Spherical CR Geometry and Dehn Surgery (AM-165)

by Richard Evan Schwartz

Overview

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.

Product Details

ISBN-13:	9781400837199
Publisher:	Princeton University Press
Publication date:	01/29/2007
Series:	Annals of Mathematics Studies , #165
Sold by:	Barnes & Noble
Format:	eBook
Pages:	200
File size:	2 MB

About the Author

Richard Evan Schwartz is Professor of Mathematics at Brown University.

Table of Contents

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