Spherical CR Geometry And Dehn Surgery

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  • Format: Paperback
  • Copyright: 2007-01-29
  • Publisher: Princeton Univ Pr

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This book proves an analogue of William Thurston's celebrated hyperbolic Dehn surgery theorem in the context of complex hyperbolic discrete groups, and then derives two main geometric consequences from it. The first is the construction of large numbers of closed real hyperbolic 3-manifolds which bound complex hyperbolic orbifolds--the only known examples of closed manifolds that simultaneously have these two kinds of geometric structures. The second is a complete understanding of the structure of complex hyperbolic reflection triangle groups in cases where the angle is small. In an accessible and straightforward manner, Richard Evan Schwartz also presents a large amount of useful information on complex hyperbolic geometry and discrete groups. Schwartz relies on elementary proofs and avoids quotations of preexisting technical material as much as possible. For this reason, this book will benefit graduate students seeking entry into this emerging area of research, as well as researchers in allied fields such as Kleinian groups and CR geometry.

Table of Contents

Prefacep. xi
Basic Materialp. 1
Introductionp. 3
Dehn Filling and Thurston's Theoremp. 3
Definition of a Horotube Groupp. 3
The Horotube Surgery Theoremp. 4
Reflection Triangle Groupsp. 6
Spherical CR Structuresp. 7
The Goldman-Parker Conjecturep. 9
Organizational Notesp. 10
Rank-One Geometryp. 12
Real Hyperbolic Geometryp. 12
Complex Hyperbolic Geometryp. 13
The Siegel Domain and Heisenberg Spacep. 16
The Heisenberg Contact Formp. 19
Some Invariant Functionsp. 20
Some Geometric Objectsp. 21
Topological Generalitiesp. 23
The Hausdorff Topologyp. 23
Singular Models and Spinesp. 24
A Transversality Resultp. 25
Discrete Groupsp. 27
Geometric Structuresp. 28
Orbifold Fundamental Groupsp. 29
Orbifolds with Boundaryp. 30
Reflection Triangle Groupsp. 32
The Real Hyperbolic Casep. 32
The Action on the Unit Tangent Bundlep. 33
Fuchsian Triangle Groupsp. 33
Complex Hyperbolic Trianglesp. 35
The Representation Spacep. 37
The Ideal Casep. 37
Heuristic Discussion of Geometric Fillingp. 41
A Dictionaryp. 41
The Tree Examplep. 42
Hyperbolic Case: Before Fillingp. 44
Hyperbolic Case: After Fillingp. 45
Spherical CR Case: Before Fillingp. 47
Spherical CR Case: After Fillingp. 48
The Tree Example Revisitedp. 49
Proof of the HSTp. 51
Extending Horotube Functionsp. 53
Statement of Resultsp. 53
Proof of the Extension Lemmap. 54
Proof of the Auxiliary Lemmap. 55
Transplanting Horotube Functionsp. 56
Statement of Resultsp. 56
A Toy Casep. 56
Proof of the Transplant Lemmap. 59
The Local Surgery Formulap. 61
Statement of Resultsp. 61
The Canonical Markingp. 62
The Homeomorphismp. 63
The Surgery Formulap. 64
Horotube Assignmentsp. 66
Basic Definitionsp. 66
The Main Resultp. 67
Corollariesp. 69
Constructing the Boundary Complexp. 72
Statement of Resultsp. 72
Proof of the Structure Lemmap. 73
Proof of the Horotube Assignment Lemmap. 75
Extending to the Insidep. 78
Statement of Resultsp. 78
Proof of the Transversality Lemmap. 79
Proof of the Local Structure Lemmap. 81
Proof of the Compatibility Lemmap. 82
Proof of the Finiteness Lemmap. 83
Machinery for Proving Discretenessp. 85
Chapter Overviewp. 85
Simple Complexesp. 86
Chunksp. 86
Geometric Equivalence Relationsp. 87
Alignment by a Simple Complexp. 88
Proof of the HSTp. 91
The Unperturbed Casep. 91
The Perturbed Casep. 92
Defining the Chunksp. 94
The Discreteness Proofp. 96
The Surgery Formulap. 97
Horotube Group Structurep. 97
Proof of Theorem 1.11p. 99
Dealing with Ellipticsp. 100
The Applications103
The Convergence Lemmasp. 105
Statement of Resultsp. 105
Preliminary Lemmasp. 106
Proof of the Convergence Lemma Ip. 107
Proof of the Convergence Lemma IIp. 108
Proof of the Convergence Lemma IIIp. 111
Cusp Flexibilityp. 113
Statement of Resultsp. 113
A Quick Dimension Countp. 114
Constructing The Diamond Groupsp. 114
The Analytic Diskp. 115
Proof of the Cusp Flexibility Lemmap. 116
The Multiplicity of the Trace Mapp. 118
CR Surgery on the Whitehead Link Complementp. 121
Trace Neighborhoodsp. 121
Applying the HSTp. 122
Covers of the Whitehead Link Complementp. 124
Polygons and Alternating Pathsp. 124
Identifying the Cuspsp. 125
Traceful Elementsp. 126
Taking Rootsp. 127
Applying the HSTp. 128
Small-Angle Triangle Groupsp. 131
Characterizing the Representation Spacep. 131
Discretenessp. 132
Horotube Group Structurep. 132
Topological Conjugacyp. 133
Structure of Ideal Triangle Groupsp. 137
Some Spherical CR Geometryp. 139
Parabolic R-Conesp. 139
Parabolic R-Spheresp. 139
Parabolic Elevation Mapsp. 140
A Normality Conditionp. 141
Using Normalityp. 142
The Golden Triangle Groupp. 144
Main Constructionp. 144
The Proof modulo Technical Lemmasp. 145
Proof of the Horocusp Lemmap. 148
Proof of the Intersection Lemmap. 150
Proof of the Monotone Lemmap. 151
Proof of The Shrinking Lemmap. 154
The Manifold at Infinityp. 156
A Model for the Fundamental Domainp. 156
A Model for the Regular Setp. 160
A Model for the Quotientp. 162
Identification with the Modelp. 164
The Groups near the Critical Valuep. 165
More Spherical CR Geometryp. 165
Main Constructionp. 167
Horotube Group Structurep. 169
The Loxodromic Normality Conditionp. 170
The Groups far from the Critical Valuep. 176
Discussion of Parametersp. 176
The Clifford Torus Picturep. 176
The Horotube Group Structurep. 177
Bibliographyp. 181
Indexp. 185
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