viii Contents
§17.6. Special Maps on a Translation Surface 217
§17.7. Existence of Periodic Billiard Paths 219
Chapter 18. Translation Surfaces and the Veech Group 221
§18.1. Affine Automorphisms 221
§18.2. The Diffential Representation 223
§18.3. Hyperbolic Group Actions 224
§18.4. Proof of Theorem 18.1 226
§18.5. Triangle Groups 228
§18.6. Linear and Hyperbolic Reflections 229
§18.7. Behold, The Double Octagon! 232
Part 5. The Totality of Surfaces
Chapter 19. Continued Fractions 239
§19.1. The Gauss Map 239
§19.2. Continued Fractions 241
§19.3. The Farey Graph 242
§19.4. Structure of the Modular Group 244
§19.5. Continued Fractions and the Farey Graph 245
§19.6. The Irrational Case 247
Chapter 20. Teichm¨ uller Space and Moduli Space 251
§20.1. Parallelograms 251
§20.2. Flat Tori 252
§20.3. The Modular Group Again 254
§20.4. Moduli Space 256
§20.5. Teichm¨ uller Space 258
§20.6. The Mapping Class Group 260
Chapter 21. Topology of Teichm¨ uller Space 263
§21.1. Pairs of Pants 263
§21.2. Pants Decompositions 265
§21.3. Special Maps and Triples 267
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