The arithmetic of hyperbolic 3-manifolds:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2003
|
| Series: | Graduate texts in mathematics
219 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIII, 463 S. graph. Darst. |
| ISBN: | 0387983864 |
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MARC
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| 100 | 1 | |a Maclachlan, Colin |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The arithmetic of hyperbolic 3-manifolds |c Colin Maclachlan ; Alan W. Reid |
| 264 | 1 | |a New York [u.a.] |b Springer |c 2003 | |
| 300 | |a XIII, 463 S. |b graph. Darst. | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 219 | |
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Record in the Search Index
| _version_ | 1804129311760842752 |
|---|---|
| adam_text | Contents
Preface v
0 Number Theoretic Menagerie 1
0.1 Number Fields and Field Extensions 2
0.2 Algebraic Integers 6
0.3 Ideals in Rings of Integers 11
0.4 Units 20
0.5 Class Groups 22
0.6 Valuations 24
0.7 Completions 29
0.8 Adeles and Ideles 35
0.9 Quadratic Forms 39
1 Kleinian Groups and Hyperbolic Manifolds 47
1.1 PSL(2,C) and Hyperbolic 3 Space 47
1.2 Subgroups of PSL(2,C) 50
1.3 Hyperbolic Manifolds and Orbifolds 55
1.4 Examples 57
1.4.1 Bianchi Groups 58
1.4.2 Coxeter Groups 59
1.4.3 Figure 8 Knot Complement 59
1.4.4 Hyperbolic Manifolds by Gluing 60
1.5 3 Manifold Topology and Dehn Surgery 62
1.5.1 3 Manifolds 63
x Contents
1.5.2 Hyperbolic Manifolds 64
1.5.3 Dehn Surgery 65
1.6 Rigidity 67
1.7 Volumes and Ideal Tetrahedra 69
1.8 Further Reading 74
2 Quaternion Algebras I 77
2.1 Quaternion Algebras 77
2.2 Orders in Quaternion Algebras 82
2.3 Quaternion Algebras and Quadratic Forms 87
2.4 Orthogonal Groups 91
2.5 Quaternion Algebras over the Reals 92
2.6 Quaternion Algebras over P adic Fields 94
2.7 Quaternion Algebras over Number Fields 98
2.8 Central Simple Algebras 101
2.9 The Skolem Noether Theorem 105
2.10 Further Reading 108
3 Invariant Trace Fields 111
3.1 Trace Fields for Kleinian Groups of Finite Covolume .... Ill
3.2 Quaternion Algebras for Subgroups of SL(2,C) 114
3.3 Invariant Trace Fields and Quaternion Algebras 116
3.4 Trace Relations 120
3.5 Generators for Trace Fields 123
3.6 Generators for Invariant Quaternion Algebras 128
3.7 Further Reading 130
4 Examples 133
4.1 Bianchi Groups 133
4.2 Knot and Link Complements 134
4.3 Hyperbolic Fibre Bundles 135
4.4 Figure 8 Knot Complement 137
4.4.1 Group Presentation 137
4.4.2 Ideal Tetrahedra 137
4.4.3 Once Punctured Torus Bundle 138
4.5 Two Bridge Knots and Links 140
4.6 Once Punctured Torus Bundles 142
4.7 Polyhedral Groups 143
4.7.1 Non compact Tetrahedra 144
4.7.2 Compact Tetrahedra 146
4.7.3 Prisms and Non integral Traces 149
4.8 Dehn Surgery Examples 152
4.8.1 J0rgensen s Compact Fibre Bundles 152
4.8.2 Fibonacci Manifolds 153
4.8.3 The Weeks Matveev Fomenko Manifold 156
Contents xi
4.9 Fuchsian Groups 159
4.10 Further Reading 162
5 Applications 165
5.1 Discreteness Criteria 165
5.2 Bass s Theorem 168
5.2.1 Tree of SL(2, Kv) 169
5.2.2 Non integral Traces 170
5.2.3 Free Product with Amalgamation 171
5.3 Geodesies and Totally Geodesic Surfaces 173
5.3.1 Manifolds with No Geodesic Surfaces 173
5.3.2 Embedding Geodesic Surfaces 174
5.3.3 The Non cocompact Case 176
5.3.4 Simple Geodesies 178
5.4 Further Hilbert Symbol Obstructions 180
5.5 Geometric Interpretation of the Invariant Trace Field .... 183
5.6 Constructing Invariant Trace Fields 189
5.7 Further Reading 194
6 Orders in Quaternion Algebras 197
6.1 Integers, Ideals and Orders 197
6.2 Localisation 200
6.3 Discriminants 205
6.4 The Local Case I 207
6.5 The Local Case II 209
6.6 Orders in the Global Case 214
6.7 The Type Number of a Quaternion Algebra 217
6.8 Further Reading 223
7 Quaternion Algebras II 225
7.1 Adeles and Ideles 226
7.2 Duality 229
7.3 Classification of Quaternion Algebras 233
7.4 Theorem on Norms 237
7.5 Local Tamagawa Measures 238
7.6 Tamagawa Numbers 244
7.7 The Strong Approximation Theorem 246
7.8 Further Reading 250
8 Arithmetic Kleinian Groups 253
8.1 Discrete Groups from Orders in Quaternion Algebras .... 254
8.2 Arithmetic Kleinian Groups 257
8.3 The Identification Theorem 261
8.4 Complete Commensurability Invariants 267
xii Contents
8.5 Algebraic Integers and Orders 272
8.6 Further Reading 274
9 Arithmetic Hyperbolic 3 Manifolds and Orbifolds 275
9.1 Bianchi Groups 275
9.2 Arithmetic Link Complements 277
9.3 Zimmert Sets and Cuspidal Cohomology 281
9.4 The Arithmetic Knot 285
9.5 Fuchsian Subgroups of Arithmetic Kleinian Groups 287
9.6 Fuchsian Subgroups of Bianchi Groups
and Applications 292
9.7 Simple Geodesies 297
9.8 Hoovering Up 299
9.8.1 The Finite Subgroups A4, S4 and A5 299
9.8.2 Week s Manifold Again 300
9.9 Further Reading 302
10 Discrete Arithmetic Groups 305
10.1 Orthogonal Groups 306
10.2 SO(3,1) and SO(2,1) 310
10.3 General Discrete Arithmetic Groups and
Margulis Theorem 315
10.4 Reflection Groups 322
10.4.1 Arithmetic Polyhedral Groups 325
10.4.2 Tetrahedral Groups 326
10.4.3 Prismatic Examples 327
10.5 Further Reading 329
11 Commensurable Arithmetic Groups and Volumes 331
11.1 Covolumes for Maximal Orders 332
11.2 Consequences of the Volume Formula 338
11.2.1 Arithmetic Kleinian Groups with
Bounded Covolume 338
11.2.2 Volumes for Eichler Orders 340
11.2.3 Arithmetic Manifolds of Equal Volume 341
11.2.4 Estimating Volumes 342
11.2.5 A Tetrahedral Group 343
11.3 Fuchsian Groups 345
11.3.1 Arithmetic Kleinian Groups with
Bounded Covolume 345
11.3.2 Totally Real Fields 346
11.3.3 Fuchsian Triangle Groups 346
11.3.4 Signatures of Arithmetic Fuchsian Groups 350
11.4 Maximal Discrete Groups 352
11.5 Distribution of Volumes 356
Contents xiii
11.6 Minimal Covolume 358
11.7 Minimum Covolume Groups 363
11.8 Further Reading 368
12 Length and Torsion in Arithmetic Hyperbolic Orbifolds 371
12.1 Loxodromic Elements and Geodesies 371
12.2 Geodesies and Embeddings in Quaternion Algebras 373
12.3 Short Geodesies, Lehmer s and Salem s Conjectures 377
12.4 Isospectrality 383
12.5 Torsion in Arithmetic Kleinian Groups 394
12.6 Volume Calculations Again 405
12.7 Volumes of Non arithmetic Manifolds 410
12.8 Further Reading 413
13 Appendices 415
13.1 Compact Hyperbolic Tetrahedra 415
13.2 Non compact Hyperbolic Tetrahedra 416
13.2.1 Arithmetic Groups 416
13.2.2 Non arithmetic Groups 417
13.3 Arithmetic Fuchsian Triangle Groups 418
13.4 Hyperbolic Knot Complements 419
13.5 Small Closed Manifolds 423
13.6 Small Cusped Manifolds 431
13.7 Arithmetic Zoo 436
13.7.1 Non compact Examples 436
13.7.2 Compact Examples, Degree 2 Fields 439
13.7.3 Compact Examples, Degree 3 Fields 440
13.7.4 Compact Examples, Degree 4 Fields 441
Bibliography 443
Index 459
|
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| dewey-ones | 510 - Mathematics |
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| dewey-search | 510 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| isbn | 0387983864 |
| language | English |
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| series2 | Graduate texts in mathematics |
| spelling | Maclachlan, Colin Verfasser aut The arithmetic of hyperbolic 3-manifolds Colin Maclachlan ; Alan W. Reid New York [u.a.] Springer 2003 XIII, 463 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 219 Includes bibliographical references and index Three-manifolds (Topology) Kleinsche Gruppe (DE-588)4164159-0 gnd rswk-swf Dimension 3 (DE-588)4321722-9 gnd rswk-swf Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd rswk-swf Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 s Dimension 3 (DE-588)4321722-9 s Kleinsche Gruppe (DE-588)4164159-0 s DE-604 Reid, Alan W. Sonstige oth Graduate texts in mathematics 219 (DE-604)BV000000067 219 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009883379&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Maclachlan, Colin The arithmetic of hyperbolic 3-manifolds Graduate texts in mathematics Three-manifolds (Topology) Kleinsche Gruppe (DE-588)4164159-0 gnd Dimension 3 (DE-588)4321722-9 gnd Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd |
| subject_GND | (DE-588)4164159-0 (DE-588)4321722-9 (DE-588)4161044-1 |
| title | The arithmetic of hyperbolic 3-manifolds |
| title_auth | The arithmetic of hyperbolic 3-manifolds |
| title_exact_search | The arithmetic of hyperbolic 3-manifolds |
| title_full | The arithmetic of hyperbolic 3-manifolds Colin Maclachlan ; Alan W. Reid |
| title_fullStr | The arithmetic of hyperbolic 3-manifolds Colin Maclachlan ; Alan W. Reid |
| title_full_unstemmed | The arithmetic of hyperbolic 3-manifolds Colin Maclachlan ; Alan W. Reid |
| title_short | The arithmetic of hyperbolic 3-manifolds |
| title_sort | the arithmetic of hyperbolic 3 manifolds |
| topic | Three-manifolds (Topology) Kleinsche Gruppe (DE-588)4164159-0 gnd Dimension 3 (DE-588)4321722-9 gnd Hyperbolische Mannigfaltigkeit (DE-588)4161044-1 gnd |
| topic_facet | Three-manifolds (Topology) Kleinsche Gruppe Dimension 3 Hyperbolische Mannigfaltigkeit |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009883379&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
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