Cover

Title

Copyright

Dedication

Contents

Preface

PART 1. BASIC MATERIAL

Chapter 1. Introduction

1.1 Dehn Filling and Thurston’s Theorem

1.2 Definition of a Horotube Group

1.3 The Horotube Surgery Theorem

1.4 Reflection Triangle Groups

1.5 Spherical CR Structures

1.6 The Goldman-Parker Conjecture

1.7 Organizational Notes

Chapter 2. Rank-One Geometry

2.1 Real Hyperbolic Geometry

2.2 Complex Hyperbolic Geometry

2.3 The Siegel Domain and Heisenberg Space

2.4 The Heisenberg Contact Form

2.5 Some Invariant Functions

2.6 Some Geometric Objects

Chapter 3. Topological Generalities

3.1 The Hausdorff Topology

3.2 Singular Models and Spines

3.3 A Transversality Result

3.4 Discrete Groups

3.5 Geometric Structures

3.6 Orbifold Fundamental Groups

3.7 Orbifolds with Boundary

Chapter 4. Reflection Triangle Groups

4.1 The Real Hyperbolic Case

4.2 The Action on the Unit Tangent Bundle

4.3 Fuchsian Triangle Groups

4.4 Complex Hyperbolic Triangles

4.5 The Representation Space

4.6 The Ideal Case

Chapter 5. Heuristic Discussion of Geometric Filling

5.1 A Dictionary

5.2 The Tree Example

5.3 Hyperbolic Case: Before Filling

5.4 Hyperbolic Case: After Filling

5.5 Spherical CR Case: Before Filling

5.6 Spherical CR Case: After Filling

5.7 The Tree Example Revisited

PART 2. PROOF OF THE HST

Chapter 6. Extending Horotube Functions

6.1 Statement of Results

6.2 Proof of the Extension Lemma

6.3 Proof of the Auxiliary Lemma

Chapter 7. Transplanting Horotube Functions

7.1 Statement of Results

7.2 A Toy Case

7.3 Proof of the Transplant Lemma

Chapter 8. The Local Surgery Formula

8.1 Statement of Results

8.2 The Canonical Marking

8.3 The Homeomorphism

8.4 The Surgery Formula

Chapter 9. Horotube Assignments

9.1 Basic Definitions

9.2 The Main Result

9.3 Corollaries

Chapter 10. Constructing the Boundary Complex

10.1 Statement of Results

10.2 Proof of the Structure Lemma

10.3 Proof of the Horotube Assignment Lemma

Chapter 11. Extending to the Inside

11.1 Statement of Results

11.2 Proof of the Transversality Lemma

11.3 Proof of the Local Structure Lemma

11.4 Proof of the Compatibility Lemma

11.5 Proof of the Finiteness Lemma

Chapter 12. Machinery for Proving Discreteness

12.1 Chapter Overview

12.2 Simple Complexes

12.3 Chunks

12.4 Geometric Equivalence Relations

12.5 Alignment by a Simple Complex

Chapter 13. Proof of the HST

13.1 The Unperturbed Case

13.2 The Perturbed Case

13.3 Defining the Chunks

13.4 The Discreteness Proof

13.5 The Surgery Formula

13.6 Horotube Group Structure

13.7 Proof of Theorem 1.11

13.8 Dealing with Elliptics

PART 3. THE APPLICATIONS

Chapter 14. The Convergence Lemmas

14.1 Statement of Results

14.2 Preliminary Lemmas

14.3 Proof of the Convergence Lemma I

14.4 Proof of the Convergence Lemma II

14.5 Proof of the Convergence Lemma III

Chapter 15. Cusp Flexibility

15.1 Statement of Results

15.2 A Quick Dimension Count

15.3 Constructing The Diamond Groups

15.4 The Analytic Disk

15.5 Proof of the Cusp Flexibility Lemma

15.6 The Multiplicity of the Trace Map

Chapter 16. CR Surgery on the Whitehead Link Complement

16.1 Trace Neighborhoods

16.2 Applying the HST

Chapter 17. Covers of the Whitehead Link Complement

17.1 Polygons and Alternating Paths

17.2 Identifying the Cusps

17.3 Traceful Elements

17.4 Taking Roots

17.5 Applying the HST

Chapter 18. Small-Angle Triangle Groups

18.1 Characterizing the Representation Space

18.2 Discreteness

18.3 Horotube Group Structure

18.4 Topological Conjugacy

PART 4. STRUCTURE OF IDEAL TRIANGLE GROUPS

Chapter 19. Some Spherical CR Geometry

19.1 Parabolic R-Cones

19.2 Parabolic R-Spheres

19.3 Parabolic Elevation Maps

19.4 A Normality Condition

19.5 Using Normality

Chapter 20. The Golden Triangle Group

20.1 Main Construction

20.2 The Proof modulo Technical Lemmas

20.3 Proof of the Horocusp Lemma

20.4 Proof of the Intersection Lemma

20.5 Proof of the Monotone Lemma

20.6 Proof of The Shrinking Lemma

Chapter 21. The Manifold at Infinity

21.1 A Model for the Fundamental Domain

21.2 A Model for the Regular Set

21.3 A Model for the Quotient

21.4 Identification with the Model

Chapter 22. The Groups near the Critical Value

22.1 More Spherical CR Geometry

22.2 Main Construction

22.3 Horotube Group Structure

22.4 The Loxodromic Normality Condition

Chapter 23. The Groups far from the Critical Value

23.1 Discussion of Parameters

23.2 The Clifford Torus Picture

23.3 The Horotube Group Structure

Bibliography

Index