Spherical CR Geometry And Dehn Surgery

by Schwartz, Richard Evan

• ISBN13: 9780691128108
• ISBN10: 0691128103
• Format: Paperback
• Copyright: 2007-01-29
• Publisher: Princeton Univ Pr

Summary

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

Preface	p. xi
Basic Material	p. 1
Introduction	p. 3
Dehn Filling and Thurston's Theorem	p. 3
Definition of a Horotube Group	p. 3
The Horotube Surgery Theorem	p. 4
Reflection Triangle Groups	p. 6
Spherical CR Structures	p. 7
The Goldman-Parker Conjecture	p. 9
Organizational Notes	p. 10
Rank-One Geometry	p. 12
Real Hyperbolic Geometry	p. 12
Complex Hyperbolic Geometry	p. 13
The Siegel Domain and Heisenberg Space	p. 16
The Heisenberg Contact Form	p. 19
Some Invariant Functions	p. 20
Some Geometric Objects	p. 21
Topological Generalities	p. 23
The Hausdorff Topology	p. 23
Singular Models and Spines	p. 24
A Transversality Result	p. 25
Discrete Groups	p. 27
Geometric Structures	p. 28
Orbifold Fundamental Groups	p. 29
Orbifolds with Boundary	p. 30
Reflection Triangle Groups	p. 32
The Real Hyperbolic Case	p. 32
The Action on the Unit Tangent Bundle	p. 33
Fuchsian Triangle Groups	p. 33
Complex Hyperbolic Triangles	p. 35
The Representation Space	p. 37
The Ideal Case	p. 37
Heuristic Discussion of Geometric Filling	p. 41
A Dictionary	p. 41
The Tree Example	p. 42
Hyperbolic Case: Before Filling	p. 44
Hyperbolic Case: After Filling	p. 45
Spherical CR Case: Before Filling	p. 47
Spherical CR Case: After Filling	p. 48
The Tree Example Revisited	p. 49
Proof of the HST	p. 51
Extending Horotube Functions	p. 53
Statement of Results	p. 53
Proof of the Extension Lemma	p. 54
Proof of the Auxiliary Lemma	p. 55
Transplanting Horotube Functions	p. 56
Statement of Results	p. 56
A Toy Case	p. 56
Proof of the Transplant Lemma	p. 59
The Local Surgery Formula	p. 61
Statement of Results	p. 61
The Canonical Marking	p. 62
The Homeomorphism	p. 63
The Surgery Formula	p. 64
Horotube Assignments	p. 66
Basic Definitions	p. 66
The Main Result	p. 67
Corollaries	p. 69
Constructing the Boundary Complex	p. 72
Statement of Results	p. 72
Proof of the Structure Lemma	p. 73
Proof of the Horotube Assignment Lemma	p. 75
Extending to the Inside	p. 78
Statement of Results	p. 78
Proof of the Transversality Lemma	p. 79
Proof of the Local Structure Lemma	p. 81
Proof of the Compatibility Lemma	p. 82
Proof of the Finiteness Lemma	p. 83
Machinery for Proving Discreteness	p. 85
Chapter Overview	p. 85
Simple Complexes	p. 86
Chunks	p. 86
Geometric Equivalence Relations	p. 87
Alignment by a Simple Complex	p. 88
Proof of the HST	p. 91
The Unperturbed Case	p. 91
The Perturbed Case	p. 92
Defining the Chunks	p. 94
The Discreteness Proof	p. 96
The Surgery Formula	p. 97
Horotube Group Structure	p. 97
Proof of Theorem 1.11	p. 99
Dealing with Elliptics	p. 100
The Applications103
The Convergence Lemmas	p. 105
Statement of Results	p. 105
Preliminary Lemmas	p. 106
Proof of the Convergence Lemma I	p. 107
Proof of the Convergence Lemma II	p. 108
Proof of the Convergence Lemma III	p. 111
Cusp Flexibility	p. 113
Statement of Results	p. 113
A Quick Dimension Count	p. 114
Constructing The Diamond Groups	p. 114
The Analytic Disk	p. 115
Proof of the Cusp Flexibility Lemma	p. 116
The Multiplicity of the Trace Map	p. 118
CR Surgery on the Whitehead Link Complement	p. 121
Trace Neighborhoods	p. 121
Applying the HST	p. 122
Covers of the Whitehead Link Complement	p. 124
Polygons and Alternating Paths	p. 124
Identifying the Cusps	p. 125
Traceful Elements	p. 126
Taking Roots	p. 127
Applying the HST	p. 128
Small-Angle Triangle Groups	p. 131
Characterizing the Representation Space	p. 131
Discreteness	p. 132
Horotube Group Structure	p. 132
Topological Conjugacy	p. 133
Structure of Ideal Triangle Groups	p. 137
Some Spherical CR Geometry	p. 139
Parabolic R-Cones	p. 139
Parabolic R-Spheres	p. 139
Parabolic Elevation Maps	p. 140
A Normality Condition	p. 141
Using Normality	p. 142
The Golden Triangle Group	p. 144
Main Construction	p. 144
The Proof modulo Technical Lemmas	p. 145
Proof of the Horocusp Lemma	p. 148
Proof of the Intersection Lemma	p. 150
Proof of the Monotone Lemma	p. 151
Proof of The Shrinking Lemma	p. 154
The Manifold at Infinity	p. 156
A Model for the Fundamental Domain	p. 156
A Model for the Regular Set	p. 160
A Model for the Quotient	p. 162
Identification with the Model	p. 164
The Groups near the Critical Value	p. 165
More Spherical CR Geometry	p. 165
Main Construction	p. 167
Horotube Group Structure	p. 169
The Loxodromic Normality Condition	p. 170
The Groups far from the Critical Value	p. 176
Discussion of Parameters	p. 176
The Clifford Torus Picture	p. 176
The Horotube Group Structure	p. 177
Bibliography	p. 181
Index	p. 185
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