Conformal Geometry of Discrete Groups and Manifolds (Degruyter Expositions in Mathematics)

Preface......Page 0
1.1. (X, G)-structures on manifolds  ......Page 13
1.2. Conformal geometry on the sphere ......Page 16
1.3. The hyperbolic space H^n......Page 23
1.4. Lie subgroups of the MSbius group ......Page 31
1.5. Structure developments and holonomy homomorphisms ......Page 34
1.6. The eight 3-dimensional geometries ......Page 37
1.7. Four-dimensional geometries ......Page 43
1.8. Geometry of orbifolds ......Page 45
Notes ......Page 51
2.1. Convergence groups ......Page 52
2.2. Group action on the discontinuity set ......Page 66
2.3. Fundamental domains ......Page 74
2.4. Convex polyhedra and reflection groups ......Page 89
2.5. Discrete group action on the limit set ......Page 98
Notes......Page 112
3.1. Margulis's Lemma and splittings of hyperbolic manifolds  ......Page 113
3.2. Injectivity radius of hyperbolic manifolds ......Page 119
3.3. Thin cusp submanifolds ......Page 123
3.4. Precisely invariant horoballs ......Page 129
3.5. Group action on the set of horoballs  ......Page 133
3.6. Convex hull constructions ......Page 135
3.7. Tessellations of manifolds by ideal hyperbolic polyhedra ......Page 141
3.8. Hyperbolic arithmetics  ......Page 142
3.9. Arithmetic groups generated by reflections  ......Page 145
3.10. Non-arithmetic groups of Gromov and Piatetski-Shapiro ......Page 152
3.11. Fibonacci manifolds ......Page 156
Notes......Page 163
4.1. Classical finiteness for planar Kleinian groups ......Page 165
4.2. Geometrical finiteness in higher dimensions  ......Page 168
4.3. Equivalent definitions of geometrical finiteness  ......Page 178
4.4. Geometrically finite ends and coverings  ......Page 187
4.5. Geometry of tessellations ......Page 190
4.6. Cayley graphs and geometric isomorphisms of discrete groups ......Page 203
4.7. Geometrical finiteness for discontinuity set components ......Page 216
Notes......Page 227
5.1. Basic topology related to Kleinian manifolds ......Page 229
5.2. Topological aspects of combination theorems ......Page 237
5.3. Universal groups and Poincare Conjecture  ......Page 253
5.4. Ends of Kleinian manifolds, their compactification and Ahlfors's Conjecture ......Page 256
5.5. Kleinian n-manifolds and hyperbolic cobordisms  ......Page 268
5.6. Finiteness problems for Kleinian n-manifolds ......Page 288
Notes ......Page 300
6. Uniformization ......Page 301
6.1. Classical uniformization ......Page 302
6.2. Modem concepts of uniformization  ......Page 303
6.3. Hyperbolization of manifolds and hyperbolic volumes ......Page 320
6.4. Uniformizable conformal structures  ......Page 336
6.5. Conformal uniformization of "flat" connected sums ......Page 347
6.6. Conformal uniformization of Seifert manifolds ......Page 354
6.7. Torus sums of conformal structures  ......Page 362
6.8. Canonical Riemannian metric on conformal manifolds ......Page 367
Notes ......Page 389
7.1. Deformations of geometric structures  ......Page 391
7.2. Rigidity of hyperbolic structures  ......Page 404
7.3. Quasi-Fuchsian structures: bendings ......Page 420
7.4. Quasi-Fuchsian structures: cone deformations  ......Page 437
7.5. Bendings along surfaces with boundaries......Page 446
7.6. Global properties of deformation spaces  ......Page 453
Notes ......Page 469
Bibliography  ......Page 473
Index......Page 525