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топология / Farb, Margalit, A primer on mapping class groups

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Index

(−1, 1)–form, 338 4g + 2 theorem, 188

proof, 196

84(g − 1) theorem, 187 proof, 195

9g − 9 lemma, 356 Sg –tile, 277

I(Sg )–equivalent, 231 ǫ–thick part, 361

kth Torelli group, 244 1–pronged singularity, 346 2–chain relation, 129

Alexander Lemma, 47 Alexander Method, 58, 59

proof, 63 statement, 59

Alexander trick, 47

algebraic intersection number, 207 on the torus, 28

algebraic topology on Teich(Sg ), 276 algebraic topology on Teichm¨uller space,

274

ambient isotopy, 34 Anosov, 373

Anosov diffeomorphism, 392 arc, 35

simple, 35

arc complex, 156 area of a lattice, 272 Artin group, 149

axis of a hyperbolic isometry, 19

belt trick, 251

Beltrami differential, 338 Beltrami differentials

versus quadratic differentials, 341 Beltrami equation, 339

Bers' constant, 363 big diagonal, 248

bigon, 29

bigon criterion, 29 first proof, 31 second proof, 31

biholomorphic, 303

Birman Exact Sequence, 105, 118 forgetful map, 118 non-splitting of, 176

point pushing map, 119 Birman exact sequence

for the Torelli group, 230 generalized, 250 statement, 119

Birman–Craggs–Johnson homomorphisms, 241

Birman–Hilden theorem, 259, 260 for closed surfaces, 263

Borel construction, 158 boundary components, 16 bounding pair, 39, 228 bounding pair map, 228 braid, 245

braid diagram, 246 braid group, 246

abelianization, 253

as a mapping class group, 249 center, 253

is torsion free, 253

low complexity cases, 252 modulo center, 254 presentation, 252

pure, 255

roots of central elements, 254 standard generators, 247

braid group on 3 strands, 82 braid groups, 245

braid relation, 83, 261 converse, 84

branched cover, 312

degree of ramification at a point of, 322

450

ramification point of, 322 of a Riemann surface, 322

of a topological surface, 312 bundle isomorphism, 176 Burkhardt generators, 206

canonical reduction system, 377 capping, 79

capping boundary components, 123 capping the boundary

in the Torelli group, 229 Cayley graph, 92

center, 75

of the mapping class group, 75 central extension of a group, 169 chain, 129

chain of simple closed curves, 40, 101 nonseparating, 40

chain relation, 129, 262

Change of Coordinates Principle, 36, 38 examples, 39

characteristic classes, 178 characteristic subgroup, 225, 243 classification of mapping classes

proof, 385 restatement, 380 statement, 379

classification of simple closed curves, 37 classification of surfaces, 16

classifying map, 177 click, 188

click homeomorphism, 45 closed curve, 21

essential, 21 multiple, 22 collar lemma, 384

complex derivatives, 302 complex dilatation, 337

complex hyperplane arrangement, 248 complex of curves, 113, 365

connectedness, 113 sporadic cases, 115

complex of nonseparating curves, 116 complex of spaces, 159

complex structures vs. hyperbolic structures, 301

configuration space, 247 conformal map, 303 conformal structures, 301

INDEX

congruence subgroup

of the mapping class group, 222 of the symplectic group, 220

congruence subgroups of Sp(2g, Z), 220

conjugacy classes of finite subgroups of

Mod(S), 200 conjugacy separable, 223

coordinate system of curves, 284 curve, 21

lift of, 22 separating, 37 simple closed, 24

curves isotopic, 33

cut system complex, 150 cut systems, 150

cutting a surface, 36 cylinder decomposition, 290

degenerate star relation, 154 degree at a point

of an orbifold covering, 192 Dehn twists, 65

action on homology, 211

action on simple closed curves, 66 and free groups, 85

and intersection number, 69 basic properties, 72 conjugates, 73

definition, 64

groups generated by, 87 have infinite order, 69 left versus right, 66 nontriviality, 67

on the torus, 65 powers, 74

relations between, 82 roots, 130

via cutting and gluing, 65 Dehn–Lickorish Theorem, 110

proof, 127 Dehn–Nielsen–Baer Theorem, 91

analytical proof, 108

for punctured surfaces, 92, 104 quasi-isometry proof, 100 topological proof, 107

dilatation, 303 dilatation at a point, 302

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