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Spherical CR Geometry and Dehn Surgery (AM-165)
Richard Evan Schwartz

Book Description
Chapter 1 [in PDF format]

TABLE OF CONTENTS:

Preface xi

PART 1. BASIC MATERIAL 1

Chapter 1. Introduction 3
1.1 Dehn Filling and Thurston's Theorem 3
1.2 Definition of a Horotube Group 3
1.3 The Horotube Surgery 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

Chapter 2. Rank-One Geometry 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 Invariant Functions 20
2.6 Some Geometric Objects 21

Chapter 3. 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 Fundamental Groups 29
3.7 Orbifolds with Boundary 30

Chapter 4. Reflection Triangle Groups 32
4.1 The Real Hyperbolic Case 32
4.2 The Action on the Unit 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

Chapter 5. Heuristic Discussion of Geometric Filling 41
5.1 A Dictionary 41
5.2 The Tree Example 42
5.3 Hyperbolic Case: Before Filling 44
5.4 Hyperbolic Case: After Filling 45
5.5 Spherical CR Case: Before Filling 47
5.6 Spherical CR Case: After Filling 48
5.7 The Tree Example Revisited 49

PART 2. PROOF OF THE HST 51

Chapter 6. Extending Horotube Functions 53
6.1 Statement of Results 53
6.2 Proof of the Extension Lemma 54
6.3 Proof of the Auxiliary Lemma 55

Chapter 7. Transplanting Horotube Functions 56
7.1 Statement of Results 56
7.2 A Toy Case 56
7.3 Proof of the Transplant Lemma 59

Chapter 8. The Local Surgery Formula 61
8.1 Statement of Results 61
8.2 The Canonical Marking 62
8.3 The Homeomorphism 63
8.4 The Surgery Formula 64

Chapter 9. Horotube Assignments 66
9.1 Basic Definitions 66
9.2 The Main Result 67
9.3 Corollaries 69

Chapter 10. Constructing the Boundary Complex 72
10.1 Statement of Results 72
10.2 Proof of the Structure Lemma 73
10.3 Proof of the Horotube Assignment Lemma 75

Chapter 11. Extending to the Inside 78
11.1 Statement of Results 78
11.2 Proof of the Transversality Lemma 79
11.3 Proof of the Local Structure Lemma 81
11.4 Proof of the Compatibility Lemma 82
11.5 Proof of the Finiteness Lemma 83

Chapter 12. Machinery for Proving Discreteness 85
12.1 Chapter Overview 85
12.2 Simple Complexes 86
12.3 Chunks 86
12.4 Geometric Equivalence Relations 87
12.5 Alignment by a Simple Complex 88

Chapter 13. Proof of the HST 91
13.1 The Unperturbed Case 91
13.2 The Perturbed Case 92
13.3 Defining the Chunks 94
13.4 The Discreteness Proof 96
13.5 The Surgery Formula 97
13.6 Horotube Group Structure 97
13.7 Proof of Theorem 1.11 99
13.8 Dealing with Elliptics 100

PART 3. THE APPLICATIONS 103

Chapter 14. The Convergence Lemmas 105
14.1 Statement of Results 105
14.2 Preliminary Lemmas 106
14.3 Proof of the Convergence Lemma I 107
14.4 Proof of the Convergence Lemma II 108
14.5 Proof of the Convergence Lemma III 111

Chapter 15. Cusp Flexibility 113
15.1 Statement of Results 113
15.2 A Quick Dimension Count 114
15.3 Constructing The Diamond Groups 114
15.4 The Analytic Disk 115
15.5 Proof of the Cusp Flexibility Lemma 116
15.6 The Multiplicity of the Trace Map 118

Chapter 16. CR Surgery on the Whitehead Link Complement 121
16.1 Trace Neighborhoods 121
16.2 Applying the HST 122

Chapter 17. Covers of the Whitehead Link Complement 124
17.1 Polygons and Alternating Paths 124
17.2 Identifying the Cusps 125
17.3 Traceful Elements 126
17.4 Taking Roots 127
17.5 Applying the HST 128

Chapter 18. Small-Angle Triangle Groups 131
18.1 Characterizing the Representation Space 131
18.2 Discreteness 132
18.3 Horotube Group Structure 132
18.4 Topological Conjugacy 133

PART 4. STRUCTURE OF IDEAL TRIANGLE GROUPS 137

Chapter 19. Some Spherical 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 Using Normality 142

Chapter 20. The Golden Triangle Group 144
20.1 Main Construction 144
20.2 The Proof 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 Shrinking Lemma 154

Chapter 21. The Manifold at Infinity 156
21.1 A Model for the Fundamental Domain 156
21.2 A Model for the Regular Set 160
21.3 A Model for the Quotient 162
21.4 Identification with the Model 164

Chapter 22. The Groups near the Critical Value 165
22.1 More Spherical CR Geometry 165
22.2 Main Construction 167
22.3 Horotube Group Structure 169
22.4 The Loxodromic Normality Condition 170

Chapter 23. The Groups far from the Critical Value 176
23.1 Discussion of Parameters 176
23.2 The Clifford Torus Picture 176
23.3 The Horotube Group Structure 177

Bibliography 181
Index 185

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File created: 4/21/2017

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