A PRIMER ON MAPPING CLASS GROUPS

PART 1. MAPPING CLASS GROUPS 15
1. Curves, Surfaces, and Hyperbolic Geometry 17
1.1 Surfaces and Hyperbolic Geometry 17
1.2 Simple Closed Curves 22
1.3 The Change of Coordinates Principle 36
1.4 Three Facts about Homeomorphisms 41
2. Mapping Class Group Basics 44
2.1 Definition and First Examples 44
2.2 Computations of the Simplest Mapping Class Groups 47
2.3 The Alexander Method 58
3. Dehn Twists 64
3.1 Definition and Nontriviality 64
3.2 Dehn Twists and Intersection Numbers 69
3.3 Basic Facts about Dehn Twists 73
3.4 The Center of the Mapping Class Group 75
3.5 Relations between Two Dehn Twists 77
3.6 Cutting, Capping, and Including 82
4. Generating the Mapping Class Group 89
4.1 The Complex of Curves 92
4.2 The Birman Exact Sequence 96
4.3 Proof of Finite Generation 104
4.4 Explicit Sets of Generators 107
5. Presentations and Low-dimensional Homology 116
5.1 The Lantern Relation and H1(Mod(S); Z) 116
5.2 Presentations for the Mapping Class Group 124
5.3 Proof of Finite Presentability 134
5.4 Hopf’s Formula and H2(Mod(S); Z) 140
5.5 The Euler Class 146
5.6 Surface Bundles and the Meyer Signature Cocycle 153
6. The Symplectic Representation and the Torelli Group 162
6.1 Algebraic Intersection Number as a Symplectic Form 162
6.2 The Euclidean Algorithm for Simple Closed Curves 166
6.3 Mapping Classes as Symplectic Automorphisms 168
6.4 Congruence Subgroups, Torsion-free Subgroups, and Residual Finiteness 176
6.5 The Torelli Group 181
6.6 The Johnson Homomorphism 190
7. Torsion 200
7.1 Finite-order Mapping Classes versus Finite-order Homeomorphisms 200
7.2 Orbifolds, the 84(g - 1) Theorem, and the 4g + 2 Theorem 203
7.3 Realizing Finite Groups as Isometry Groups 213
7.4 Conjugacy Classes of Finite Subgroups 215
7.5 Generating the Mapping Class Group with Torsion 216
8. The Dehn–Nielsen–Baer Theorem 219
8.1 Statement of the Theorem 219
8.2 The Quasi-isometry Proof 222
8.3 Two Other Viewpoints 236
9. Braid Groups 239
9.1 The Braid Group: Three Perspectives 239
9.2 Basic Algebraic Structure of the Braid Group 246
9.3 The Pure Braid Group 248
9.4 Braid Groups and Symmetric Mapping Class Groups 253

PART 2. TEICHMULLER SPACE AND MODULI SPACE ¨ 261
10. Teichm ¨uller Space 263
10.1 Definition of Teichm¨uller Space 263
10.2 Teichm¨uller Space of the Torus 265
10.3 The Algebraic Topology 269
10.4 Two Dimension Counts 272
10.5 The Teichm¨uller Space of a Pair of Pants 275
10.6 Fenchel–Nielsen Coordinates 278
10.7 The 9g - 9 Theorem 286
11. Teichm ¨uller Geometry 294
11.1 Quasiconformal Maps and an Extremal Problem 294
11.2 Measured Foliations 300
11.3 Holomorphic Quadratic Differentials 308
11.4 Teichm¨uller Maps and Teichm¨uller’s Theorems 320
11.5 Gr¨otzsch’s Problem 325
11.6 Proof of Teichm¨uller’s Uniqueness Theorem 327
11.7 Proof of Teichm¨uller’s Existence Theorem 330
11.8 The Teichm¨uller Metric 337
12. Moduli Space 342
12.1 Moduli Space as the Quotient of Teichm¨uller Space 342
12.2 Moduli Space of the Torus 345
12.3 Proper Discontinuity 349
12.4 Mumford’s Compactness Criterion 353
12.5 The Topology at Infinity of Moduli Space 359
12.6 Moduli Space as a Classifying Space 362

PART 3. THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY 365
13. The Nielsen–Thurston Classification 367
13.1 The Classification for the Torus 367
13.2 The Three Types of Mapping Classes 370
13.3 Statement of the Nielsen–Thurston Classification 376
13.4 Thurston’s Geometric Classification of Mapping Tori 379
13.5 The Collar Lemma 380
13.6 Proof of the Classification Theorem 382
14. Pseudo-Anosov Theory 390
14.1 Five Constructions 391
14.2 Pseudo-Anosov Stretch Factors 403
14.3 Properties of the Stable and Unstable Foliations 408
14.4 The Orbits of a Pseudo-Anosov Homeomorphism 414
14.5 Lengths and Intersection Numbers under Iteration 419
15. Thurston’s Proof 424
15.1 A Fundamental Example 424
15.2 A Sketch of the General Theory 434
15.3 Markov Partitions 442
15.4 Other Points of View 445
Bibliography 447
Index 465

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.
A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.