The Geometry and Topology of Three-Manifolds
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TOC
1. Geometry and three-manifolds
2. Elliptic and hyperbolic geometry
2.1. The Poincar´e disk model
2.2. The southern hemisphere
2.3. The upper half-space model
2.4. The projective model
2.5. The sphere of imaginary radius
2.6. Trigonometry
3. Geometric structures on manifolds
3.1. A hyperbolic structure on the figure-eight knot complement
3.2. A hyperbolic manifold with geodesic boundary
3.3. The Whitehead link complement
3.4. The Borromean rings complement
3.5. The developing map
3.8. Horospheres
3.9. Hyperbolic surfaces obtained from ideal triangles
3.10. Hyperbolic manifolds obtained by gluing ideal polyhedra
4. Hyperbolic Dehn surgery
4.1. Ideal tetrahedra in H3
4.2. Gluing consistency conditions
4.3. Hyperbolic structure on the figure-eight knot complement
4.4. The completion of hyperbolic three-manifolds obtained from ideal polyhedra
4.5. The generalized Dehn surgery invariant
4.6. Dehn surgery on the figure-eight knot
4.8. Degeneration of hyperbolic structures
4.10. Incompressible surfaces in the figure-eight knot complement
5. Flexibility and rigidity of geometric structures
5.2.
5.3.
5.4. Special algebraic properties of groups of isometries of H3
5.5. The dimension of the deformation space of a hyperbolic three-manifold
5.7.
5.8. Generalized Dehn surgery and hyperbolic structures
5.9. A Proof of Mostow’s Theorem
5.10. A decomposition of complete hyperbolic manifolds
5.11. Complete hyperbolic manifolds with bounded volume
5.12. Jorgensen’s Theorem
6. Gromov’s invariant and the volume of a hyperbolic manifold
6.1. Gromov’s invariant
6.3. Gromov’s proof of Mostow’s Theorem
6.5. Manifolds with Boundary
6.6. Ordinals
6.7. Commensurability
6.8. Some Examples
7. Computation of volume
7.1. The Lobachevsky function l()
7.2.
7.3.
7.4.
8. Kleinian groups
8.1. The limit set
8.2. The domain of discontinuity
8.3. Convex hyperbolic manifolds
8.4. Geometrically finite groups
8.5. The geometry of the boundary of the convex hull
8.6. Measuring laminations
8.7. Quasi-Fuchsian groups
8.8. Uncrumpled surfaces
8.9. The structure of geodesic laminations: train tracks
8.10. Realizing laminations in three-manifolds
8.11. The structure of cusps
8.12. Harmonic functions and ergodicity
9. Algebraic convergence
9.1. Limits of discrete groups
9.3. The ending of an end
9.4. Taming the topology of an end
9.5. Interpolating negatively curved surfaces
9.6. Strong convergence from algebraic convergence
9.7. Realizations of geodesic laminations for surface groups with extra cusps, with a digression on stereographic
coordinates
9.9. Ergodicity of the geodesic flow
11. Deforming Kleinian manifolds by homeomorphisms of the sphere at infinity
11.1. Extensions of vector fields
13. Orbifolds
13.1. Some examples of quotient spaces
13.2. Basic definitions
13.3. Two-dimensional orbifolds
13.4. Fibrations
13.5. Tetrahedral orbifolds
13.6. Andreev’s theorem and generalizations
13.7. Constructing patterns of circles
13.8. A geometric compactification for the Teichm¨uller spaces of polygonal orbifolds
13.9. A geometric compactification for the deformation spaces of certain Kleinian groups
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