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List of Illustrations | |
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Preface | |
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Hyperbolic space and its isometries | |
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Mobius transformations | |
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Hyperbolic geometry | |
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The circle or sphere at infinity | |
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Gaussian curvature | |
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Further properties of Mobius transformations | |
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Exercises and explorations | |
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Discrete groups | |
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Convergence of Mobius transformations | |
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Discreteness | |
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Elementary discrete groups | |
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Kleinian groups | |
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Quotient manifolds and orbifolds | |
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Two fundamental algebraic theorems | |
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Introduction to Riemann surfaces and their uniformization | |
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Fuchsian and Schottky groups | |
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Riemannian metrics and quasiconformal mappings | |
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Exercises and explorations | |
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Properties of hyperbolic manifolds | |
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The Ahlfors Finiteness Theorem | |
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Tubes and horoballs | |
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Universal properties | |
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The thick/thin decomposition of a manifold | |
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Fundamental polyhedra | |
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Geometric finiteness | |
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Three-manifold surgery | |
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Quasifuchsian groups | |
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Geodesic and measured geodesic laminations | |
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The convex hull of the limit set | |
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The convex core | |
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The compact and relative compact core | |
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Rigidity | |
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Exercises and explorations | |
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Algebraic and geometric convergence | |
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Algebraic convergence | |
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Geometric convergence | |
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Polyhedral convergence | |
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The geometric limit | |
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Convergence of limit sets and regions of discontinuity | |
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New parabolics | |
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Acylindrical manifolds | |
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Dehn surgery | |
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The prototypical example | |
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Manifolds of finite volume | |
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The Dehn surgery theorems for finite volume manifolds | |
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Exercises and explorations | |
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Deformation spaces and the ends of manifolds | |
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The representation variety | |
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Homotopy equivalence | |
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The quasiconformal deformation space boundary | |
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The three great conjectures | |
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Ends of hyperbolic manifolds | |
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Tame manifolds | |
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Quasifuchsian spaces | |
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The quasifuchsian space boundary | |
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Geometric limits at boundary points | |
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Exercises and explorations | |
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Hyperbolization | |
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Hyperbolic manifolds that fiber over a circle | |
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Automorphisms of surfaces | |
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The Double Limit Theorem | |
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Manifolds fibered over the circle | |
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The Skinning Lemma | |
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Hyperbolic manifolds with totally geodesic boundary | |
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Skinning the manifold (Part II) | |
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The Hyperbolization Theorem | |
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Knots and links | |
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Geometrization | |
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The Orbifold Theorem | |
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Exercises and Explorations | |
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Line geometry | |
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Half-rotations | |
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The Lie product | |
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Square roots | |
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Complex distance | |
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Complex distance and line geometry | |
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Exercises and explorations | |
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Right hexagons and hyperbolic trigonometry | |
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Generic right hexagons | |
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The sine and cosine laws for generic right hexagons | |
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Degenerate right hexagons | |
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Formulas for triangles, quadrilaterals, and pentagons | |
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Exercises and explorations | |
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Bibliography | |
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Index | |