Series C. Spaces of Kleinian groups

Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
1. Introduction......Page 10
2.1. One-parameter families of metrics......Page 11
2.3. Hyperbolic metrics on a manifold......Page 12
2.4. Developing maps......Page 13
2.5. Holonomy representations......Page 14
2.6. Complex projective structures......Page 15
3.2. Deformations of hyperbolic cone-manifolds......Page 17
4. The drilling theorems......Page 21
5. Geometric inflexibility......Page 23
6. Applications to the Bers’ slice......Page 25
6.1. The Bers’ density conjecture......Page 26
6.2. Cusps are dense......Page 30
6.3. The ending lamination conjecture......Page 31
References......Page 33
1. Introduction......Page 38
2.2. Ends and End invariants......Page 40
3.1. Bounded geometry condition......Page 41
4. Limit of Kleinian groups......Page 49
5. Iterations of hyperbolic actions......Page 51
6. Third bounded cohomology groups......Page 52
References......Page 55
1. Introduction......Page 58
2.1. Geodesic Laminations......Page 59
2.2. Real trees......Page 60
2.3. Train tracks......Page 61
2.4. 3-manifolds......Page 62
2.5. Pleated surfaces......Page 63
2.6. Masur domain......Page 64
3. Relations between O(M), D(M) and P(M)......Page 65
4. Topological properties of D(M)......Page 69
5. Pleated surfaces......Page 71
6. Action on R-trees......Page 74
7. Conclusion......Page 78
References......Page 79
1. Introduction......Page 84
2.1. Measured laminations......Page 86
2.3. The Thurston boundary......Page 87
3. Pleating varieties......Page 88
3.1. Scaling functions......Page 90
4. Global geometry......Page 91
5. Monotonicity of angle for fixed length......Page 92
6. The constant angle variety......Page 94
7. Proof of the bending measure conjecture......Page 95
References......Page 96
1. Introduction......Page 100
2.1. Definitions......Page 103
2.3. The filling property......Page 105
3. Closed census......Page 106
3.1. The first 7 geometries......Page 107
3.2. Hyperbolic manifolds......Page 108
3.3. Non-geometric manifolds......Page 110
4. Census of hyperbolic manifolds......Page 111
4.2. Manifolds with χ < 0......Page 112
5.1. Ideal tetrahedra and octahedra......Page 113
5.2.1. Closed case......Page 114
5.3. Matveev-Fomenko conjecture......Page 115
6.1. The closed case......Page 116
6.2. Families of hyperbolic manifolds with boundary of known complexity......Page 117
7.2. Cutting dead branches......Page 118
7.3. Local non-minimality criteria......Page 119
8. Bricks......Page 120
8.1. A quick introduction......Page 121
8.3. Manifolds with marked boundary......Page 122
8.5. Bricks......Page 123
8.6. The algorithm that finds the bricks......Page 124
8.7. Bricks with c ≤ 10.......Page 125
References......Page 127
1. Introduction......Page 130
1.1. Main Theorem......Page 131
1.2.2. Approximating Peano curves by Polygonal curves......Page 132
1.2.3. Generalised Hausdorff measures of sets......Page 133
2.2. Cannon–Thurston maps and their basic properties......Page 134
3. Distortions......Page 135
3.1.2. Definition of Distortions......Page 136
3.1.3. A Property of Distortions......Page 137
4.1. Model manifolds......Page 140
4.4. Geometry of M̃ and F......Page 142
5. Proof of Main Theorem......Page 144
5.2.2. Reduction: The second stage......Page 145
5.2.3. Reduction: The third stage......Page 146
5.3. Estimation of modulus of continuity......Page 147
5.3.1. A lemma from hyperbolic geometry......Page 148
5.3.2. Geometries on surfaces......Page 149
5.3.3. Proof of Proposition 5.5......Page 150
6.1. Continuity of Cannon–Thurston maps......Page 151
6.3. Iterations under pseudo-Anosov actions......Page 152
8. Generalised Hausdorff measures......Page 154
References......Page 156
1. Introduction......Page 160
2. Quasifuchsian groups and the widths of the limit sets......Page 163
3. Hyperbolic punctured surface bundles......Page 168
4. McShane’s analysis of G......Page 176
5. Absolute convergence......Page 177
6. Proof of Theorems 2.2 and 2.3......Page 179
7. Proof of Theorem 3.2......Page 181
8. Proof of Proposition 7.6......Page 185
9. Proof of Lemma 8.4......Page 189
References......Page 191
1. Introduction......Page 196
2. The train track complex......Page 199
3. Proof of the theorem......Page 207
References......Page 214
1. Definitions and statement of the main theorem......Page 218
2. The set of handle curves is connected......Page 219
3. Subsurface projections and distances......Page 222
4. Proof of the theorem......Page 225
References......Page 226
1. Introduction......Page 228
2. Preliminaries......Page 231
3. Proof of the Theorem......Page 235
References......Page 239
1. Introduction......Page 242
2. The Hessian of geodesic-length......Page 243
3. Convexity results......Page 246
4. The CAT(0) geometry of the augmented Teichmüller space......Page 248
References......Page 252
1. Introduction......Page 256
3. Mappings homotopic to the identity......Page 257
4. Deformations......Page 261
5. Ahlfors’ finiteness theorem......Page 262
References......Page 264
1. Introduction......Page 268
2. Construction of the Fuchsian group G of genus zero......Page 271
3. Construction of a generator of the field of G-automorphic functions on H2......Page 273
4.1. Notations......Page 277
4.3. The space of rational functions Rq(Λ) (q ≥ 2)......Page 278
4.4. The Poicaré series operator Θq (q ≥ 2)......Page 279
4.5. The Eichler cohomology......Page 280
4.6. The Eichler integral......Page 281
4.7. The Bers map β∗......Page 282
4.8. The proof of Theorem 4.2......Page 283
4.9. Kra’s algorithm for the construction of a non-trivial Poincaré series......Page 284
5. Construction of a non-trivial Poincaré series. Main result......Page 285
References......Page 289
1.1. Two Parabolics after Lyndon–Ullman......Page 292
1.2. Classical and Non-classical groups......Page 293
1.3. Group Theoretic statement......Page 296
2. Definitions for Schottky groups......Page 297
3. The Geometry and Topology: pinching and nodes......Page 298
4.1. The Representation Variety......Page 301
4.2. Spaces of two-generator groups......Page 302
5.1. H3 geometry......Page 303
5.2. Definitions and basic facts......Page 304
5.3. Methods......Page 305
References......Page 306
1. Introduction......Page 310
2. The first example: cusps on the boundary of Maskit’s T1,1......Page 311
3. Cusp groups and spiral circle packings......Page 317
4. Double cusp groups on the space of pairs of punctured tori......Page 329
5. Cusp groups on Riley’s slice......Page 335
6. Maximal cusps in the Schottky space of genus g......Page 338
References......Page 343
Circle packings on surfaces with projective structures: a survey......Page 346
1. Introduction......Page 364
2.2. Space of projective structures......Page 366
2.3. Grafting......Page 367
2.4. Sequences of quasi-fuchsian representations......Page 368
2.5. ACM-sequences......Page 370
2.6. Pull-backs of limit sets......Page 371
3.1. Grafting for boundary groups......Page 372
3.2. Continuity of grafting maps......Page 373
4.1. Exotic components bump to the standard one......Page 374
4.3. Self-bumping of exotic components......Page 375
5. Additional observations......Page 378
References......Page 380
1. Introduction......Page 384
2. Markoff maps of type preserving representations......Page 385
3.1. Holonomy of projective structures......Page 387
3.2. Jørgensen’s theory of once punctured tori......Page 389
3.3. A problem in the original algorithm......Page 390
4.1. Bowditch’s theory of Markoff triples and quasifuchsian groups......Page 391
4.2. Modified algorithm......Page 392
5.1. Three-dimensional slice of quasifuchsian space......Page 394
5.2. Overhang in the boundary of Bers slice......Page 395
5.3. Final remark......Page 397
References......Page 398