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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 | |