Spherical CR Geometry and Dehn Surgery

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:	9780691128108
Publisher:	Princeton University Press
Publication date:	02/18/2007
Series:	Annals of Mathematics Studies , #165
Pages:	200
Product dimensions:	6.00(w) x 9.25(h) x (d)
Age Range:	18 Years

About the Author

Richard Evan Schwartz is Professor of Mathematics at Brown University.

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