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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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