A Primer on Mapping Class Groups (Princeton Mathematical)

Cover......Page 1
Contents......Page 8
Preface......Page 12
Acknowledgments......Page 14
Overview......Page 18
PART 1. MAPPING CLASS GROUPS......Page 32
1.1 Surfaces and Hyperbolic Geometry......Page 34
1.2 Simple Closed Curves......Page 39
1.3 The Change of Coordinates Principle......Page 53
1.4 Three Facts about Homeomorphisms......Page 58
2.1 Definition and First Examples......Page 61
2.2 Computations of the Simplest Mapping Class Groups......Page 64
2.3 The Alexander Method......Page 75
3.1 Definition and Nontriviality......Page 81
3.2 Dehn Twists and Intersection Numbers......Page 86
3.3 Basic Facts about Dehn Twists......Page 90
3.4 The Center of the Mapping Class Group......Page 92
3.5 Relations between Two Dehn Twists......Page 94
3.6 Cutting, Capping, and Including......Page 99
4. Generating the Mapping Class Group......Page 106
4.1 The Complex of Curves......Page 109
4.2 The Birman Exact Sequence......Page 113
4.3 Proof of Finite Generation......Page 121
4.4 Explicit Sets of Generators......Page 124
5.1 The Lantern Relation and H[sub(1)] (Mod(S); Z)......Page 133
5.2 Presentations for the Mapping Class Group......Page 141
5.3 Proof of Finite Presentability......Page 151
5.4 Hopf’s Formula and H[sub(2)] (Mod(S); Z)......Page 157
5.5 The Euler Class......Page 163
5.6 Surface Bundles and the Meyer Signature Cocycle......Page 170
6.1 Algebraic Intersection Number as a Symplectic Form......Page 179
6.2 The Euclidean Algorithm for Simple Closed Curves......Page 183
6.3 Mapping Classes as Symplectic Automorphisms......Page 185
6.4 Congruence Subgroups, Torsion-free Subgroups, and Residual Finiteness......Page 193
6.5 The Torelli Group......Page 198
6.6 The Johnson Homomorphism......Page 207
7.1 Finite-order Mapping Classes versus Finite-order Homeomorphisms......Page 217
7.2 Orbifolds, the 84(g – 1) Theorem, and the 4g + 2 Theorem......Page 220
7.3 Realizing Finite Groups as Isometry Groups......Page 230
7.4 Conjugacy Classes of Finite Subgroups......Page 232
7.5 Generating the Mapping Class Group with Torsion......Page 233
8.1 Statement of the Theorem......Page 236
8.2 The Quasi-isometry Proof......Page 239
8.3 Two Other Viewpoints......Page 253
9.1 The Braid Group: Three Perspectives......Page 256
9.2 Basic Algebraic Structure of the Braid Group......Page 263
9.3 The Pure Braid Group......Page 265
9.4 Braid Groups and Symmetric Mapping Class Groups......Page 270
PART 2. TEICHMÜLLER SPACE AND MODULI SPACE......Page 278
10.1 Definition of Teichmüller Space......Page 280
10.2 Teichmüller Space of the Torus......Page 282
10.3 The Algebraic Topology......Page 286
10.4 Two Dimension Counts......Page 289
10.5 The Teichmüller Space of a Pair of Pants......Page 292
10.6 Fenchel–Nielsen Coordinates......Page 295
10.7 The 9g – 9 Theorem......Page 303
11.1 Quasiconformal Maps and an Extremal Problem......Page 311
11.2 Measured Foliations......Page 317
11.3 Holomorphic Quadratic Differentials......Page 325
11.4 Teichmüller Maps and Teichmüller\'s Theorems......Page 337
11.5 Grötzsch’s Problem......Page 342
11.6 Proof of Teichmüller\'s Uniqueness Theorem......Page 344
11.7 Proof of Teichmüller\'s Existence Theorem......Page 347
11.8 The Teichmüller Metric......Page 354
12.1 Moduli Space as the Quotient of Teichmüller Space......Page 359
12.2 Moduli Space of the Torus......Page 362
12.3 Proper Discontinuity......Page 366
12.4 Mumford’s Compactness Criterion......Page 370
12.5 The Topology at Infinity of Moduli Space......Page 376
12.6 Moduli Space as a Classifying Space......Page 379
PART 3. THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY......Page 382
13.1 The Classi.cation for the Torus......Page 384
13.2 The Three Types of Mapping Classes......Page 387
13.3 Statement of the Nielsen–Thurston Classification......Page 393
13.4 Thurston’s Geometric Classification of Mapping Tori......Page 396
13.5 The Collar Lemma......Page 397
13.6 Proof of the Classification Theorem......Page 399
14. Pseudo-Anosov Theory......Page 407
14.1 Five Constructions......Page 408
14.2 Pseudo-Anosov Stretch Factors......Page 420
14.3 Properties of the Stable and Unstable Foliations......Page 425
14.4 The Orbits of a Pseudo-Anosov Homeomorphism......Page 431
14.5 Lengths and Intersection Numbers under Iteration......Page 436
15.1 A Fundamental Example......Page 441
15.2 A Sketch of the General Theory......Page 451
15.3 Markov Partitions......Page 459
15.4 Other Points of View......Page 462
Bibliography......Page 464
B......Page 482
E......Page 483
H......Page 484
M......Page 485
Q......Page 487
T......Page 488
Z......Page 489