Hyperbolic manifolds and discrete groups:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2001
|
| Schriftenreihe: | Progress in mathematics
183 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009057564&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXV, 467 S. graph. Darst. |
| ISBN: | 0817639047 3764339047 |
Internformat
MARC
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| 245 | 1 | 0 | |a Hyperbolic manifolds and discrete groups |c Michael Kapovich |
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| 300 | |a XXV, 467 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1819318051593519104 |
|---|---|
| adam_text | Contents
Preface v
1 Three Dimensional Topology 1
1.1 Basic definitions and facts 1
1.2 Incompressible surfaces 3
1.3 Existence of incompressible surfaces 7
1.4 Homeomorphisms of Haken manifolds 9
1.5 Pared 3 manifolds 10
1.6 Seifert manifolds 11
1.7 The decomposition theorem 12
1.8 Characteristic submanifolds 14
1.9 3 manifolds fibered over S1 16
1.10 Baumslag Solitar relations 18
2 Thurston Norm 23
2.1 Norms defined over Z 23
2.2 Variation of fiber bundle structure 25
2.3 Application to incompressible surfaces 26
3 Geometry of Hyperbolic Space 31
3.1 General definitions and notation 31
3.2 CA7*(A.) spaces 32
3.3 Basic properties of hyperbolic space 36
3.4 Models of hyperbolic space 39
3.5 Isometries of hyperbolic space 41
xxii Contents
3.6 The convergence property 43
3.7 Convex polyhedra in hyperbolic space 46
3.8 Cayley graphs of finitely generated groups 47
3.9 Quasiisometries 48
3.10 Quasiconformal mappings 53
3.11 Distortion of distance by quasiconformal maps 55
3.12 Harmonic functions on hyperbolic space 56
4 Kleinian Groups 57
4.1 Nilpotent groups 57
4.2 Residual finiteness and the Selberg lemma 59
4.3 Representation varieties 60
4.4 Cohomology of groups and sheaves 65
4.5 Group cohomology and representation varieties 69
4.6 Basics of discrete groups 72
4.7 Properties of limit sets 76
4.8 Quotient spaces of discrete groups 76
4.9 Conical and parabolic limit points 77
4.10 Fundamental domains 78
4.11 Poincare theorem on fundamental polyhedra 80
4.12 The Kazhdan Margulis Zassenhaus theorem 84
4.13 Geometry of Margulis tubes and cusps 87
4.14 Geometrically finite groups 92
4.15 Criteria of geometric finiteness 93
4.16 Kleinian groups and Riemann surfaces 95
4.17 The convex hull and domain of discontinuity 98
4.18 Combination theorems 102
4.19 Ahlfors finiteness theorem 110
4.20 Extensions of the Ahlfors finiteness theorem 112
4.21 Limit sets of geometrically finite groups 115
4.22 Groups with Kleinian subgroups 116
4.23 Ends of hyperbolic manifolds 116
5 Teichmiiller Theory of Riemann Surfaces 119
5.1 Tensor calculus on Riemann surfaces 119
5.2 Properties of quasiconformal maps 122
5.3 Geometry of quadratic differentials 124
5.4 Teichmiiller spaces of Riemann surfaces 125
5.5 The Weil Petersson metric on T(5) 131
5.6 Torsion in Mods 131
6 Introduction to Orbifold Theory 135
6.1 Definitions and examples 135
Contents xxiii
6.2 Two dimensional orbifolds 141
6.3 Three dimensional locally reflective orbifolds 144
6.4 Glossary of orbifolds: The good, the bad, and 147
6.5 A homeomorphism theorem for 3 orbifolds 154
7 Complex Projective Structures 161
7.1 Basic definitions 161
7.2 Grafting 166
8 Sociology of Kleinian Groups 169
8.1 Algebraic convergence of representations 169
8.2 Geometric convergence 171
8.3 Isomorphisms of geometrically finite groups 175
8.4 The Douady Earle extension 178
8.5 The Mostow rigidity theorem 186
8.6 The Sullivan rigidity theorem 189
8.7 Bers isomorphism 190
8.8 Smoothness of representation varieties 193
8.9 Applications to quasiconformal stability 201
8.10 The Calabi Weil infinitesimal rigidity theorem 203
8.11 Space of quasi Fuchsian representations 204
8.12 Distortion of the translation length 205
8.13 Proof of the recurrence theorem 205
8.14 Proof of the Ahlfors finiteness theorem 206
8.15 A generalization of the Bers isomorphism 211
8.16 Totally degenerate groups 212
8.17 Algebraic topology versus geometric topology 214
8.18 Justification of the Poincare continuity method 216
9 Ultralimits of Metric Spaces 219
9.1 Ultrafilters 219
9.2 Ultralimits of metric spaces 220
9.3 The asymptotic cone of a metric space 224
10 Introduction to Group Actions on Trees 227
10.1 Basic definitions and properties 227
10.2 Actions on simplicial trees 232
10.3 Limits of isometric actions on CA7 (0) spaces 236
10.4 Compactification of character varieties 236
10.5 Proper actions 240
11 Laminations, Foliations, and Trees 243
11.1 Euclidean motivation 243
11.2 Geodesic currents 245
xxiv Contents
11.3 Measured foliations on hyperbolic surfaces 246
11.4 Interval exchange transformations 253
11.5 Train tracks 254
11.6 Measured geodesic laminations 255
11.7 Topology on measured laminations 262
11.8 From foliations to laminations 263
11.9 From laminations to foliations 264
11.10 Action of Mods on geodesic laminations 266
11.11 The geometric intersection number 266
11.12 From laminations to trees 267
11.13 An application of Skora s theorem 269
11.14 Characterization of aperiodic homeomorphisms 270
11.15 Dynamics of aperiodic homeomorphisms 274
11.16 Compactification of the Teichmiiller space 276
12 Rips Theory 279
12.1 Stable trees 280
12.2 Unions of bands and band complexes 282
12.3 Pushing 285
12.4 Transversal measure on a union of bands 287
12.5 Dynamical decomposition of unions of bands 287
12.6 Band complexes 289
12.7 Holonomy of vertical paths in X 294
12.8 The Kazhdan Margulis theorem for actions on trees 295
12.9 The moves 299
12.10 Preliminary motions of the Rips machine 304
12.11 The machine 306
12.12 The machine output 313
12.13 Proof of the decomposition theorem 319
12.14 Proof of Skora s duality theorem 321
12.15 Geometric actions on trees 322
12.16 Nongeometric actions on trees 326
12.17 Compactness of representation varieties 329
13 Brooks Theorem and Circle Packings 333
13.1 Orbifolds and patterns of circles 334
13.2 Brooks theorem 338
13.3 A packing invariant of patterns of disks 339
13.4 A packing invariant of Kleinian groups 345
13.5 Proof of Brooks theorem 348
14 Pleated Surfaces and Ends of Hyperbolic Manifolds 351
14.1 Singular hyperbolic metrics 351
Contents xxv
14.2 Existence theorem for singular pleated maps 354
14.3 Compactness theorem for pleated maps 355
14.4 Geometrically tame ends of hyperbolic manifolds 357
14.5 Ending laminations 361
14.6 Infinite coverings of hyperbolic manifolds 362
14.7 Geometric profile of algebraic convergence 365
15 Outline of the Proof of the Hyperbolization Theorem 369
15.1 Case A: The generic case 372
15.2 Case B: Manifolds fibered over the circle 375
16 Reduction to the Bounded Image Theorem 377
16.1 Step 1: The Maskit combination 377
16.2 Step 2: Formulation of the fixed point theorem 377
16.3 Step 3: Proof of the contraction theorem 379
16.4 Step 4: End of the proof of the fixed point theorem 381
17 The Bounded Image Theorem 383
17.1 Limit exists 383
17.2 Geometry of the limit 385
18 Hyperbolization of Fibrations 397
18.1 Compactness theorem for aperiodic homeomorphisms 397
18.2 The Double Limit Theorem: An outline 398
18.3 Proof of Theorem 15.17 400
18.4 An alternative approach 401
19 The Orbifold Trick 403
19.1 Manifold coverings of bipolar orbifolds 403
19.2 Building hyperbolic orbifolds of finite type 404
19.3 Gluing orbifolds of zero Euler characteristic 409
19.4 Hyperbolization of locally reflective orbifolds 411
19.5 Hyperbolic design 412
20 Beyond the Hyperbolization Theorem 417
20.1 Thurston s geometrization conjecture 417
20.2 Other structures 425
20.3 Higher dimensions 428
20.4 Hyperbolic groups 428
References 433
Index 461
|
| any_adam_object | 1 |
| author | Kapovich, Michael 1963- |
| author_GND | (DE-588)134100409 |
| author_facet | Kapovich, Michael 1963- |
| author_role | aut |
| author_sort | Kapovich, Michael 1963- |
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| callnumber-first | Q - Science |
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| ctrlnum | (OCoLC)44467443 (DE-599)BVBBV013285525 |
| dewey-full | 516.9 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.9 |
| dewey-search | 516.9 |
| dewey-sort | 3516.9 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV013285525 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:44:32Z |
| institution | BVB |
| isbn | 0817639047 3764339047 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009057564 |
| oclc_num | 44467443 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-384 DE-29T DE-19 DE-BY-UBM DE-703 DE-634 DE-83 DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-384 DE-29T DE-19 DE-BY-UBM DE-703 DE-634 DE-83 DE-11 |
| physical | XXV, 467 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spellingShingle | Kapovich, Michael 1963- Hyperbolic manifolds and discrete groups Progress in mathematics Espaces hyperboliques Groupes discrets Discrete groups Hyperbolic spaces Funktionentheorie (DE-588)4018935-1 gnd Hyperbolischer komplexer Raum (DE-588)4138418-0 gnd Diskrete Gruppe (DE-588)4135541-6 gnd Dimension 3 (DE-588)4321722-9 gnd Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4138418-0 (DE-588)4135541-6 (DE-588)4321722-9 (DE-588)4161044-1 (DE-588)4037379-4 |
| title | Hyperbolic manifolds and discrete groups |
| title_auth | Hyperbolic manifolds and discrete groups |
| title_exact_search | Hyperbolic manifolds and discrete groups |
| title_full | Hyperbolic manifolds and discrete groups Michael Kapovich |
| title_fullStr | Hyperbolic manifolds and discrete groups Michael Kapovich |
| title_full_unstemmed | Hyperbolic manifolds and discrete groups Michael Kapovich |
| title_short | Hyperbolic manifolds and discrete groups |
| title_sort | hyperbolic manifolds and discrete groups |
| topic | Espaces hyperboliques Groupes discrets Discrete groups Hyperbolic spaces Funktionentheorie (DE-588)4018935-1 gnd Hyperbolischer komplexer Raum (DE-588)4138418-0 gnd Diskrete Gruppe (DE-588)4135541-6 gnd Dimension 3 (DE-588)4321722-9 gnd Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd |
| topic_facet | Espaces hyperboliques Groupes discrets Discrete groups Hyperbolic spaces Funktionentheorie Hyperbolischer komplexer Raum Diskrete Gruppe Dimension 3 Hyperbolische Mannigfaltigkeit Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009057564&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
| work_keys_str_mv | AT kapovichmichael hyperbolicmanifoldsanddiscretegroups |