Outer circles: an introduction to hyperbolic 3-manifolds

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
List of Illustrations......Page 13
Preface......Page 15
1.1 Möbius transformations......Page 21
1.2 Hyperbolic geometry......Page 26
The hyperbolic plane......Page 28
Hyperbolic space......Page 29
1.3 The circle or sphere at infinity......Page 31
1.4 Gaussian curvature......Page 35
Commutativity......Page 38
Isometric circles and planes......Page 39
Trace identities......Page 42
1.6 Exercises and explorations......Page 43
2.1 Convergence of Möbius transformations......Page 69
2.2 Discreteness......Page 71
2.3 Elementary discrete groups......Page 75
Infinite elementary discrete groups......Page 76
2.4 Kleinian groups......Page 78
2.5 Quotient manifolds and orbifolds......Page 82
2.5.1 Two fundamental algebraic theorems......Page 88
2.6 Introduction to Riemann surfaces and their uniformization......Page 89
What is uniformization?......Page 91
Fuchsian groups......Page 94
Schottky groups......Page 95
2.8 Riemannian metrics and quasiconformal mappings......Page 98
Teichmüller spaces of Riemann surfaces......Page 100
2.9 Exercises and explorations......Page 103
3.1 The Ahlfors Finiteness Theorem......Page 125
3.2 Tubes and horoballs......Page 126
3.3 Universal properties......Page 128
Historical remarks......Page 134
3.4 The thick/thin decomposition of a manifold......Page 135
3.5 Fundamental polyhedra......Page 136
The Ford fundamental region and polyhedron......Page 140
Poincaré’s Theorem......Page 142
Additional remarks......Page 143
3.6 Geometric finiteness......Page 144
Finite volume......Page 148
3.7 Three-manifold surgery......Page 149
Application of Dehn’s Lemma and the Loop Theorem......Page 150
Equivariant extensions ∂M→M......Page 152
3.8 Quasifuchsian groups......Page 154
3.9 Geodesic and measured geodesic laminations......Page 156
Measured laminations......Page 159
3.10 The convex hull of the limit set......Page 164
Examples......Page 165
The bending measure......Page 166
Pleated surfaces......Page 167
3.11 The convex core......Page 171
Length estimates......Page 172
Existence of bending measures......Page 173
3.12 The compact and relative compact core......Page 175
3.13 Rigidity......Page 176
3.14 Exercises and explorations......Page 181
4.1 Algebraic convergence......Page 207
4.2 Geometric convergence......Page 213
4.3 Polyhedral convergence......Page 214
4.4 The geometric limit......Page 217
Hausdorff and Carathéodory convergence......Page 220
Sequences of limit sets and regions of discontinuity......Page 221
4.6 New parabolics......Page 223
4.7 Acylindrical manifolds......Page 225
4.8 Dehn surgery......Page 227
4.9 The prototypical example......Page 228
4.10 Manifolds of finite volume......Page 231
4.11 The Dehn surgery theorems for finite volume manifolds......Page 232
Well ordering of volumes of hyperbolic manifolds......Page 235
The well ordering......Page 236
Volumes of higher-dimensional manifolds......Page 237
4.12 Exercises and explorations......Page 238
5.1 The representation variety......Page 259
The discreteness locus......Page 261
The quasiconformal deformation space......Page 262
5.2 Homotopy equivalence......Page 264
Rigidity of hyperbolic manifolds under homotopy equivalences......Page 265
Components of the discreteness locus......Page 266
5.3 The quasiconformal deformation space boundary......Page 268
5.4 The three great conjectures......Page 270
5.5 Ends of hyperbolic manifolds......Page 271
5.6 Tame manifolds......Page 272
The Tameness Theorem......Page 273
The Ending Lamination Theorem......Page 275
The Density Theorem......Page 280
5.7 Quasifuchsian spaces......Page 281
Bers slices......Page 282
5.8 The quasifuchsian space boundary......Page 285
Collapsing mappings......Page 286
The Bers boundary......Page 289
5.9 Geometric limits at boundary points......Page 291
The Jørgensen picture of the once punctured torus case......Page 292
Geometric limits at the Bers boundary......Page 294
The totality of geometric limits at the quasifuchsian space boundary......Page 297
The Thurston boundary......Page 299
5.10 Exercises and explorations......Page 302
6.1.1 Automorphisms of surfaces......Page 332
6.1.2 The Double Limit Theorem......Page 334
6.1.3 Manifolds fibered over the circle......Page 335
6.2.1 Hyperbolic manifolds with totally geodesic boundary......Page 337
6.2.2 Skinning the manifold (Part II)......Page 339
6.3 The Hyperbolization Theorem......Page 342
6.3.1 Knots and links......Page 345
6.4 Geometrization......Page 347
6.5 The Orbifold Theorem......Page 349
6.6 Exercises and Explorations......Page 351
7.1 Half-rotations......Page 368
7.2 The Lie product......Page 369
7.4 Complex distance......Page 373
7.5 Complex distance and line geometry......Page 375
7.6 Exercises and explorations......Page 376
8.1 Generic right hexagons......Page 386
8.2 The sine and cosine laws for generic right hexagons......Page 388
8.3 Degenerate right hexagons......Page 390
Right triangles......Page 392
Planar pentagons with four right angles......Page 393
Quadrilaterals with three right angles......Page 394
8.5 Exercises and explorations......Page 395
Bibliography......Page 413
Index......Page 431