint, nonparallel, nontrivial simple closed curves. Hence, for some positive integer n, L and ¢n L are parallel closed geodesics, hence equal. But then

0< /, dy = ( L

J1>n(L)

dy = k n

/, L

dy
-6. 8-+0

Proof. If S ;:> -6, then the result follows directly from the fact that A(98) ;:> inf R 8, so we may assume that S < -6. Let U8 be the eigenfunction of -4,6.go + R8 with minimal eigenvalue A(98) < 0, such that U8 is strictly positive and IIu8112 ~ 1. We can use the Rayleigh quotient method to estimate A(98) from below. We may estimate

r (IV8U812 + R8U~)dg8;:> lx-(rexl-8,8]) r R8u~dg8 + r R8u~dg8 lrexl-8,81 ;:> -6 r u~d98 + S r u~d98 ~ -6 + (S + 6) r u~d98. lx-(rexl-8,8]) lrexl-8,81 lrexl-8,81

A(98)

~

lx

Thus, we need only show that frexl-8,81 u~d98 -+ 0 as 0 -+ O. Applying Holder's inequality, we see that

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The right hand term converges to 0, since Vol(~ X [-0,0],98) -+ 0 as 0 -+ 0 and IIu8116 is bounded by the following. Since the metrics g8 approximate 9 in CO norm, there is a uniform Sobolev constant C (see the justification in Lemma 4.3) such that

D Rescaling the metrics g8 by factors converging to 1, we may approximate (X,g) by metrics g~ such that A(g~) ;:> -6. 6. RICCI FLOW WITH

CO INPUT

Let (X, g) be the smooth closed 3-manifold equipped with the Lipschitz Riemannian metric g from the beginning of section 4, defined by equation (2) and satisfying the curvature conditions (1). Let {98 h>o be a family of smooth Riemannian metrics on X converging uniformly to g. We do not yet place any other geometric assumptions on the approximates g8. The goal of this section is to establish facts about flowing the singular metric g in the direction of its Ricci curvature. The analytic tools of this section were proven by M. Simon [40]' where the dual Ricci harmonic map heat flow is studied with nonsmooth initial data. To apply Simon's machinery, it is necessary to fix a "background metric". Concretely this means fixing a smooth Riemannian metric h on X so that the identity map (X, h) --+ (X,g) is K-bilipschitz, where K > 1 is a fixed constant depending only on the dimension. For our purposes, it would suffice to pick a metric g8 for sufficiently small b. Let us choose such a metric and call it h. The following theorem follows from statements in section 5 of [40].

Theorem 6.1. There exists a constant T > 0 depending only on the background metric h, a sequence bj --+ 0, a sequence of families of smooth metrics {g(b j , t)}, 0 :s; t c; T,j E N, and a family of metrics g(O, t), 0 c; t c; T, such that the following hold. (1) The metric g(O, 0) ~ 9 and g(Oj, 0) ~ g8,. (2) The family {g(Oj, t) }o

1

2: V31IDMII·

Equality holds if and only if M has constant sectional curvature -1 and the boundary of M is totally geodesic. Proof. Let (M X {+, - },g) be two identical copies of (M, g), with boundary components ~± ~ 3M X ± and isometry : ~+ -+ ~_ defined by (x,+) ~ (x,-) and DM ~ (M X {+,-})/. 8ince H(~±,g±)(x) ~ 0, the curvature inequality hypothesis of Theorem 7.1 is satisfied. Thus, Vol(M) ~ ~V31IDMII. If we have equality, then the metric on DM is hyperbolic. This implies that 3M is totally geodesic, since the metric on DM will be singular if the second fundamental form of 8M is nonzero anywhere. D 8. HYPERBOLIC CONVEX CORES

In this section we will use Theorem 7.2 to study convex cores of hyperbolic 3manifolds. More specifically we can quickly prove a conjecture of Bonahon stating that the volume of a compact hyperbolic 3-manifold M with nonempty convex boundary is at least half the simplicial volume of the doubled manifold DM. In the incompressible boundary setting this was proved in [42]. We will also prove the relative version of the inequality for cusped hyperbolic 3-manifolds.

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If N is a complete (infinite volume) hyperbolic 3-manifold without boundary, then the convex core C N 2.02988 by [9]' so plugging R ~ log(3)/2 into the above D inequality, we see that Vol(M) ;:> .67. Remark. Theorem 10.1 allows us to assail the problem of finding the minimal volume hyperbolic 3-manifold on various fronts, by either classifying cusped orientable

hyperbolic 3-manifolds with volume < 2.852 (for example, by extending the arguments ofCao and Meyerhoff [9]) or by classifying closed 3-manifolds with a minimal length geodesic with tube radius < 0.7. 11. UNIVERSAL MANIFOLD PAIRlNGS

The paper [14] defines the notion of a universal (n + 1)-manifold pairing. This

consists of linear combinations of (n + I)-manifolds with marked boundary, together with a pairing on these manifolds which have the same n-dimensional marked boundary which takes values in formal linear combinations of closed (n + 1 )-manifolds. A unitary TQFT may be thought of as a "representation" of such a pairing, assigning a Hermitian vector space to each closed n-manifold and a number to each

closed (n + I)-manifold and satisfying some extra axioms [3]. Conjecture 2.2 of [14] states that the (2+ 1)-dimensional universal manifold pairing faithfully detects linear combinations of 3-manifolds (there are no Hermitian isotropic vectors). We provide some partial evidence for this conjecture in a special case. Let :E be a closed oriented surface, and let MI:; denote the formal combinations of compact oriented 3-manifolds M with 8M = :E and coefficients in C. Define the pairing (,) : M" X M" -+ M by (L: i aiMi, L: j bjNj) ~ L:i,j aibjMi U" N j

(where M

~

M , and N j denotes Nj with the reversed orientation). The following

conjecture generalizes in a natural way Conjecture 2.2 of [14].

Conjecture 11.1. If there exists Xl, ... ,X m E M" such that L:i(Xi,Xi) ~ 0 E M, then

Xl = ... = Xm =

O.

We provide some evidence for this conjecture in a special case.

Corollary 11.2. Suppose that each

MI:; is a formal linear combi~ 0 E M. Let is minimal amongst Xi such that

Xl, ... , Xm E

nation of compact, acylindricaI3-manifolds, such that L:i(Xi,Xi) Xi

=

LjE1i xijMj ,

and suppose that

L:i(Xi,Xi) ~o. Thenforj,k

E

Li IIil

2

Ii, IIMju"Mkll ~ IIMju"Mjll.

Proof. Let gj denote the hyperbolic metric on M j with totally geodesic boundary. Denote the manifold Mjk ~ M j U" M k , j, k E I;. Let L;jk C Mjk be a least area representative of :E c Mjk in the canonical hyperbolic metric on M jk . Then IIMjkl1 ;:> ~(IIMjjll + IIMkkll) by Theorem 7.2. Since L:i,j IXijl2Mjj f 0, in order for Li(Xi, Xi) = 0, we must have Maa = M bc , for some b f:- c, b, c E Ii. Moreover,

we may assume that IIMaal1 is minimal over IIMjjll,j E U,Ii. Thus, IIMaal1 ~ liMbe II ;:> ~(IIMbbll + IIM"II) ;:> IIMaal1 (the second inequality follows from the

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assumed minimality of IIMaall), and thus IIMaal1 ~ IIMbbll ~ IIM"II ~ liMbe II· Since we have equality, we must have that :Ebc is totally geodesic in Mbc by Theorem 7.2, and thus gb and gc induce the same metric up to isotopy on:E. Take the maximal subpartition J ~ {J" ... ,Jn } of 1~ {I" ... ,1m } such that for all i,j E J"

IIMij II

~

IIMaa II· This partition is a subset of the partition by the isotopy class of the

metric on :E = 8Mj induced by gj, and therefore it is easy to see that UJ includes all j such that IIMjjl1 ~ IIMaall· Then for i not in U J, i f j, IIMijl1 > IIMaall, and thus all the cancellation among the terms of Li(Xi, Xi) with Maa must occur among the terms of the partition J. Associated to J z, there is a vector yz, where y, ~ 'L jE ], xijMj, where J, c I;. Then 'L,(y" y,) ~ 0, and 'L,IJ,1 2 c; 'Li l1il 2 By the minimality hypothesis on I, we have J = I, and the result follows. D This reduces this special case of Conjecture 2.2 of [14] to a geometric question about how many distinct ways a hyperbolic manifold may be obtained by gluing together two copies of a manifold with geodesic boundary by the identity on the boundary. It seems promising that this question may yield to geometric techniques. 12. CONCLUSION

The results in this paper give rise to many interesting questions. A natural question is whether one may prove the main result without using Ricci flow.

Conjecture 12.1. If a finite volume hyperbolic 3-manifold with minimal surface boundary is locally minimal among such manifolds, then the boundary is totally geodesic. If this conjecture were true, then one should be able to reprove Theorem 7.2 by deforming a hyperbolic manifold with minimal surface boundary to have geodesic boundary, while decreasing volume. Another possible approach would be to try to use the methods of the natural

map [61. If (M,g) is a Riemannian metric, let h(M,g) denote the volume entropy of g. Conjecture 12.2. If (M,g) is a closed Riemannian 3-manifold with R(g) ;:> -6, then h(M,g) c; h(1HI 3 ). This would enable one to reprove the main theorem using the techniques of the natural map. One nagging point we were unable to resolve is to prove the main theorem in the case that the manifold is noncompact.

Conjecture 12.3. Let (M,g) be a complete finite volume 3-manifold with minimal surface boundary and scalar curvature R(g) ;:> -6. Then Vol(M,g) ;:> !IIDMII, with equality if and only if M has geodesic boundary. This conjecture might be useful in an attempt to prove Conjecture 2.2 of [14],

by extending the arguments of Corollary

11.2.

ApPENDIX A. VOLUME CHANGE UNDER DRILLING: THEORY VS. EXPERIMENT

Let M be a closed hyperbolic 3-manifold, and let r be a simple geodesic in M. Consider the complement M, = M \ r. This appendix focuses on the following question: how are the volumes of M and M, related? In general, Thurston showed

that Vol(M)
0 such that one side of F t has volume less than Volume(M)/2 if t E (-I, -I + plU [I - p, I). In particular F -1+fJ bounds a 3--{;hain C_1+fJ of volume less than Volume(M)/2 and similarly F!-fJ bounds such a chain C!-fJ' Consider the

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2040

3--{;ycle Z E H3(M; 2 2 ) formed by the union of the 3--{;hains keF x [-I + p, I - I'll), C-1+!l and Cl-!l' We say the family of maps k has degree one if Z represents the fundamental homology class of M (with Z 2 --{;oefficients) and in that case we say k is a sweepout of M. Note that changing the value of p leads to a continuous change of the volume of Z, while a change in the homology class of Z would cause the volume to change by a multiple of the volume of M. Thus the choice of p does not affect whether k has degree one. We digress somewhat to point out that harmonic maps gives a new approach to constructing sweepouts of bounded area. Pitts and Rubinstein showed the existence of an unstable minimal surface in a 3-manifold using a minimax argument [13]. Starting with a strongly irreducible Heegaard splitting, they obtain a minimal surface that has maximal area in a I-parameter family of surfaces obtained from a sweepout. Since a genus- g minimal surface in a hyperbolic manifold has area less than 4n (g - I), this implies the same area bound for each surface in the sweepout. As noted by Rubinstein [16] and by Bachman, Cooper and White [1], the existence of a sweepout by bounded area surfaces has implications on the geometry and Heegaard genus of a 3-manifold. Bachman, Cooper and White constructed a sweepout using piecewise-geodesic surfaces with two vertices, which gives a somewhat weaker area bound. Harmonic maps give an alternate way to obtain a sweepout by bounded area surfaces, with the same area bound implied by Pitts-Rubinstein, but without assuming that the Heegaard splitting is strongly irreducible. The harmonic sweepout has the drawback of allowing singular surfaces, like [1], but unlike [13], where a sweep out by embedded surfaces is implied. Theorem 6.1 If M is a hyperbolic 3-manifold with a genus-g Heegaard splitting then M has a sweepout in which each surface has area bounded above by 4n (g - I). Proof Pntting the induced metric on each surface associated to a Heegaard splitting sweepout gives a continuous family of isometric embeddings {fo,t: (F, gt) --+ M, -I < t < I},

with gt the induced metric on F under the map /o,t. By Lemma 5.1, this family of maps is homotopic to a family of harmonic maps {/l,t: (F, gt) --+ M, -I < t < I}. Lemma 5.3 implies that during the homotopy from /o,t to /l,t, the area of (fs,t stays below the initial area of /o,t. So for each s we have lim Area(fs t) = O.

t---+±l

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Stabilization of Heegaard splittings

Since [s,t changes continuously with s the degree remains one and [s,t forms a sweepout for each fixed value of s, Each map [l,t in the final sweepout has area D bounded above by 4n (g - I), by Lemma 5,2, Given a sweepout of a 3-manifold M and two subsets L and R with disjoint interiors, we can characterize the direction of the sweepout relative to L and R, describing which of L and R is first engulfed, Let fJ be a constant that satisfies the conditions of Lemma 45 , so that for -I < s :': -I + fJ, Fs bounds a 3-chain of volume less than Volume(L U R)/2, If -I < t < -I + fJ define K t to be the 22 3-chain C t with boundary F t , given by the side of F t having smaller volume, If t ::: -I + fJ define K t to be the 3 --{:hain with boundary F t obtained by adding the 3 --{:hains keF x [-I + fJ, t]) and C-1+!l (mod 2), As t increases on (-I, I) the volume of K t changes continuously, while lim Volume(K t

n (L U R»

= 0

lim Volume(K t

n (L U R»

= Volume(L U R).

t ---+- l

and

t-+ 1

Note that we are using 22 --{:hains, so that the volume of K t need not grow monotonically with t. Let Y denote the set of points t half the volume of L U R,

Y =

E

(-I, I) where the volume of K t

{t :Volume(K

t

n (L U R»

Volume(L U R)

=

n (L U R) equals

~}. 2

For a sweepout of disjoint embedded surfaces coming from a Heegaard splitting, Y contains exactly one point. For a generic sweep out Y c is finite and contains an odd number of points. Definition A point t

E

Y is an L point if

Volume(K t n L) > Volume(K t n R) and an R point otherwise. A sweepout is an LR-sweepout if it has an odd number of L points and an RL-sweepout if it has an odd number of R points. The Heegaard splitting Eo gives rise to an LR -sweepout that begins with surfaces near a graph at the spine of H L, sweeps out L U R with embedded slices, and ends with surfaces that collapse to a graph at the spine of HR. The stabilized Heegaard splitfing Go gives rise to an LR-sweepout by embedded surfaces of genus 2g - I. A stabilization adds a loop to each of the graphs forming the spines of Eo, the two loops Qeometry & Topology, Volume 13 (2009)

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linking once in M g , and each crossing the separating surface S. Between the resulting graphs is a product region which can be tilled with Heegaard surfaces of the stabilized splitting. See Figure 3, which shows two spines and a Heegaard surface that has been stabilized once. Note that for arbitrarily large n and any J > 0, the stabilizations can be chosen so that the added area in going from the surfaces of Eo to those of Go is less than J. When Go and G 1 are equivalent, composing the Heegaard sweepout of Go with the diffeomorphisms {Is, 0:': s :': I} gives a family of genus 2g - I Heegaard sweepouts connecting Go and G 1.

i :~" "' ': : : ::~~:: : : : : : : : : : : : : : : : : ::::::::{~:: : : : ':.: : : : : ~~:: : : : : : : : : : : :::::::~fi.f) 1

[

Figure 3: Stabilized spines

Lemma 6.1 Suppose that Eo is equivalent to El after (g -I) stabilizations. Then there is a constant A o , independent of the number of blocks n, and a family of embedded surfaces

satisfying the following conditions: (1)

F has genus 2g - I.

(2) For each s, {/s,t: F --+ M g , -I < t < I} is a sweepout. (3) {fo,t (F), -I < t < I} is an LR -sweepout. (4) {fl,t(F), -I < t < I} is an RL-sweepout. (5) Each surface in the two sweepouts /o ,t (F) and /1,t (F) has area bounded above by A o.

Proof M g is formed from the union of a handlebody H L, n -blocks forming L , n-blocks forming R, and a second handlebody HR. The geometry of a single block, and of each of HL, HR, does not depend on n. Pick a spine for each of HL, HR and foliate the complement of this spine in each handlebody by embedded Heegaard surfaces, connecting the spine to the slice that forms the boundary of each handlebody. Then till L and R with interpolating slices. This Heegaard sweepout is a foliation of the complement of the two spines in M g by genus- g leaves {L t , -I < t < I}. Now stabilize by adding (g - I) loops to each spine and (g - I) handles to each surface between the two spines, giving a Heegaard sweepout of Go. Let ao > 0 and Vo > 0 Qeometry & Topology, Volume 13 (2009)

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be two constants. By adding thin handles in the stabilization, we can arrange that the area of any surface increases by less than ao and so that the volume bounded by the additional (g - I) I-handles is less than Vo. For our purpose it suffices to take

Vo = ao = min{l, Volume(B)/2}. Let F be a surface of genus 2g - I and construct maps {/o,t: F --+ M g , -I < t < I} that smoothly parametrize the stabilized surfaces. We will refer to lo,t (F) as Fo,t. Set the value of the constant Ao to be the largest area of the surfaces {Fo,t, -I < t < I}. Note that our construction gives a value for Ao that is determined by g, the area of the slices in a single block, and the geometry of H L and Hr. In particular A 0 is independent of the number of blocks n used to construct Mg. The Heegaard splitting Go = (HI, H 2 , 5) obtained by stabilizing Eo (g -I) times is equivalent to the reverse~ splitting G 1 = (H2 , HI, -5), where 5 = Fo,o and -5 indicates the orientation of S has reversed. So an isotopy Is, 0:': s :': I from the identity map 10 to a diffeomorphism II carries (HI, H 2 , 5) to (H2 , HI, -5). Construct the family of surfaces Is,t: F --+ Mg by defining

Is,t = Is

0

lo,t: F --+ Mg

and let Fs,t denote Is,t(F). For any constant 0 < a < I, we can arrange, by stretching out a collar around the invariant surface Fo,o, that II carries Fo,t to Fo,-t for each t E [-I +a, I-a]. For a sufficiently small the embedded surfaces

{Fs,t;t E (0,-1 +a]U[I-a, I)} lie in small neighborhoods of the images of the spines of Go under Is, have area uniformly bounded above by ao, and bound submanifolds having volume less than Vo. The Heegaard splitting Eo gives rise to a sweepout {L t, -I < t < I} of M g by genus- g surfaces, starting near the spine of H L and ending near the spine of HR. This sweepout foliates the complement of the two spines. The surface L t bounds a 3--{;hain K t that fills up the side containing {U Lr : t' < t}. Ko bisects the volume of L U R and contains L but not R. The surfaces L t, when parametrized, give an LR-sweepout with Volume(Ko n L) = nVolume(B), while Volume(K o n R) = O. The surface Fo,t, obtained by stabilizing L t , bounds a 3--{;hain Ko,t whose volume of intersection with Land R differs by less than Vo from that of K t . Therefore Fo,t also gives an LR -sweepout. By the same argument applied to the stabilization of E 1 , we have that F 1,t gives an RL-sweepout. All properties now follow. 0

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We now show that a path of sweepouts in M g whose surfaces have area uniformly bounded by a constant A 0 cannot start with an LR -sweepout and end with RLsweepout if n is sufficiently large. Define

VL = Volume (L) = Volume (R) so that 2VL = Volume (L U R). Lemma 6.2 Given a constant Ao there is a constant no such that if n > no and Mg is constructed with 2n blocks then there does not exist a smooth family of maps

{hs,t: F --+ M g ,

0::: s::: I,

-I < t < I}

from a surface F to Mg satisfying the following conditions: (I)

F has genus less than 2g.

(2) Foreachtixeds, {hs,t: F--+Mg -I 1 such that whenever P is a closed, unknotted, polygonal curve in ]R3 having n edges and length L , the following hold:

(1) There exists an embedded P L disk spanning P whose number of 1 (C 4 )n2 ; triangles is at most 32

(2) Such an embedded P L disk can be chosen to lie inside a ball of radius 4L , and its area then satisfies

A cos(2 arcsin a/2)

>0

over this whole range, since 0 < a S 1. The curve", itself also lies inside the cone with axis in the direction T(O) and based at ",(0). For if the curve ever left this cone, then it would also leave a cone with a slightly larger cone angle, and would exit the boundary of this slightly wider cone transversely. The tangent vector of '" at such a point would not lie within the original cone. q.e.d. We next com pare the distance of two points on a curve with specified tubular neighborhood as measured along the curve and as measured in 1R 3 Lemma 3.3. Let 0 < a < 1/2 and let", : [0, ar] -+ 1R 3 be a smooth , properly embedded unit speed C 2-curve with curvature everywhere bounded by l/r. Let e denote the line segment in 1R 3 connecting ",(0) and ",(ar). Then", is contained in the right circular cone with cone point ",(0) , axis e, and that subtends the angle 2Ba = 4arcsin(a/2) to its axis. Moreover

(length (e))2 ;:> cos( 2Ba) (length (",))2 Proof. Lemma 3.2 shows that two tangent vectors on "'( 8) differ by an angle at most 2Ba = 4arcsin(a/2), so that their inner product satisfies T(81) . T(82) ;:> cos(2Ba) > 0, using 0 < a S 1/2 in the last inequality. Let T denote the unit tangent vector pointing from ",(0) to ",(ar), so that T

442

=

1 ( ) (",(ar) - ",(0)) length e

=

1 ( ) length e

COLLECTED WORKS WITH COMMENTARY

lar 0

T(8)d8 .

AREA INEQUALITIES FOR EMBEDDED DISKS

13

Then for any 0 SiS ar, the inner product 1 [r T(s) . T(SI)ds length (e)

T· T(SI)

> >

ar length (e) cos( 2lia).

cos( 2lia)

It follows that the entire curve lies in the right circular cone with vertex at ",(0), axis the line segment e pointing in direction T, subtending angle 2lia = 4arcsin(aI2), and with its endpoint ",(ar) being the center of the base of the cone. Now, since length ("') = ar, we obtain length (e)

=

11",(ar) - ",(0)11

I[r

=

I[r

T(s) . T(sl)dsl

T(/). Cen;h(e) [r T(S)dS)

ru

r far 1 > length (e) )0 )0 IT(sl). T(s)ldsds l >

d/I

(length ("')? cos( 2lia). length (e)

This yields (length (e)? ;:> cos(2lia)(length(",)?, as asserted.

q.e.d.

In what follows, we let, denote a C 2 -curve having a normal tubular neighborhood of radius r. The existence of the tubular neighborhood for , implies that the absolute value of the curvature at each point of, is at most 1/r. (The points where the normal exponential map develops singularities are called the focal set. They occur at distance r for a curve at a point of (absolute) curvature 1/r, see [11 , p. 232].) Therefore, Lemmas 3.2 and 3.3 are applicable. We construct a particular polygonal curve pi approximating the C 2 _ curve, as follows. Let pi in lR 3 have vertices on the curve, taking n = l32L I r J +1 equally spaced points Zo, Zl, ... ,Zn-1 along, (measured by arclength of ,) and inscribing line segments connecting successive points. Let , j be the arc of, between Zj and Zj+1 and let ej = [Zj, Zj+1] be the line segment of pi between these points in lR 3 , with last segment e n -1 = [Zn-1, zo]. All arcs , j are of equal length Lin, and we define the quantity aj by ajr = length(rj) = Lin, which implies that 1/33 S aj S 1/32. We will verify below that pi is embedded in lR 3

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Let Cj denote the right circular cone with cone point Zj, axis the line segment ej, subtending the angle 4liaj = 8arcsin(aj/2). Note that Cj has base diameter no larger than 2sin(4liaj )(ajr) since length (ej) S ajr, and the center of its base is Zj+1. Lemma 3.4. Let, be a closed C 2 -curve with tubular neighborhood of radius r. In the construction above, the arc rj lies inside the closed cone Cj. Distinct cones Cj and Ck with j < k are disjoint unless j = k + 1 or j = n - 1, k = 0, and in this case, the two cones intersect in the single point Zj+l, resp. Z = 0 if j = n - 1. The polygonal curve pi is embedded in lR 3

,j

Proof. Lemma 3.3 applies to the arc to show that it is contained inside the cone C j . The cone C j subtends an angle to its axis of

4liaj

=

8 arcsin aj /2 S 8 arcsin(1/64)

< 0.14

and the base of this cone has diameter at most 2sin(4liaj)(ajr) S 2(8arcsin(1/64))r/32

< O.01r.

If two cones C j and Ck intersect, then they necessarily contain points Vj E ,j and Vk E ,k separated by distance at most (0.02)r. Let e be the line segment between Vj and Vk. We claim that since, has an r-tubular neighborhood, the shorter of the two arcs of, connecting Vj and Vk has length at most (0.03)r. To see this, consider the arc of, starting from Vj = ,(so) to a variable point w = ,(Sl), where i is an arclength parametrization of, that increases from s = So as ,(Sf) moves along the curve in one direction. The distance IIVj -,(sl)11 is initially increasing. Let ,(S2) be the first maximum point reached. At that point, the tangent vector T(s2) is orthogonal to the line [Vj, wI. and part of this line lies in the normal disk of radius r around w, so we must have IIVj - wll ;:> r. Similarly, allowing Sl to decrease from s, we find that 1I,(sl) - Vjll increases monotonically to the first point Sl, where IIVj -,(sl)11 reaches a maximum, and this maximum is at least r by the same reasoning. Now, the remaining part of the curve falls outside the normal tubular neighborhood of radius r determined by the arc {,(s) : Sl S s S S2}. This normal tubular neighborhood includes all points within distance r/4 of Vj, for if v is a point with Ilv - Vjll S r/4, then the closest point w = ,(s*) on the arc {,(s) : Sl S s S S2} to v has Ilw - vii S r/4 from it, and attains a local minimum for the distance function from v. It is at distance at least r /2> r /4 from each of the endpoints ,(Sl), ,(S2), since otherwise

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a contradiction. The local minimality implies that v-w is perpendicular to the tangent vector T(s*), so lies in the disk of radius r at w = ,(s*) normal to T(s*). We conclude that all points on the curve, , within distance r /2 of Vj must lie in that part of the curve, where IIVj -,(i) II is monotone increasing as lSi -sol increases. The remark above shows that this holds at least for lSi - sol Sr. Since IIVj - vkll < 0.02r, the point Vk must fall on this part of the curve. Now, we can apply Lemma 3.3 to observe that on taking a = 0.03, the segment e = i,(0),,(0.03r)] already satisfies (length(e))2;:> COS(2lia)(ar)2;:> cos(4arcsinO.015)(0.03r)2

> (0.02)2r2

We conclude that the shorter of the two arcs of, connecting Vj and Vk has length at most (0.03)r, as asserted. Since the length of each segment is at least r /33 > 0.03r, it follows that Cj and Ck are adjacent cones, i.e., k = j + 1 or j = n - 1, k = O. Adjacent cones however intersect in a single point, the center of the base of one and the cone point of the other. Finally, since each segment ej of pi lies in a different cone, and overlaps are only possible at endpoints of the ejs, the closed polygonal curve pi is embedded in lR 3 q.e.d.

,j

Lemmas 3.5 and 3. 6 will be used to construct a Lipschitz homeomorphism inside the tubular neighborhood that carries pi to Q. We start with a two-dimensional map. Given a two-dimensional Euclidean disk D(d) = D(O,d) of radius d centered at (0,0) and a point q = (x,y) in the disk, with Ilqll < d, let "q,d : D(d) -+ D(d) be the map that sends the origin to q, and that maps each line segment from the origin to a boundary point of D(d) linearly to the line segment from q to the same boundary point. It is explicitly given by

"q,d(W)

=

W + (l-llwll/d)q.

Lemma 3.5. The 17ULp "q,d : D(d) -+ D(d) is a Lipschitz homeomorphism with Lipschitz constant at most 1 + Ilqll/d S 2. Its inverse map ,,-~ is Lipschitz with Lipschitz constant at most 1/(1- Ulqll/d)). q, If ql and q2 lie in the disk D(d), then for any point wE D(d)

lI"q"d(W) - "q2,d(W)11 S Ilql - q211· Proof. The map" q,d is clearly a homeomorphism, which leaves the boundary of D(d) fixed. It remains to estimate a Lipschitz bound. For wE D(d), we have

"q,d(W)

=

W + (l-llqll/d)q.

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We calculate that Ilqll + d(llw111-llw211) W211(1 + Ilqll/d).

IIWI - W211

< IIWI -

Similarly II 1.

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We first smooth the braids", and ",-1 by replacing neighborhoods of their vertices by smooth arcs, getting a smooth braid (3 of thickness k > 0, and curvature bounded above by a constant Ko. We do this smoothing so that (3 still consists of vertical arcs near its endpoints. Then, there is a constant r1 > with the property that the thickness of (3 is greater than r1, and each vertical segment near an endpoint of (3 has length greater than r1. Form {3" and (3-n as before by stacking copies of (3 or (3-1. Since copies of (3 are identified along points that have neighborhoods coinciding with vertical segments, the resulting curve is Coo. Moreover, the thickness of (3n and (3-n is still greater than r1, since distinct copies of (3 have r1-tubular neighborhoods with disjoint interiors. Scaling by a homothety gives (3n = (1/3n){3", with length 1/3, and similarly (3-n. The thickness of each of (3n and (3-n is greater than rl/3n. We can also smooth the curves an, ... ,hn in neighborhoods of their internal vertices, to obtain nearby arcs a~, ... ,h~ that are smooth with arcs near the endpoints remaining as straight vertical segments, with the length of each straight segment no less than a constant r2 (which does not depend on n). The approximating arcs are chosen sufficiently close to maintain their disjointness from Cyl(ro), and with their total length remaining less than 1/3. Then, combining (3n, (3-n with a~, ... , h~ gives a closed curve Lin of length one that is isotopic to Kn in the complement of Cyl(ro), and the thickness of Lin is given by k/3n, where k is the smaller of r1, r2. We now establis_h a lower bound on the area A of any embedded spanning disk for 6 n . Lemma 5. 1 implies that there is a Co > 1 such that for large enough n, any embedded disk with boundary 6 n has area A greater than Con. Note that k < 1 since Lin has length one and thickness less than k. Thus, we can rescale n to eliminate the constant _k appearing in the thickness bound by taking a subsequence 6 n := 6[kn[+1' and setting C1 = (Co)k We then have minimal area A > (C1)n. q.e.d.

°

In the next result, we show how to deform Kn to construct a similar sequence of smooth curves r n having uniformly bounded curvature. Recall that the curvature function for a C 2-curve embedded in ]R3 is computed by taking an arclength parameterization I : [0, L] -+ ]R3, so that lI:tl(t)11

=

1,andthecurvaturefunctionisthenK(t)

=

11::2 1 (t)ll.

Proof of Theorem 1.3 . The construction of the family {rn}n~l"oo is similar to that of {6n}n~1"oo. We again use smoothings a~, ... , h~ of the arcs an, ... ,hn to curves that lie nearby them, and assert that

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the approximating curves can be chosen to have curvature at each point bounded above by a fixed constant K. To see this, we use the fact that a sequence of smooth curves converging to a smooth limit curve have curvatures converging pointwise to the curvature of the limit curve. For any curve with monotonically increasing z-component, a deformation of the curve by a diffeomorphism 9),(x, y, z) := (AX, Ay, z) carries it to a curve that, for A small, lies very close to to the z-axis. For A sufficiently small, the curvature of the image curve under g>. is uniformly close to zero. We choose An -+ 0 to be a sequence converging to 0 sufficiently fast so that the curvature of (3n = g>'n ((3n) -+ 0, where (3n is as before, and similarly, we define (3-n. We also smooth the curves an, ... , hn in neighborhoods of their internal vertices, to obtain nearby arcs a~, ... ,h~ that are smooth with arcs near the endpoints remaining straight vertical segments. While these vertical segments all converge to the z-axis as n --+ (X), the curvature of each of a~, ... ,h~ remains uniformly bounded. One way to see this explicitly is to choose smooth limiting curves a~, ... ,h~ of bounded curvature that are not embedded, but rather have arcs near their endpoints that agree with the z-axis. Any sequence of smooth curves a~, ... , h~ that converge to these curves smoothly will have uniformly bounded curvature. Then, the curvature of the curves r n actually converges to 0 at points corresponding to the braids pn, (3-n, and is uniformly bounded at other points. The curvature is zero along the straight segments where a~, ... , h~ join (3n and (3-n· Since the required area for a spanning disk goes to infinity as n -+ co, we can rechoose r n to be a suitable subsequence to have the required property A > n. This concludes the proof of Theorem 1.3. q.e.d. 6. Appendix: Isoperimetric theorems for curves in ]R3

This appendix establishes two versions of the isoperimetric theorem in]R3 as stated as (1) and (2) in Section 1. Result (1) follows from the solution of the Plateau problem, due to Douglas and Rado, [12], [26], and on a result of Carleman on isoperimetric inequalities for minimal disks [9]. See also [27]. We deduce (2) from (1), using standard cut and paste methods of 3-manifold topology, which can be used to convert an immersed surface to an embedded surface, possibly with an increase in genus, but maintaining orientability. Such a cut and paste can be achieved without increasing area by a "rounding of creases". Alternate approaches to proving (2) are possible based on work of Blaschke [5, p. 247], which was the original approach to (2). The discussion in [25, p. 1202] indicates that Blaschke's argument is heuristic, but can be

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made valid for area-minimizing immersed surfaces having the curve as boundary. Theorem 6,1, Let / be a simple closed C 2 -curve in]R3 of length L. Then, there exists an immersed disk having area A in ]R3 , with / as boundary, such that If the curve is not a circle, then there exists such an immersed disk for which strict inequality holds. Proof of Theorem 6.1. Span / by a least area disk D, whose existence is guaranteed for rectifiable / by the solution of the Plateau problem [12], [26]. The regularity results of Osserman [24 ] and Gulliver [15] show that D is a smooth immersion in its interior. (The interior of D may transversely intersect /, as indeed it must if / is knotted.) The argument in [27] (see also [9]) shows that 471"A g. PROOF: We know already that the state of

A after input of w is

atail for [w).

IT [w] = xg, where 9 E D , then from 5.2,A IS MONOTONE it follows that SA(W) SA(SA(X),g))- SA(g) = g. Therefore, SA(W) is a maximal tail in D, as claimed.

=

/5.5,A FINDS MAXIMAL DELTA FACTOR/

§6.

CANONICAL FORMS

Since the output and the final state of the transducer A' depend only on the element of P , and not on the particular word representing that element, this leads to a way to define canonical words representing elements of P . We have seen that [w] can be rewritten

, where the two factors depend only on the element of P represented by w. We apply this process recursively to the left. factor, obtaining a factorization

[w] = g1g 2 ... gm , when:: each {Ji is in D , and where gl< is t he maximal dement of D which satisfies 9192 ... gk lgk ·

Figure 6.1. Positive canonical form. left-greedy cano nical form

This figure illU3irate3 an element of A m

PROPOSITION 6 .2. LEFT AND RIGHT CANCELLATION. The semigroup P satisfies the left and right cancellation laws, that is, if ab = acthen b c and if ba = co then b = c.

=

PROOF: It suffices to prove this in the case that a is a generator. Since each generator gk satisfies gk --< 6. and also 6. l- gk, it will su.+fice to show that cancellation holds for left and right multiplication by 6..

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Cancellation for right multiplication by fl. is trivial: given two elements b and c, represent them as words Wb and We in canonical form . Then wbfl. and wefl. are still in canonical form, so they are equal only if they are identical; but if they are identical, so are Wb and We·

Left multiplication by fl. is almost as trivial. One method is to observe that P is isomorphic to the reversal of P, that is, the group where the multiplications ab = c are replaced by ba = c instead. If P has right cancellation, then its reversal has left cancellation. Here is another, "direct" proof along the lines of the proof for right multiplication. For any word w, define its flip tV to be obtained by flipping it over, that is, replacing each gl: with gn-I:. Flipping is a semi group automorphism, and it preserves canonical form. For an element 9 E D, define g* to be the unique element of D such that gg* = fl.. In Sn, the corresponding operation is a* = -,a- 1 a-I fl. . From the identity (a·)" = fl.- 1afl. in the symmetric group , we see that (g*)" = g, so that

=

gg* = gOg = fl..

It follows that if gl . .. gm is .a word in canonical form, then fl.g 1 ... gm

=

gliigl . .. gm = g1 fl.g 2 .. . gm

=[h92 ... 9mfl.. This is in canonical form, and it contains the information of the original word, so it follows that if lib .6.c then b c. 1 6.2 ,LEFT AND RIGHT CANCELLATION I

=

=

The semigroup P has the directed system property: that for any two two elements p and q there is a..'1 element r such that r >- p a..'1d r >- q. In fact, the element r may be taken as a power of fl.: it is easy to see how to premultiply any element by something to make it a power of fl., using the two operations above. From this property, along with left and right cancellation, it follows t hat P has a unique embedding in a group P which it generates: geometrically, these three properties guarantee that in the Cayley graph for P, large neighborhoods of vertices far in the "interior" look as if they are large neighborhoods in the Cayley graph of a group. There is a directed system, made up of the Cayley graph for P based at various vertices x. Whenever x >- y, there is a map of the graph based at y to the graph based at x, obtained by prefixing w where wy = x. Left ca.."1cellation says that these maps are injective. Right cancellation says that each vertex has at most one incoming edge for each generator, so that in the direct limit, each vertex has exactly one incoming and one outgoing edge for each generator. This implies that the maps are surjective onto larger and larger neighborhoods, as one goes far out in the directed system. It follows that the direct limit labeled graph is homogeneous, so it is the Cayley graph of a group P. The uniqueness of a group containing and generated by P follows from the fact that arbitrarily large neighborhoods of the Cayley graph in such a group look like neighborhoods of elements in the Cayley graph of P . Version 1.1

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THEOREM 6.3. GROUP COMPLETION OF P IS THE BRAID GROUP. The group completion is the braid group Bn. P is the positive braid semigroup.

P of the semigroup P

PROOF: Since the presentation of the semigroup P was the presentation for the braid group Bn (when regarded as a presentation for a group), it follows that P satisfies the relations of B n , so it is a quotient of Bn. Therefore, the map of Pinto Bn is an embeddin . It follows that Bn = P. 6.3,GROUP COMPLETION OF P IS THE BRAID GROUP §7. RECOGNIZING CANONICAL FORM

BY

A FINITE STATE AUTOMATON

THEOREM 7.1. CANONICAL FORMS ARE REGULAR. The set of words in canonical form in the positive braid semigroup is a regular set. PROOF: We will show how to construct a finite state a.utoma.ton which tells whether or not a word is in canonical form. Actually, it is more convenient to work with a left-greeedy canonical form, which is backward from the right-greedy canonical form we used above. We will show later that the two questions are equivalent. A word W is in left-greedy canonical form if it can be expressed as the product W

= WjW2 .. • Wm

where each Wi E D, Wi is the canonical representative of its element in D (that is, high numbered strands make their moves as late as possible), and if Wi is the longest head of W;Wi+1 •.• Wm such that Wi -( D.. We will construct an automaton B which will ['-"cognize left-greedy ca..TlOnlcal form. The first problem in constructing B is to determine the boundaries between the subwords Wi. This part is easy. At any time, the automaton will keep track of the part of Wi which it has seen so far. When a new generator gi is input, B determines whether p(gi)-l flp(Wi) = 1, or equivalently, whether the two strands involved have already crossed within Wi. If they have crossed, the new generator can only be part of the next word Wi+1, while if not, it must be part of Wi. While B is still working on Wi, it can check whether the current word is the canonical representative for its element of D, since there are only finitely many possibilities. In addition, B keeps track of whether p(wi-d fI P(Wi)-l = 1, that is, whether any crossings in Wi could have heen made earlier in Wi-I. To do this, B can remember Wi-1 until it encounters the boundary of Wi+1 . Clearly the criteria recogrized by B are necessary for W to be in canonical form. We need to show also that they are sufficient. This can be done by backwards induction, on the tails Wi = Wi ... 11.7". Obviously, Wn is in canonical form . Suppose that W i +1 is in canonical form, and consider Wi = Wi TVi + 1 . The inductive hypothesis implies, in particular, that the state of A after reading lVi + j backwards is the reverse of Wi+1. From 5.3,ACTION ON INPUT OF D, it follows that the state of A reading Wi backwards is the reverse of Wi, as required. If R is any regular set, then the set R' consisting of strings s' obtained from strings s E R by reading them backwards is also regular. This is easy to see when R is described

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by a regular expressions: the rules of formation are symmetric on interchange of left and right, and the reverse regular expression, with corrections for writing unary operators in postfix rather than prefix notation, describes the reverse set of strings. Since every regular set is described by a regular expression, the assertion follows. The operation of reversing order in the string is not quite so immediate in terms of finite state automata, however: it can be done by first turning a FSA into what is called a non-deterministic FSA - it keeps track of all possibilities for the state that a FSA reading a word backward might have been in when it reached the current point. It follows that there is a FSA B' which recognizes right-greedy canonical form. We avoided the direct construction, because the recognition of sub word boundaries does not seem simple. One way to construct B' is to let it keep track of the implications of several hypotheses as to the chain of sub word boundaries. Any new hypothesis as to a new subword boundary, is consistent with at most one chain of previous hypotheses. Subword boundaries must come at least every n( n-1 )/2 generators, and the information propagated in any hypothetical chain is similar to the information propagated by B, that is, the set of pairs of adjacent strands at the previous hypothetical subword boundary which had crossed in the previous hypothetical subword, so B' requires a finite (but much larger) 17.1,CANONICAL FORMS ARE REGULAR memory.

I

§8. SYMMETRIC CANONICAL FORM

Let N denote the negative braid semigroup N = P-l. Every element b of the braid group can be expressed as a product b = np, wit h n E N and p E· P . This is equivalent to the fp.ct that there are positive braids q which have arbitrarily large neighborhoods in the Cayley graph of Bn consisting of positive braids: write l; = q-lqb, .for any sufficient q such that qb is positive. It is also easy to do this by hand: given a braid, if you grasp the section between the top and any negative twist, and give it a negative half-twist , the negative twist is replaced by a positive word, a t the expense of introducing a negative half-twist at the top . THEOREM 8.1. MINIMAL FACTORIZATION. There is a unique factorization b = np where n and p have minima11ength, with n E N and pEP. The set of words of tbe form uv represen ting minimal factorizations is regular, where u is a negative word and v is a positive word. When p is represented as a word in righ t-greedy form, and n is in left-greedy form (the inverse of a word in righ t-greedy form), the resulting word U! is upjque, and tbe set of such w is regular. PROOF: There is an FSA A-I which will recognize the largest element of D-l which is a tail of n = [ul, and there is also a FSA Arwhich will recognize the largest element of D which is a head of p = [v]. (The second FSA comes from the fact that the reversal of any regular set is regular: A r accepts the reversal of any string accepted by A.) We can make an FSA E which runs A-I as long as its input is negative, and then runs A r as long as the input is positive, remembering all the while the final state of A-I. If E ever sees a negative generator after it has seen the first positive generator, it goes to the fail state. It also goes to the fail state if there is ever any cancellation between the largest Version 1.1

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tail in D-l it found for u and the largest head in D it has established for v. Otherwise, it accepts uv. To prove that the factorization b = 71p is unique, we will make use of the canonical form as described in the second paragraph of the statement: write b = [w] = 711 ... 71kPi" . PI,

where each 71i E D-l and Pi E D, and where (i) any pair of stra..'1ds adjacent at the end of 71j which have crossed in 71i also cross in 71i+l, for i < k (ii) no pair of strands adjacent at the end of 71,1: cross in both 71k and in Pi, and (iii) any pair of strands adjacent at the end of Pi which cross in Pi-l also cross in Pi, for i > O. Of course, these conditions can be alternatively expressed in terms of the lattice operations - but this description seems more immediate. A word meeting these conditions is in canonical mixed form. The idea is that crossings are pushed toward the negative-positive boundary. Associated to any word w in canorucal mixed form is another word

p( w) =

ffiJ. ..• mkPj ... PI

tn,

representing Ll.k[w], where [mil = LI.(Ll.k-i[71;}LI. - .(k-.i)) a.nd E D . (Note that the conjugate of 71; by a power of LI. only depends on the parity of the power, and this operation takes D-l to itself; therefore t he result is positive and in D). The pairs of adjacent strands at the end of mi which have crossed in mi are precisely those at the end of LI. k-injLl. -(k-i) which did not cross there. The pairs of strands adjaceni at the beginning of mi+l which cross in mi+l are precisely those which do not cross in Ll.k-i71i+1D.-(k-i). Therefore, the left-greedy condition on the 71i translates to a right-greedy condition on the mi. (Another way to say this is that mi a..Tld mi+l are either t:,.ni and 71i+1Ll., or they are nit:,. and t:,.ni+l. In either case, the ieft-greedy condition on the ni is complemented to the rightgreedy condition on the mi.) Simjlarly, the pairs of strands which are adjacent at the end of mk which crossed in mk are the compiement of those wruch crossed in 71k . Since those which cross in nk are disjoint from those wrDch cross in Ph those which cross in mk contain those which cross in Pi' Therefore, the word pC w) is in left-greedy canonical form. There is an inverse process, provided we specify also the number of subwords to be converted to negative. Thai is, given a positive word x in left-greedy canonical form and an integer k, there is associated a. word Sk(X) in canonical mixed form representing t:,.kx, as follows: If k is positive, prefix x by the given power of t:,. . If k is negative but does not exceed the number of subwords of x, convert k of these subwords negative sub words by reversing the procedure above. If k exceeds the number of subwords of x, the extra LI.'s are suffixed to the end. By reasoning as in the previous paragraphs, the resulting words are in canonical mixed form. Version 1.1

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It easily follows that if WI and W2 are in canonical Jnixed form, and if [WIJ = [W2J, then WI = W2. For, if WI and W2 have respectively ki andk2 negative terms, set k = max(k!, k2 ) and consider the positive words in left-greedy form =.lk-kip(Wi) representing .lk[WiJ. k2 and p(wd = p(wz), so WI = S-k(P( W2» = Wz. Since left-greedy form is unique, k = kl IS.1,MINIMAL FACTORIZATION

=

I

Tbe braid group is a lattice witb respect to tbe partial ordering b -< c if tbere is a positive braid p sucb tbat bp = c. Tbat is, for any two braids bI and bz, there is a unique greatest braid c = bI /\ bz such tbat c -< bi and c -< bz, and there is a unique least braid d sucb tbat bi -< d and b2 -< d. Tbe partial ordering -< and tbe lattice structure is invariant under multiplication on tbe left: bi -< b2 if and only if cb I -< bz , and COROLLARY 8.2. BRAID GROUP IS LATTICE .

c( bi

/\

bz) = (cb l )

/\

(cb z )

c(bi V b2 ) = (cbI) V (cbz) .

Note that -< is an extension of the previously-defined partial ordering of the same name on P , coming from the head reiation. PROOF : Given bI and bz , consider an arbitrary expression b;-lb2 = np as the product of a negative times a positive braid. Let d = bi n = bzp-I. Then d -< bl = dn- I and d -< bz = dp. Put n in left-greedy caIlonical form, and p in right-greedy ca..'1onical form. If there is any cancellation in the middle, this gives rise to an element d -< e such th".t e -< bi. Since cancellation short.ens the wordlength, it can only occur a finite number of times. When there is no cancellation, then np is in canonical mixed form: in this case, d is the unique ma..ximal element meeting the inequalities d -< bi. The join operation /\ is similar, using a positive-negative canonical form, rather than negative-positive. braid group is lattice

I

I

§9. THE BRAID GROUP IS AUTOMATIC Any element of En can be expressed as a product of a power (positive or negative) of 6 wi t h a positive braid. To phrase this a different way, any braid becomes a positive braid when it is multiplied by a sufficiently large power of 6. This follows from our description of the group completion P in terms of P, and it is also easy to see directly. Therefore, we can define a canonical word representing an element bEEn to be an expression b = WIU'Z ... Wm, which is either a left-greedy canonical form for an element of the positive braid seruigroup, or else a sequence of canonical words representing .l-l followed by a left-greedy canonical form for an element of the positive braid seruigroup which is 6-free. The existence and uniqueness of this representation for b follows from the existence and uniqueness of the representation for t1 N b, where N is a large positive power, and from the fact that the leftgreedy canonical form changes only by prefixing ~ when an element of P is premultiplied by.l. This canonical form is also recognized by a finite state automaton C. Version 1.1

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THEOREM 9.1. LEFT-GREEDY AUTOMATION.

The braid group has an automatic structure

wbose regular set of words is tbe set of words in Jeft-greedy canonical form. PROOF: A finite state automaton to test equality is trivial, since two words in Rc represent the same element of En if and only if they are identical. We must construct n - 1 finite state automata, C ll ... ,Cn - I , which read pairs (w, x) of words in Rc synchronously letter by letter, where Ci accepts (w, x) if and only if x = wg;. Consider the process of adjusting a word wg; into left-greedy canonical form, assuming that W WI •.. Wm is already in canonical form. For definiteness, suppose that W is a positive word. If Wm cannot accept the crossing of the ith and i + 1st strands and still remain in D, then the product is already left-greedy canonical. Otherwise, "om can absorb gi, so the first approximation is to set the new Wm = W m 9;, rewritten appropriately. However, Wm-l may now be able to accept a crossing from Wm: this will happen if the ith and i + 1st strands were adjacent at the beginning of W m , and if they had 'not crossed in Wm - I. Once Wm-l accepts such a crossing, it can happen that it is now able to accept additional crossings from W m , since new pairs of strands are now adjacent at the sub word boundary which may have crossed in w"" but not in Wm-I ... an entire cascade is triggered, as crossings are ha.'1ded do'w n th~ line toward the earlier subwords, possibly allowing new crossings to flow into a certain subword and freeing others which were jammed . . ',' (But see the discussion after theorem 9.2,RIGHT-GREEDY AUTOMATION for a more precise description of the actual process of adjustment .) Despite this discouraging vision of ar. uncontrollable landslide, there is a bound to ihe fu."Uount cf material which can flow past any subword boundary. One way to see tills is to consideD"the efi'ect of first postmultiplying by gi, and then postmultiplyingby gi, so that the net effect is that of postmultiplying b y ~. If we postmultiply by ~ in one step, the process is easy to trace:

=

WI ..• Wn-I wnD.

'Wn-IWn(W~W~)

=

WI"

= =

Wl'''W n-,I(WW*)W* n n n WI . .. Wn-lD.W~

wi flows past each subword boundary. It follows that only an element can flow past any sub word boundary on postmultiplication by any generator (or for that matter, any element of D). If w begins with negative powers of ~, the phenomenon is similar, except that on postmultiplication by Ll, the first positive subword is promoted to ~ whereupon it is cancelled with the preceding negative power. The number of subwords is first increased by 1, then decreased by 2 for a net decrease of 1. It follows that if W and x are in left-greedy canonical form, -where W = xgj, and if W k and Xk denote the initial segments of length k, the set of group elements X;IWk In this case, exactly 9

-< wi

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is bounded: for if instead we took segments cut off at subword boundaries which correspond under the cascade process, the group elements would be in D. The actual truncations differ from some pair of corresponding sub word boundaries 'by a bounded distance. 19.1 ,LEFT-GREEDY AUTOMATION I We can define a similar right-greedy canonical form for Bn, by allowing negative powers of.t. at the tail. Let C I be a finite state automaton which recognizes right-greedy canonical form, and define ReI to be the set of words in right-greedy form. THEOREM 9.2. RIGHT-GREEDY AUTOMATION. The braid group has an automatic structure whose regular set is the set of words in right-greedy canonical form. PROOF: The proof is similar to the prooffor left-greedy canonical form, but simpler. When a positive word W = WIWZ ••• Wm in right-greedy canonical form is multiplied on the right by a generator, there is a cascade effect whose progress is easier to follow. The final generator gi must be absorbed by the last subword w m • This may force it to give up some positive Om -< W m , which is forcefed to W m -l. Inductively, a word Ok -< Wk is released, and so 01 O. The compooition of CW -lipschitz maps is CW-lipschitz, and the CW-lipschitz constants multiply. If the CW complexes happen to be simplicial complexes, such that each isimplex is isometric with the convex hull of the standard basis in R,+I, and if the map f is piecewise linear, then the lipschitz property in the conventional sense, together with the cellular property (suitably reinterpreted so as to apply to simplexes rather than to cells), clearly implies (and is stronger than) the property of being CW -lipschitz.

r:

We say that a CW complex is locally bounded if, for each n, there is a constant k n such that every open n-cell is contained in a subcomplex with at moot kn cells. All the CW complexes we work with will have this property, since each will arise by taking a covering space of an appropriate finite CW complex. We need to discuss lipschitz homotopies of maps. It is no good requiring the time parameter of the homotopy to vary over the unit interval, because then no space of infinite diameter could be lipschitz contractible. We need to move a point x to the basepoint in a time which is something like the distance of x to the basepoint. So We parametrize the homotopy over [0,(0), and insist that locally the homotopy becomes stationary after a finite time, and that this finite time varies in a lipschitz way. We now formalize this idea in the CW context.

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We make R into a CW complex by placing a O-cC'll at each integer point, with a I-cell connecting each adjacent pair of integers. We can make [0,00) into a C\V complex in a similar manner.

Definition 2.3. (CW-lipschitz contraction). Let X be a CW subcomplex of the CW complex Y. Let F: X x [0,00) _ Y be a CW-lipschitz map, where the usual product CW structure is taken on the domain. We say that F is a CW-lipschitz contraction of X in Y if the following conditions are satisfied. a) There is a sequence (k n ) of constants and a (not necessarily continuous)

function h; Y ----I [0,00), which we call the duration function of the homotopy, with the properties listed below. b) F(x, 0) is equal to the basepoint, for each x E X.

0) F(x, t)

~

x fo, t:> h(x),

d) If L is a connected finite subcomplex of Y of dimension n, then h(L) is contained in an interval with integral endpoints and with length ILl ko. e) If L is a finite n-dimensional connected subcomplex of Y x [0,00), then F(X x [0, 00) n L) is contained in a connected subcomplex of Y with at most ILI~'" cells. f) The restriction of F to the n-skeleton of its domain is k,,-CW-lipschitz. (Note that this does not imply the previous condition, because L can be connected, while X x [0, 00) n L is not connected.) We do not assume that F, preserves the basepoint for each t. To understand Conditions d) and e), think of a path-metric space Y, with a subset X which is connected by rectifiable paths. Our conditions are analogous to asking for a lipschitz contraction of X in Y, where we use not the intrinsic path metric on X, but the metric induced from Y. A connected I-dimensional CW complex can be metrized by giving each 1· ~ ell length 1 in the obvious way, and then using the path metric. The CW-diam.eter of a subset S of a connected CW complex is found by taking a connected subcomplex which contains 5, and then taking the diameter, in the metric sense, of the I-skeleton of this complex. There are usually many different ways of doing this, and we choose a way that gives a minimal answer. Also, the CW-diameter could turn out to be a half integer. In that case, we

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increase it by a half so that it becomes an integer. If S is connected, then the smallest sub complex I( containing S is connected and the CW -diameter of S is equal to the diameter of 1(1. A cellular i-chain (or just a chain) of X is a formal integral linear combination of cells of dimension i. The CW-volume of a (cellular) chain is defined to be the sum of the absolute values of the integer coefficients. The CW-diamder of a chain is the smallest diameter of any connected subcomplex containing the chain. The boundary homomorphism is defined by thinking of a cellular chain as an element of H,{X', X;-1), and then using the usual homology boundary of a triple. When we talk of the n-sphere, we mean the CW complex with one cell in dimension 0 and one cell in dimension n. When we talk of the n-disk, we mean the standard CW complex with one cell in dimension 0, one cell in dimension n - 1 and one cell in dimension n.

3

Combable groups

Let G be a group with a finite set of generators. The Cayley graph r of G with respect to these generators can be identified with the l-skeleton of the universal cover k of the complex ]{ described immediately after Definition 1.1 (combinatorial area). Definition 3.1. (combable). We say that G is combable if there is a CWlipschitz contraction of Gin r. Here G is identified with the O-skeleton of r == k 1 • This definition arises from the theory of automatic groups, where the automatic structure determines the combing. Thus automatic implies combable. Open Question 3.2. We believe that there exist combable groups which are not synchronously automatic, but we do not know of any examples. Theorem 3.3. (volume times distance estimate). Let G oe 4 combu,ble group. There exists 4 CW complex ]() with universal cover k, having the following properties. a) The n-skdeton of I( is finite for each n.

b) /, ;, a K(G, 1).

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c) There is a CW-lipschitz contraction of i{ to its basepoint, and the contraction keeps the basepoint fixed. d) For each n > 0 there is a constant a" > 0, such that any cellular map f from the n-sphere to i{ extends to a cellular map f' of the (n + 1) -disk, with voio+,(f') $ aodiam(f(SO))voioU). e) For each n > 0 there is a constant b" > 0, such that for any cellular ncycle z, there is a cellular (n+ 1 )-chain c, such that bc = z andvol n+l (c) ::; bndiam( z )vol( z). The above conditions hold for any J((G, 1), such that each skeleton is finite. Proof of (3.3): The proof of this theorem and of Lemmas 3.5 (lipschitz bounds) and 3.7 (lipschitz extension) will be by induction on a positive integer which we denote by n. These lemmas should be assumed to hold for all values of n less than the one currently under consideration. In addition, we assume that all the lemmas which occur earlier in the text hold for the value of n under consideration. The induction starts with the contraction F: i{0 x [0,00) ----+ i{l given by the hypotheses. We are given the duration function h on i{l. We first change h so that it is more convenient to use. We increase h on each vertex, so that its value is an integer. We then redefine h on each open l-cell, to be the larger of the two values of h at its endpoints. Since the homotopy is only defined on the O-skeleton, all the conditions for a CW-lipschitz contraction of i{0 in i{l continue to hold. All future extensions of h will also have the property that the value of h on an i-open cell (i ~ 1) is equal to the largest value of h on the smallest subcomplex containing the image of the attaching map of the cell, and we will regard h as defined, without further comment. We next change the given contraction F: i{0 x [0,00) ----+ i{l so that it fixes the basepoint *. This has the effect of increasing the CW-lipschitz constant by at most h(*). (See Definition 2.3d to understand why the constant may increase.) We may now change the definition of the duration function so that h(.)~O.

3.4. (Some induction assumptions). Now suppose that the n-skeleton of J( (n 2': 1) has been constructed and that we have inductively a kn_1-CWlipschitz contraction F: i{n-l x [0,00) ----+ i{n which fixes the basepoint. We

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further suppose that each attaching map S' j(n is kn_1-lipschitz.

--t

j{' in the CW structure of

We first prove the following lemma. Lemma 3.5. (lipschitz bounds). Let n ~ 1. Given 0: > 0, there. exists [3 = fin(O:) > with the following property. Let f,g:sn,. --t j(n,. be 0:CW-lipschitz maps. Ifn = 1, f and g are assumed to be equal in 1I'1(j(1). If n> 1, -f and 9 are assumed to define equal cellular n-chains. Then there is a [3-CW-lipschitz map F: sn x I,. x I --t j(n,. which is a homotopy bettoeen f and g. Proof of (S.5): Kn is a finite complex, which is therefore compact. It follows that the covering space j(n is locally compact, and there is a nmnber b such that the closure of any cell in j(n has a neighbourhood meeting at most b cells. Let L be the union of all connected subcomplexal of j(n containing the basepoint, and containing at most 0: open cells. By induction on 0:, this means that ILl ~ b". In particular, L is a finite complex. If X is a CW complex, let ex denote the cone on X, with the obvious CW structure. We have the maps

°

L2...L/Ln- 1 = S" V ... V sn~L U CLn-l~j(n where k is an inverse homotopy equivalence to the projection p: LUCLn-l --t L/Ln-\ j = plL, ulL is the inclusion, while ulCLn-l is induced by the kn_l-CW-lipschitz contraction of the (n -i)-skeleton to a point (assmned to exist by induction). Let H be a homotopy between ukj and the inclusion i: L --t j(n. In the commutative diagram

,"L

1

'If

nj( ..

~

• --=...

the maps between homology groups are all injective. Let f',g': S .. --t L be defined by if' = f and ig' = g. From the commutative diagram above, the maps f' and / have the same image in Hn(L, L ..- 1 ). It follows from the Hurewicz Isomorphism Theorem that, if n > 1, 1'f' ~19: "5" ,.--t

5" V ... V 5" , •.

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r ::

If n = 1, the same result is true. In fact, in that case, 9 1 , as we see by lifting to maps of the unit interval into the universal cover of [\1, The maps and j9 1 are both a-CW-lipschitz maps and they each have a standard geometric form, namely there are at most 0: disjoint closed round n-disks in the domain the complement of these disks is mapped to the basepoint, apd each of the disks is mapped by a euclidean similarity to the unit disk Dn, followed by the standard identification Dn --+ Dn/sn-l = sn with one of the n-spheres in the range. It follows that there is an a-CW-lipschitz map F:S n xI,* xl --t snV ... vS n ,*, which is a homotopy betweenjjl and J9 1•

jr

sn,

We get a homotopy between f = it and 9 = ig' by combining the homotopy between if and ukj the homotopy ukF between ukj and ukjg' and the homotopy Hg' between ukjg' and ig'. We choose I so that H and uk are "(-CW-lipschitz. Then each of the three homotopies is O'''(-CW-lipschitz. The composition of these homotopies is therefore 30'''(-CW-lipschitz. We set

H

r

{3

~

t,

t

3,.

Our next task is to construct the (n + 1 )-cells of g. If n = 1, we proceed as follows. We metrize j(t by assigning length 1 to each edge in the obvious way, and then using the path metric. For each non-constant loop in j(l. based at *, parametrized by path length and travelling at unit speed, without turning except at integer values of the parameter, and of length at most 41.0 • we create one new 2-cell in g. To define the attaching map of the boundary circle of this 2-cell, we reparametrize linearly to make the time parameter of the loop in j(l vary over [0,211'"], and we then project to gl. Definition 3.6. (construction of cells). If n > 1, we follow a different procedure. Since j{n is (n -l)-connected, the Hurewicz theorem tells us that 'ltn(I(n) ~ Hn(kn), provided n > 1. Now Hn{k n) consists of the n-cycles in the group of cellular n-chains, Let "( = Pn-lk!_t, where Pn-l is the constant which we know inductively from Lemma 3.7 (lipschitz extension). In view of the fact that j{n is the covering space of a finite CW complex, maps ·sn,. - t j{n,. with CW-lipschitz constant 0'", = 6"( + 6k~_1 can give rise to only a finite number of distinct cellular n-chains, and hence to only a finite number of distinct homotopy classes. We define the (n + l)-cells of K as follows. For each of our finite set of elements of 'ltn(I(n), we choose a

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representative with CW-lipschitz constant a". We obtain an attaching map into J(" by projecting. Lemma 3.7. (lipschitz extension). There is a constant p" with the following property. Let f: S", * --f k", * be a cellular map with n-dimensional CW-volume v and CW-diameter d and let it have a CW·lipschitz constant a. Then there is an extension to a cellular map f: D"+1 _ K,,+l with (n + 1). volume bounded by p" vd, and with CW-lipschitz constant p"a l • Proof 0/ (9.1): Let L be the smallest subcomplex of k" containing ItS"). By assumption the diameter of L1 is d. It follo","" that any two O-cells in L are joined by a sequence of at most d I-cells. Therefore the duration function for the CW-lipschitz contraction of k n- 1 in kn is bounded by kod on the l-skeleton of L. Our construction of the duration function ensures that it is bounded throughout L by kod. We will describe a basepoint preserving homotopy of maps G t : S",. _

K,,+l, *. which takes place over the interval [0, kodj, starts with the constant map of S" to the basepoint of j(n+l, and ends with the given map /. This homotopy enables us to extend f to D"+ I, by using the identification D"+l ~ S" x I IS" x O. We will discuss the associated constants below. Definition 2.1 (cellular) gives us v disjoint closed round n-disks in sn, such that the inverse images under f of the open n-cells of j(n consists of the interiors of these disks, and each disk is mapped in by a euclidean similarity with nn followed by a characteristic map. On the complement X in S" of the interiors of these round disks, we will use the map f followed by the given contraction F,,-t; Kn-1 x [0, kodl_ [(". Let D be the closure of one of the small round n disks in 5"\X and let S be its boundary. We give D a temporary CW structure as an n-cell with the usual three open cells of dimensions 0, n -1 and n, to enable us to use the induction hypothesis. By 3.4 (Some induction assumptions), tiS is k,,_I-CWlipschitz. Also F"-t is k"_l-CW-lipschitz, so the composition is k~_t-CW­ lipschitz. Using Lemma 3.7 (lipschitz extension), which, by induction has already been proved in dimension'n - I, we find a constant.., = p,,-tk!_t, I such that, for each integer i (0 < i :$; kod), 0 tiS can be extended to a )'-lipschitz map of D into kn. We also have F,,-tlk,,-t x Ii, i + 110 «/IS) x 1) for 0 :$; i < kod, which is a k!_t-CW-lipschitz map. We want to define G on

r.-

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D.B.A. EPSTEIN, W.P. THURSTON

[i,i+ 1[.

So far, we have defined G on the boundary sphere B, of D x [i, i + 1] and this sphere has dimension n. We want to use Lemma 3.5 to construct a homotopy to the attaching map of one of the (n + 1)-cells of Definition 3.6 (construction of cells). But we have to take care, as the current CW structure of Bi contains S cells (two of dimension 0, one of dimension 1, two of dimension n - 1 and three of dimension n), while Lemma 3.5 uses a CW structure ·on Sf> with two cells (one of dimension 0 and one of dimension n). The total number of cells in the smallest subcomplex containing the image of B, is bounded by 6, + 6k~_1' The first term is a bound for the subcomplex containing the images of D x ti} and D x {i + I} (recall that D has three cells). The second term comes from S x [i, i + 1] which contains six cells. The volume of G1B. is bounded by k~_l + 2)'. It follows that, if we change the CW structure of Bi to one with only two cells, then it is (6)' + 6k!_t)-CW-lipschitz. (We have not tried to achieve the best possible constants.) When we regard Bi as a CW complex with two cells (changing its CW structure), it is (6)' + 6k!_1)-CW-lipschitz. Let p = {3n(6)' + 6k!_1) be given as in Lemma 3.5 (lipschitz bounds). Then there is p-CW-lipschitz homotopy of G1B i to an attaching map of one of the (n + I)-cells in j(n+t. The homotopy takes place in [(n. We extend G to the interior of D x [i, i + 1] by using the homotopy near the boundary and then the characteristic map of the (n + 1)-cell. This completes the construction of G. If we choose the constant Pn 2: ko, then the statement in Lemma 3.7 (lipschitz extension) about the {n + I)-volume is clearly satisfied. The CWlipschitz constant of the extension of f to D n+ 1 is computed from the total number of cells in the image and the (n + 1)-volume. We have just computed the volume and it remains to compute the total number of cells.

The interval [0, kodJ contains 2kod+ 1 cells. Also sn contains two cells. Therefore the image of X x [0, kodJ in j(n-t X [0, kodJ is contained in a subcomplex containing no more than 2a(2kod + 1) cells, and so the image in j(n contains no more than 2k n_ t a(2k od + i) cells. The image of each intD x i consists of at most 3)' cells in [(no Here"'f is the constant defined in the course of the current proof, a few paragraphs back, and the coefficient 3 occurs because D contains 3 cells. Therefore the total number of cells in the image of Dn+t in

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i{n is bounded by 2k"_,a(2kod + 1)

+ 3(kod + I)V1 + kodv(6~ + 1),

where the first term bounds the number of cells in the image of X x (0, kodJ, the second term bounds the number of cells in the images of the cells of the form D x til, where i is an integer, and the third term bounds the number of cells in the image of the interiors of the (n + I)-cells. The coefficient 6 in the third term arises from the fact that the !3-CW-lipschitz homotopy given by Lemma 3.5 is defined on 5 n- 1 x I, which contains 6 cells. Since a sphere contains two cells, we have v oc:::: 2a and d oc:::: 2a. Since!3 and '"'f are constants, we can cho~e Pn so that the CW-lipschitz constant of the extension is bounded by Pn Q2 as required. Lemma 3.B. There is a CW-lipschitz contraction Fn of k" in extends the given contraction on j'{n_l.

kn+\

which

Proof of (S.8): We need to define Ft" on each n-cell. On the boundary of the n-cell, the map is already given as the attaching map followed by Ft"-1, and this is a k~_1-CW-lipschitz map of 5"-1 x [O,kod] to i{n. For each integer i, we define F,n : Dn-1 --> 1(n using Lemma 3.7 (lipschitz extension). The extension is p"k!_1-CW-lipschitz.

Let a" be as in Definition 3,6 (construction of cells). The map of sn into k" which we get from the attaching map of an (n + l)-cell of k" x (0,00), followed by the part of the contraction which is already defined, is a,,-CWlipschitz. According to Lemma 3.5 (lipschitz bounds) we can extend to the interior of the (n + I)-cell by means of a !3,,(a)-CW-lipschitz homotopy near the boundary of the cell and a standard characteristic map away from the boundary. This clearly means that there is a constant k" such that Fn is k,,-CW-lipschitz.

To prove part e) of Theorem 3.3 (volume times distance estimate), we translate i{ so that the basepoint is contained in the support of z. The duration function is bounded by a constant times the diameter of z on the support of z. The contraction of !{" gives the required chain c.

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To prove the last statement of Theorem 3.3, let L be another I\( G, 1), with every skeleton finite, and let l\. be the l\(G, 1) constructed above. Then there is a basepoint preserving homotopy equivalence between Land 1\, and the homotopy equivalence u and its inverse v can be made cellular. Also the basepoint preserving homotopies of uv and Vtl to the respective identity maps can be made cellular. The induced maps and homotopies between the universal covers are all CW-lipsc~itz. The required results in L then follow easily from the known results in l\..

The results of Theorem 3.3 (volume times distance estimate) also have an expression in terms of free Z( G)-resolutions of Z by finitely generated free Z(G)-modules. We omit the precise formulation of this result. The following is the most important corollary of the above theorem. If the reader is only interested in this special result, the proof can be simplified quite a lot. In particular, special arguments can replace Lemma 3.5 (lipschitz bounds), because I-dimensional CW complexes have such a simple structure. Corollary 3.9. (quadratic isoperimetric inequality). Let G be an automatic group with generators k Then there is a constant k > I with the property that if w is a loop in the Cayley graph of length n then it bounds a disk of combinatorial area at most kn'l. J

J.M.Alonso has proved independently that a comb able group has a free Gresolution which is finitely generated in each dimension. Related results have been proved by KS.Brown and D.Anick. Open Question 3.10. If G is a comb able group, is there a contractible space on which G acts properly discontinuously with compact quotient? The above theorem is a powerful tool for proving that various groups are not combable, and hence not automatic. For example, it can be applied to show that standard 3-dimensional nilpotent and solvable groups are not combable. It can also be used to show that SL(3,Z) is not combable. All these results can be proved with the I-dimensional isoperimetric inequality. The higher dimensional isoperimetric inequalities are required to prove that SL(n, Z) is not combable for n 2: 4. The proofs have not yet been written down, and some details remain to be filled in.

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Amongst other important matters omitted from this paper is a discussion of the properties of n-connected spaces on which a combable group acts properly discontinuously. The theory goes through in this case in almost the same way as the case we have discussed.

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J. W. CANNON, W. J. FLOYD, M. A. GRAYSON AND W. P. THURSTON

SOLVGROUPS ARE NOT ALMOST CONVEX·

ABSTRAcr. We show that no cocompact discrete group based on solvgeometry, Sol, is almost convex. This reflects the geometry of Sol, and implies that the Cayley graph of a cocompact

discrete group based on Sol cannot be efficiently constructed by finitely many local replacement rules.

1.

INTRODUCTION

We show that no cocompact discrete group based on solvgeometry, SoL is almost convex. Almost-convexity is a metric property satisfied by all cocompact hyperbolic groups, all Euclidean groups, all free products with amalgamation of finite groups, all HNN extensions of finite groups, and all small cancellation groups [1]. Intuition suggests that it should be satisfied by those cocompact groups based on geometries having unique shortest geodesics and convex metric balls. Such is certainly true in geometries having sectional curvatures.;; O. Therefore the property is likely to apply to braid groups, mapping class groups, complex hyperbolic groups, groups of higher rank symmetric spaces whose factors have convex metric balls, etc. It is likely to apply to nilgroups as well, whose metric balls, though not convex, are almost convex. The property of almost-convexity is necessary and sufficient in order that the Cayley graph of a group be efficiently constructible by means of finitely many local replacements rules [2]. There is some hope that many groups which are not almost convex can nevertheless be studied by means of such local replacement rules if one is willing to embed them in larger groups or graphs which are almost convex. In particular, solvgeometry Sol embeds in complex hyperbolic space. Local replacement rules seem closely related to rational growth functions in groups. Nevertheless, a number of cocompact groups based on solvgeometry have rational growth functions. This was first claimed in Chapter 4 of M. A. Grayson's thesis [3], but there was an error in his proof. The error was discovered by W. Parry and was corrected in [4]. Our result shows how clearly the combinatorial structure of a geometric group mirrors the properties of the geometry on which it is based: shortest geodesics in Sol are highly non unique. Our result has significance in the study of 3-manifolds and their groups. W. P. Thurston has conjectured that each

*This research was supported in part by NSF Research Grants. Geometriae Dedicata 31: 291-300, 1989. © 1989 Kluwer Academic Publishers. Printed in the Netherlands.

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low-dimensional manifold (dimension", 3) admits a unique geometric structure [6]. Clearly any package of decision algorithms designed to compute within the fundamental groups of low-dimensional manifolds and orbifolds must be able to deal with the groups from each of the standard geometries. Thurston's conjecture would imply that a package capable of dealing with that class of groups would be sufficient for the study of 3-manifold groups. Our examples show that such a package must have capabilities beyond those of local replacement rules. Perhaps the package should take into account such things as normality, or ....

2.

REVIEW OF CAVLEV GRAPHS

Let G denote a group with finite generating set C = C- 1 • Let r = r( G, C) = (V, L, E) denote the associated Cayley graph: V = G, the vertex set of r, L = C, the label set of r, E = {(v, c, vc) I v E V, eEL}, the set of directed edges of r; the two directed edges e = (v, c, vc) and e- 1 = (vc, c- 1 , v) define the same undirected edge. We view each undirected edge geometrically, as a homeomorph of the unit interval. There is a natural action of G on the graph r given by left multiplication: ¢:G x

¢(g, v)

r-.r; =

gv,

¢(g,(v,c, vc))

9 E G, v E V; =

(gv,c,gvc),

9 E G,(v,c,vc) E E.

We may parametrize each edge by means of the unit interval in such a way that this action by left multiplication respects these parametrizations. The parametrizations may then be used to define a path metric d on r which assigns to each undirected edge of r the length 1 and which is invariant under

¢. For each integer n, let B(n) denote the closed ball of radius n in r centered at the identity of G: B(n)

=

(x

Er I d(x,id) '" n}.

Let S(n) denote the sphere of radius n in ofG: S(n)

=

(x

Er I d(x,id) =

r

centered at the identity vertex

n}.

We call r almost convex (k) (written AC(k)) if there is an integer N(k) having

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the following property: if x, y E S(n)(any n) and d(x, y) .;;; k, then x and yare joined in B(n) by a path of length.;;; N(k). We call r almost convex (Ae) if r is AC(k) for every k. Theorem 1.3 of [1] shows that if ris AC(2), then it is AC. It is only property AC(2) that is used in any of our algorithms involving AC groups [1]. But in studying the soIvgroups, we shall show directly only that they do not satisfy AC(k) for k large. It follows that they do not satisfy AC(2). Hence they do not admit simple local decision algorithms. 3. REVIEW OF SOLVGEOMETRY We use [5] as our basic reference. We review the basic structures on solvgeometry, Sol. Topological structure: Sol, as a space, coincides with Euclidean 3-space, the set of ordered triples (x, y, z) with the product topology. Group structure: A group multiplication is defined on Sol by the formula (a,b,c)·(x,y,z) = (a + x·exp(-c),b + y-exp(c),c + z). The inverse of the element (a, b, c) is the element (-a·exp(c), -b-exp(-c), -c). Riemannian structure: If one chooses the standard Euclidean metric as the Riemannian metric at the origin 0 = (0,0,0) and transports this metric to other points via left multiplication, then one obtains a metric that is left-invariant under the group structure on Sol. This metric is ds 2 = exp(2z) dx 2 + exp( -2z) dy2 +dz 2. Isometry group: The group Sol is the identity component of the isometry group Isom (Sol) of Sol. It is normal of index 8 in the entire isometry group. The isotropy subgroup D(4) at the origin is dihedral of order 8 and consists of the eight elements R(e"e 2 ) and S(e"e 2),e, and e2 equal to 1 or -1, R(e" e2 )(x, y, z) = (e, • x, e2 • y, z)

and S(e" e2Xx,y, z) = (e, -y, e2 ·x, -z).

The elements of the isotropy subgroup D(4) serve as left coset representatives for Sol in Isom(Sol). Sol has a normal subgroup E2 = {(x, y, 0) Ix, y real}. This subgroup is Abelian and acts on itself as a translation group. Any discrete subgroup of E2 is isomorphic either with id, Z, or Z ® Z. We shall analyze multiplication in Sol more completely at this point. First we examine the general finite product p. = (a"b"c,) ... (a.,b.,c.).

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The coordinates of Pk are the following: k

x(Pk)

=

L {aioexp[O -

C1 -

••• -

Ci-1]};

i=l k

y(Pk) =

L

{bioexp[O

+ C1 + ... + ci-d};

i= 1

k

z(Pk ) =

L Ci' i=l

In terms of these formulae we may easily calculate powers. The coordinates of Qk = (a, b, C)k are the following: k

X(Qk)

=

ao

L

{exp[ -(i -l)c]};

i= 1

k

y(Qk) = b

o

L {exp[(i -l)c]}; i=l

The coordinates of Rk = (a, b, C)-k are the following: k

X(Rk)

L {exp[(i -l)c]};

= -aoexp(c)

i= 1 4

y(R k )

= -boexp(-c)L {exp[-(i-l)c]}; i= 1

And finally we can calculate conjugates by powers. The coordinates of Sk = (a, b, C)k ° (x, y, 0) ° (a, b, c) - k are the following:

X(Sk)

= x exp( - kc);

y(Sk)

= yexp(kc);

Z(Sk)

= O.

The coordinates of Tk

= (a, b, c)- k ° (x', y', 0) ° (a, b, C)k are the following:

X(Tk) = x' exp(kc);

= y' exp( z(Tk ) = o.

y(Tk )

kc);

Now we analyze the individual element of Isom(SoI) in terms of Sol and D(4).

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We may use the elements of D(4) as coset representatives for Sol in Isom(SoI). Thus each element g of Isom(SoI) has a unique factorization

g

=

d(g)' s(g),

where d(g) is an element of D(4) and s(g) is an element of Sol. An arbitrary product of the form

h = d,'s, ... d,'s, may be rewritten as a product of conjugates of elements of Sol and an element of of D(4):

Sd, ... d"(d 1'" d) Sd"1 Sd"d, 2 .•. k k' Since D(4) fixes the origin and since Sol is normal in Isom(SoI), we find that

That is, we may find the image vertex "'(h) = h(O) by calculating a particular product of elements of Sol, namely a product of conjugates of the elements

4.

COCOMPACT GROUPS OF ISOM(SOL)

Preparations In preparation for a proof that no cocompact solvgroup G is almost convex, we need to describe two subgroups A and B of G, a set C· of elements associated with any finite generating set C for G, an efficient element (a, b, c) of A, and two particular nontrivial elements of B. We suppose that Gis a cocompact subgroup ofIsom(Sol). We associate with this subgroup the two subgroups, A, the intersection of G with Sol, and B, the intersection of G with E2. We first note that A is a normal subgroup of G of index": 8. It acts cocompactly and properly discontinuously on Sol. We next note that B is a normal subgroup of G which acts cocompactly and properly discontinuously on the xy-plane z = 0 (this is proved in [5, pp. 471-472]). We fix a finite generating set C for G. We look at the set C· = {S(W)d

I WE C, d E D(4)}.

We note that C· is a finite subset of Sol since C and D(4) are finite. By the calculations at the end of Section 3, every element of G that can be expressed as a product of k elements of C can be expressed as a product of k elements of C· and an element of D(4). The elements of C· are not necessarily elements of G, but we will use them to estimate the position of the "'-image of vertices of r in Sol.

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We define an element (a, b, c) of A to be z-eflicient if the z-coordinate c of (a, b, c) is equal to the C-Iength of (a, b, c) times the largest z-coordinate of an element of C'. If S(W)d is an element of C· with largest z-coordinate, then one shows easily that w· is a z-eflicient element of A. We fix a particular z-efficient element s(W)d of C· with largest z-coordinate p, and let (a, b, c) = w·. Note that c = g.p. Since B acts cocompactly on E2 , it is possible to find elements (x, y, 0) and (x', y, 0) of B such that x' and yare both strictly greater than O. We are now ready to proceed to the proof that r = r(G, C) is not almost convex.

Proof that Cocompact Solvgroups are Not AC

We assume that group G, finite generating set C, subgroups A and B, subset C· of Sol, z-eflicient element (a, b, c) = w· for some WE C, and special elements (x, y, 0) and (x', y', 0) of B are given as in Section 4, y > 0, x' > O. For each large positive integer k we form the two products Sk and r. as in Section 3: Sk = (a, b, cJ" .(x, y,O) •(a, b, C)-k

and Tk = (a, b, C)-k·(X', y', 0) '(a, b, cJ'.

By the calculations of Section 3, Sk = (x·exp(-kc),y·exp(kc),O)

and Tk = (x' 'exp(kchY"exp( - kc),O).

In particular, Sk and Tk both lie in the free Abelian group B and hence commute with one another; that is, Sk' Tk = Tk 'Sk' We shorten these two products a fixed finite amount on the right: "'k = Sk Tk '(a, b, c)-1

and

Pk =

TkS .. (a, b, c)l.

Since (a, b, c) has length g in terms of the generators C, it follows that "'k and Pk lie within 16j of one another in r. Furthermore, ifj .. k then each of "'k and

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p, lies

297

within

32k - 8j + length(x, y, 0) of the identity element of G in each of ex, and p, lies within

r. If j

+ length(x', y', 0) is fairly large, then we may assume that

32k - 8J of the identity element of G in r, where J is a large but fixed positive integer (much smaller than k) which depends only on the fixed numbers j, length (x, y,O), and length (x', y', 0). Assume now that r is almost convex. Then thereis an integer N(16j) which depends only on j such that for every k there is a path from ex, to p, in r of length less than or equalto N(16j) every vertex of which lies within 32k - 8J of the identity element of G in r. The image ofthis path in Sol must contain some vertex image which lies within cj8 of the plane E2 . We obtain a contradiction by showing that every vertex u which lies within 32k - 8J of the identity element of G in r and whose image .p( u) lies within cj8 of the plane E2 is in fact very far from S,' T, and hence very far from both ex, and p,. What we in fact show is that the image vertex .p(u) is at very large Euclidean distance from the image vertex of S,' T,. We take therefore an arbitrary product of length (';;)32k - 8J of elements of C and assume the image vertex lies within cj8 of E2 The image vertex can be expressed as a product oflength (.;; )32k - 8J of elements of C· as the calculations at the end of Section 3 show. We set K = 32k - 8J. Then the product of K elements of C· has precisely the form P J( considered in Section 3: K

x(PJ() =

L:

{a,exp[O - C1 - ... - Ci-1J)

i=l K

y(PJ() =

L:

{b,exp[O + C1 +

... + Ci-1Jl·

i=l

K

z(PJ() =

L:

{cil.

i= 1

The sums appearing within the exponentials for y(P J() are either positive for less than half (half = 16k - 4J = Kj2) of the indices i or are negative for less that half of the indices i. We make the calculations only in the former case; the other case is exactly analogous. We estimate in the former case the y-coordi-

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nate y(P K)' Let b* denote the largest y-coordinate occurring in any of the elements of C*. Let c* denote the largest z-coordinate occurring in any of the elements of C*. Recall that c* = c/8, where (a, b, c) is z-efficient. Since the sums appearing within the exponentials are positive for less than half of the indices i, the maximum possible value for y(PK ) occurs when the sums are 0 for half of the indices, c* = c/8 for one index, 2c* , 3c* , ... , mc* for two indices each, m = K/4, and each bi is the largest y-coordinate appearing in any element of C*. Thus IY(PK)I.;; b*·{(KI2) = b*' {(KI2)

.;; M{K

+ 2[exp(c*) + exp(2c*) + ... + exp(mc*)]} + 2[exp(mc* + CO) - exp(c*)]/[exp(c*) - I]}

+ exp(mc* + CO)},

where M is a constant that depends only on b* and c*. It is in terms of M, y, and c or c* that we are able to decide exactly what value to pick for J. We require that the number

y - M/[exp«J - 2)c/4)] be positive. Since y is positive and the term on the right goes to zero as

J approaches infinity, such a choice of J is always possible. Recall now that K = 32k - 8J, that m = 8k - 2J, and that c* = c/8. We can now compare the y-coordinates of Sk' Tk and P K'

Iy(Sk'Tk)1 - IY(PK) ;;. lY'exp(kc)1 -Iy' 'exp( - kc)1 - M{(32k - 8J)

+ exp[(8k -

2J

+ 1)c/8]}

= {y - M/[exp«J - 2)c/4)]}exp(kc)

- Iy' 'exp( - kc)1 - 32Mk

+ 8MJ.

The first term is exponeI).tial in k with positive coefficient, the second term is negligible for k large, and the final two terms are linear in k. Hence the expression approaches infinity as k approaches infinity. This proves that r is not almost convex, as we wished to show. 6.

EXAMPLES

Perhaps the simplest examples of cocompact groups based on solvgeometry are the groups G, =