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monograph August 20, 2006 Annals of Mathematics Studies 165 monograph August 20, 2006 monograph August 20, 2006 Spherical CR Geometry and Dehn Surgery Richard Evan Schwartz PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD monograph August 20, 2006 Copyright c 2007 by Princeton University Press (cid:13) PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,New Jersey 08540 IntheUnitedKingdom: PrincetonUniversityPress,3MarketPlace,Wood- stock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-PublicationData (this section to be filled in later) (later) p. cm. — (later) Includes bibliographical references and index. ISBN: (later) (later) (later) 5360.7—dc22 2004066029 British Library Cataloging-in-PublicationData is available This book has been composed in FONT? in LATEX The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed. Printed on acid-free paper. ∞ pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 monograph August 20, 2006 I dedicate this monograph to my daughters, Lucina and Lilith. monograph August 20, 2006 monograph August 20, 2006 Contents Preface xi PART 1. BASIC MATERIAL 1 Chapter1. Introduction 3 1.1 Dehn Filling and Thurston’s Theorem 3 1.2 Definition of a HorotubeGroup 3 1.3 The HorotubeSurgery Theorem 4 1.4 Reflection Triangle Groups 6 1.5 Spherical CR Structures 7 1.6 The Goldman-Parker Conjecture 9 1.7 Organizational Notes 10 Chapter2. Rank-OneGeometry 12 2.1 Real Hyperbolic Geometry 12 2.2 Complex Hyperbolic Geometry 13 2.3 The Siegel Domain and Heisenberg Space 16 2.4 The Heisenberg Contact Form 19 2.5 Some InvariantFunctions 20 2.6 Some Geometric Objects 21 Chapter3. Topological Generalities 23 3.1 The Hausdorff Topology 23 3.2 Singular Models and Spines 24 3.3 A Transversality Result 25 3.4 Discrete Groups 27 3.5 Geometric Structures 28 3.6 Orbifold FundamentalGroups 29 3.7 Orbifolds with Boundary 30 Chapter4. Reflection Triangle Groups 32 4.1 The Real HyperbolicCase 32 4.2 The Action on theUnit Tangent Bundle 33 4.3 Fuchsian Triangle Groups 33 4.4 Complex Hyperbolic Triangles 35 4.5 The Representation Space 37 4.6 The Ideal Case 37 monograph August 20, 2006 viii CONTENTS Chapter5. Heuristic Discussion of Geometric Filling 41 5.1 A Dictionary 41 5.2 TheTree Example 42 5.3 HyperbolicCase: Before Filling 44 5.4 HyperbolicCase: After Filling 45 5.5 Spherical CR Case: Before Filling 47 5.6 Spherical CR Case: After Filling 48 5.7 TheTree ExampleRevisited 49 PART 2. PROOF OF THE HST 51 Chapter6. Extending Horotube Functions 54 6.1 Statement of Results 54 6.2 Proof of the Extension Lemma 55 6.3 Proof of the Auxiliary Lemma 56 Chapter7. Transplanting Horotube Functions 57 7.1 Statement of Results 57 7.2 A Toy Case 57 7.3 Proof of the Transplant Lemma 60 Chapter8. The Local Surgery Formula 62 8.1 Statement of Results 62 8.2 TheCanonical Marking 63 8.3 TheHomeomorphism 64 8.4 TheSurgery Formula 65 Chapter9. Horotube Assignments 67 9.1 Basic Definitions 67 9.2 TheMain Result 68 9.3 Corollaries 70 Chapter10. Constructing theBoundary Complex 73 10.1 Statement of Results 73 10.2 Proof of the StructureLemma 74 10.3 Proof of the HorotubeAssignment Lemma 76 Chapter11. Extendingto theInside 79 11.1 Statement of Results 79 11.2 Proof of the Transversality Lemma 80 11.3 Proof of the Local StructureLemma 82 11.4 Proof of the Compatibility Lemma 83 11.5 Proof of the Finiteness Lemma 84 Chapter12. Machinery for Proving Discreteness 86 12.1 ChapterOverview 86 12.2 SimpleComplexes 87 monograph August 20, 2006 CONTENTS ix 12.3 Chunks 87 12.4 Geometric EquivalenceRelations 88 12.5 Alignment by a Simple Complex 89 Chapter13. Proof of theHST 92 13.1 The UnperturbedCase 92 13.2 The Perturbed Case 93 13.3 DefiningtheChunks 95 13.4 The Discreteness Proof 97 13.5 The Surgery Formula 97 13.6 HorotubeGroup Structure 98 13.7 Proof of Theorem 1.11 100 13.8 Dealing with Elliptics 101 PART 3. THE APPLICATIONS 103 Chapter14. The Convergence Lemmas 106 14.1 Statement of Results 106 14.2 Preliminary Lemmas 107 14.3 Proof of theConvergence Lemma I 108 14.4 Proof of theConvergence Lemma II 109 14.5 Proof of theConvergence Lemma III 112 Chapter15. Cusp Flexibility 114 15.1 Statement of Results 114 15.2 A Quick Dimension Count 115 15.3 Constructing The Diamond Groups 115 15.4 The AnalyticDisk 116 15.5 Proof of theCusp Flexibility Lemma 117 15.6 The Multiplicity of the Trace Map 119 Chapter16. CR Surgery on theWhitehead Link Complement 122 16.1 Trace Neighborhoods 122 16.2 ApplyingtheHST 123 Chapter17. Covers of theWhiteheadLink Complement 125 17.1 Polygons and Alternating Paths 125 17.2 IdentifyingtheCusps 126 17.3 Traceful Elements 127 17.4 Taking Roots 128 17.5 ApplyingtheHST 129 Chapter18. Small-Angle Triangle Groups 132 18.1 Characterizing the Representation Space 132 18.2 Discreteness 133 18.3 HorotubeGroup Structure 133 18.4 Topological Conjugacy 134 monograph August 20, 2006 x CONTENTS PART 4. STRUCTURE OF IDEAL TRIANGLE GROUPS 137 Chapter19. SomeSpherical CR Geometry 139 19.1 Parabolic R-Cones 139 19.2 Parabolic R-Spheres 139 19.3 Parabolic Elevation Maps 140 19.4 A Normality Condition 141 19.5 UsingNormality 142 Chapter20. The Golden Triangle Group 144 20.1 Main Construction 144 20.2 TheProof modulo Technical Lemmas 145 20.3 Proof of the Horocusp Lemma 148 20.4 Proof of the Intersection Lemma 150 20.5 Proof of the Monotone Lemma 151 20.6 Proof of The ShrinkingLemma 154 Chapter21. The Manifold at Infinity 156 21.1 A Model for theFundamentalDomain 156 21.2 A Model for theRegular Set 160 21.3 A Model for theQuotient 162 21.4 Identification with theModel 164 Chapter22. The Groups neartheCritical Value 165 22.1 More Spherical CR Geometry 165 22.2 Main Construction 167 22.3 HorotubeGroup Structure 169 22.4 TheLoxodromic Normality Condition 170 Chapter23. The Groups far from the Critical Value 176 23.1 Discussion of Parameters 176 23.2 TheClifford Torus Picture 176 23.3 TheHorotube Group Structure 177 Bibliography 181 Index 185