The arithmetic of hyperbolic three-manifolds / Colin Maclachlan, Alan W. Reid.
Series Graduate texts in mathematics ; 219Editor: New York : Springer, c2003Descripción: xiii, 463 p. : ill. ; 24 cmISBN: 0387983864 (acidfree paper)Tema(s): Three-manifolds (Topology)Otra clasificación: 57M50 (11R52) Recursos en línea: Publisher description | Table of contents onlyPreface 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 Adèles and Idèles [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 Or bifolds [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] -- 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 [111] -- 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 Jprgensen’s Compact Fibre Bundles [152] -- 4.8.2 Fibonacci Manifolds [153] -- 4.8.3 The Weeks-Matveev-Fomenko Manifold [156] -- 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, Kp) [169] -- 5.2.2 Non-integral Traces [170] -- 5.2.3 Free Product with Amalgamation [171] -- 5.3 Geodesics 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 Geodesics [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 Adèles and Idèles [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] -- 8.5 Algebraic Integers and Orders [272] -- 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 Geodesics [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 Fuchsian 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] -- 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 Geodesics [371] -- 12.2 Geodesics and Embeddings in Quaternion Algebras [373] -- 12.3 Short Geodesics, 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 Ttetrahedra [415] -- 13.2 Non-compact Hyperbolic Tetrahedra [415] -- 13.2.1 Arithmetic Groups [415] -- 13.2.2 Non-arithrnetic Groups [417] -- 13.3 Arithmetic Fuchsian lYiangle 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 --
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
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Incluye referencias bibliográficas e índice.
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 Adèles and Idèles [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 Or bifolds [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] --
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 [111] --
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 Jprgensen’s Compact Fibre Bundles [152] --
4.8.2 Fibonacci Manifolds [153] --
4.8.3 The Weeks-Matveev-Fomenko Manifold [156] --
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, Kp) [169] --
5.2.2 Non-integral Traces [170] --
5.2.3 Free Product with Amalgamation [171] --
5.3 Geodesics 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 Geodesics [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 Adèles and Idèles [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] --
8.5 Algebraic Integers and Orders [272] --
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 Geodesics [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 Fuchsian 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] --
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 Geodesics [371] --
12.2 Geodesics and Embeddings in Quaternion Algebras [373] --
12.3 Short Geodesics, 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 Ttetrahedra [415] --
13.2 Non-compact Hyperbolic Tetrahedra [415] --
13.2.1 Arithmetic Groups [415] --
13.2.2 Non-arithrnetic Groups [417] --
13.3 Arithmetic Fuchsian lYiangle 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 --
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