The Geometry and Topology of Three-Manifolds

Front Cover
Contents
Publisher's Note
Editor’s Preface
Introduction
CHAPTER 1 Geometry and three-manifolds
CHAPTER 2 Elliptic and hyperbolic geometry
2.1. The Poincare 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
CHAPTER 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
CHAPTER 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 3-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
CHAPTER 5 Flexibility and rigidity of geometric structures
5.1. Deformations of geometric structure
5.2. A crude dimension count
5.3. Teichmuller space
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. Mostow's Theorem
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
CHAPTER 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
6.4. Strict version of Gromov's Theorem
6.5. Manifolds with boundary
6.6. Ordinals
6.7. Commensurability
6.8. Some examples
CHAPTER 7 Computation of volume
7.1. The Lobachevsky function π(θ)
7.2. Volumes of some polyhedra
7.3. Some manifolds
7.4. Arithmetic examples
References
CHAPTER 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
CHAPTER 9 Algebraic convergence
9.1. Limits of discrete groups
9.2. Geometric tameness
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 ow
NOTE
CHAPTER 11 Deforming Kleinian manifolds by homeomorphisms of the sphere at infinity
11.1. Extensions of vector fields
CHAPTER 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 TeichmŁuller spaces of polygonal orbifolds
13.9. A geometric compactification for the deformation spaces of certain Kleinian groups.
Index