Table Of Contentmonograph 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
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Includes bibliographical references and index.
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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
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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
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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
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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
