Richard Evan Schwartz is Professor of Mathematics at Brown University.

• 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

Richard Evan Schwartz is Professor of Mathematics at Brown University.

• 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

Richard Evan Schwartz is Professor of Mathematics at Brown University.

• 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