Foundations of Hyperbolic Manifolds

✍ Scribed by John Ratcliffe

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 794
• Series: Graduate Texts in Mathematics
• Edition: 2nd
• Category: Library

✦ Synopsis

This heavily class-tested book is an exposition of the theoretical foundations of hyperbolic manifolds. It is a both a textbook and a reference. A basic knowledge of algebra and topology at the first year graduate level of an American university is assumed. The first part is concerned with hyperbolic geometry and discrete groups. The second part is devoted to the theory of hyperbolic manifolds. The third part integrates the first two parts in a development of the theory of hyperbolic orbifolds. Each chapter contains exercises and a section of historical remarks. A solutions manual is available separately.

✦ Table of Contents

Preface to the First Edition......Page 7
Preface to the Second Edition......Page 9
Contents......Page 10
1.1. Euclid’s Parallel Postulate......Page 13
1.2. Independence of the Parallel Postulate......Page 19
1.3. Euclidean n-Space......Page 25
1.4. Geodesics......Page 34
1.5. Arc Length......Page 40
1.6. Historical Notes......Page 44
2 Spherical Geometry......Page 47
2.1. Spherical n-Space......Page 48
2.2. Elliptic n-Space......Page 53
2.3. Spherical Arc Length......Page 55
2.4. Spherical Volume......Page 56
2.5. Spherical Trigonometry......Page 59
2.6. Historical Notes......Page 64
3 Hyperbolic Geometry......Page 66
3.1. Lorentzian n-Space......Page 67
3.2. Hyperbolic n-Space......Page 73
3.3. Hyperbolic Arc Length......Page 85
3.4. Hyperbolic Volume......Page 89
3.5. Hyperbolic Trigonometry......Page 92
3.6. Historical Notes......Page 110
4.1. Reflections......Page 112
4.2. Stereographic Projection......Page 119
4.3. M¨obius Transformations......Page 122
4.4. Poincar´e Extension......Page 128
4.5. The Conformal Ball Model......Page 134
4.6. The Upper Half-Space Model......Page 143
4.7. Classi.cation of Transformations......Page 148
4.8. Historical Notes......Page 154
5.1. Topological Groups......Page 156
5.2. Groups of Isometries......Page 162
5.3. Discrete Groups......Page 169
5.4. Discrete Euclidean Groups......Page 177
5.5. Elementary Groups......Page 188
5.6. Historical Notes......Page 197
6.1. The Projective Disk Model......Page 200
6.2. Convex Sets......Page 206
6.3. Convex Polyhedra......Page 213
6.4. Geometry of Convex Polyhedra......Page 224
6.5. Polytopes......Page 235
6.6. Fundamental Domains......Page 246
6.7. Convex Fundamental Polyhedra......Page 258
6.8. Tessellations......Page 265
6.9. Historical Notes......Page 273
7.1. Re.ection Groups......Page 275
7.2. Simplex Re.ection Groups......Page 288
7.3. Generalized Simplex Re.ection Groups......Page 308
7.4. The Volume of a Simplex......Page 315
7.5. Crystallographic Groups......Page 322
7.6. Torsion-Free Linear Groups......Page 334
7.7. Historical Notes......Page 344
8.1. Geometric Spaces......Page 346
8.2. Cli.ord-Klein Space-Forms......Page 353
8.3. (X,G)-Manifolds......Page 359
8.4. Developing......Page 366
8.5. Completeness......Page 373
8.6. Curvature......Page 383
8.7. Historical Notes......Page 385
9.1. Compact Surfaces......Page 387
9.2. Gluing Surfaces......Page 390
9.3. The Gauss-Bonnet Theorem......Page 402
9.4. Moduli Spaces......Page 403
9.5. Closed Euclidean Surfaces......Page 413
9.6. Closed Geodesics......Page 416
9.7. Closed Hyperbolic Surfaces......Page 423
9.8. Hyperbolic Surfaces of Finite Area......Page 431
9.9. Historical Notes......Page 444
10.1. Gluing 3-Manifolds......Page 447
10.2. Complete Gluing of 3-Manifolds......Page 456
10.3. Finite Volume Hyperbolic 3-Manifolds......Page 460
10.4. Hyperbolic Volume......Page 474
10.5. Hyperbolic Dehn Surgery......Page 492
10.6. Historical Notes......Page 517
11.1. Gluing n-Manifolds......Page 520
11.2. Poincar´e’s Theorem......Page 528
11.3. The Gauss-Bonnet Theorem......Page 535
11.4. Simplices of Maximum Volume......Page 544
11.5. Di.erential Forms......Page 555
11.6. The Gromov Norm......Page 567
11.7. Measure Homology......Page 576
11.8. Mostow Rigidity......Page 592
11.9. Historical Notes......Page 609
12.1. Limit Sets......Page 612
12.2. Limit Sets of Discrete Groups......Page 616
12.3. Limit Points......Page 629
12.4. Geometrically Finite Discrete Groups......Page 639
12.5. Nilpotent Groups......Page 656
12.6. The Margulis Lemma......Page 666
12.7. Geometrically Finite Manifolds......Page 678
12.8. Historical Notes......Page 689
13.1. Orbit Spaces......Page 693
13.2. (X,G)-Orbifolds......Page 703
13.3. Developing Orbifolds......Page 713
13.4. Gluing Orbifolds......Page 736
13.5. Poincar´e’s Theorem......Page 752
13.6. Historical Notes......Page 755
Bibliography......Page 757
Index......Page 780