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Chapter A．Hyperbolic Space 1

A.1 Models for Hyperbolic Space 1

A.2 Isometries of Hyperbolic Space:Hyperboloid Model 3

A.3 Conformal Geometry 7

A.4 Isometries of Hyperbolic Space:Disc and Half-space Models 22

A.5 Geodesics,Hyperbolic Subspaces and Miscellaneous Facts 25

A.6 Curvature of Hyperbolic Space 37

Chapter B．Hyperbolic Manifolds and the Compact Two-dimensional Case 45

B.1 Hyperbolic,Elliptic and Flat Manifolds 45

B.2 Topology of Compact Oriented Surfaces 55

B.3 Hyperbolic,Elliptic and Flat Surfaces 58

B.4 Teiehmüller Space 61

Chapter C．The Rigidity Theorem(Compact Case) 83

C.1 First Step of the Proof:Extension of Pseudo-isometries 84

C.2 Second Step of the Proof:Volume of Ideal Simplices 94

C.3 Gromov Norm of a Compact Manifold 103

C.4 Third Step of the Proof:the Gromov Norm and the Volume Are Proportional 105

C.5 Conclusion of the Proof,Corollaries and Generalizations 121

Chapter D．Margulis'Lemma and its Applications 133

D.1 Margulis'Lemma 133

D.2 Local Geometry of a Hyperbolic Manifold 140

D.3 Ends of a Hyperbolic Manifold 143

Chapter E．The Space of Hyperbolic Manifolds and the Volume Function 159

E.1 The Chabauty and the Geometric Topology 160

E.2 Convergence in the Geometric Topology:Opening Cusps The Case of Dimension at least Three 174

E.3 The Case of Dimension Different from Three Conclusions and Examples 184

E.4 The Three-dimensional Case:Jorgensen's Part of the So-called Jorgensen-Thurston Theory 190

E.5 The Three-dimensional Case.Thurston's Hyperbolic Surgery Theorem:Statement and Preliminaries 196

E.5-ⅰ Definition and First Properties of T3(Non-compact Three-manifolds with"Triangulation"Without Vertices) 198

E.5-ⅱ Hyperbolic Structures on an Element of T3 and Realization of the Complete Structure 201

E.5-ⅲ Elements of T3 and Standard Spines 207

E.5-ⅳ Some Links Whose Complements are Realized as Elements of T3 210

E.6 Proof of Thurston's Hyperbolic Surgery Theorem 223

E.6-ⅰ Algebraic Equations of H(M)(Hyperbolic Structures Supported by M∈T3) 224

E.6-ⅱ Dimension of H(M):General Case 234

E.6-ⅲ The Case M is Complete Hyperbolic:the Space of Deformations 251

E.6-ⅳ Completion of the Deformed Hyperbolic Structures and Conclusion of the Proof 256

E.7 Applications to the Study of the Volume Function and Complements about Three-dimensional Hyperbolic Geometry 267

Chapter F．Bounded Cohomology,a Rough Outline 273

F.1 Singular Cohomology 273

F.2 Bounded Singular Cohomology 277

F.3 Flat Fiber Bundles 280

F.4 Euler Class of a Flat Vector Bundle 287

F.5 Flat Vector Bundles on Surfaces and the Milnor-Sullivan Theorem 294

F.6 Sullivan's Conjecture and Amenable Groups 303

Subject Index 321

Notation Index 324

Referenees 326