Combinatorial and Geometric Group Theory: Dortmund and Ottawa-Montreal conferences

✍ Scribed by Oleg Bogopolski, Inna Bumagin, Olga Kharlampovich, Enric Ventura

• Publisher: Birkhäuser Basel
• Year: 2010
• Tongue: English
• Leaves: 325
• Series: Trends in Mathematics
• Edition: 1st Edition.
• Category: Library

✦ Synopsis

This volume assembles several research papers in all areas of geometric and combinatorial group theory originated in the recent conferences in Dortmund and Ottawa in 2007. It contains high quality refereed articles developping new aspects of these modern and active fields in mathematics. It is also appropriate to advanced students interested in recent results at a research level.

✦ Table of Contents

Cover......Page 1
Trends in Mathematics......Page 3
Combinatorial
and Geometric
Group Theory......Page 4
ISBN 9783764399108......Page 5
Table of contents......Page 6
Preface......Page 8
1. Definitions, problems and motivations......Page 11
2. A sketch of the proof of F. Grunewald and A. Lubotzky that Aut(F3) has no Kazhdan’s property (T)......Page 15
4. Finite index subgroups of Aut(Fn) containing IA(Fn)......Page 17
5. Congruence subgroups SCong(n, k) in SAut(Fn)......Page 18
6. A subgroup K(n) of index 2 in SCong(n, 2)......Page 20
7. K(3) and some its overgroups with infinite abelianization......Page 23
8. The group K(n) for n 4......Page 25
References......Page 26
Introduction......Page 29
1. Improved relative train track maps......Page 31
2. More on train tracks......Page 35
3. Terminology and examples......Page 39
4. Strata of superlinear growth......Page 43
5. Polynomially growing automorphisms......Page 47
6. Proof of the main result......Page 52
References......Page 62
1. Introduction......Page 65
2.2. Rewriting in monoids......Page 69
2.3. Convergent rewriting systems......Page 70
2.4. Computing with infinite systems......Page 73
3.1. Finite length-reducing systems......Page 74
3.2. Infinite length-reducing systems......Page 75
4.1. General results......Page 76
5.1. Geodesic systems......Page 79
5.2. Geodesically perfect systems......Page 83
6. Knuth-Bendix completion for geodesically perfect systems......Page 85
7.1. Graph groups......Page 87
7.2. Coxeter groups......Page 88
7.3. HNN-extensions......Page 89
7.4. Free products with amalgamation......Page 90
8. Stallings’ pregroups and their universal groups......Page 91
8.1. Rewriting systems for universal groups......Page 93
8.2. Characterisation of pregroups in terms of geodesic systems......Page 96
References......Page 98
1. Introduction......Page 103
2.1. Asymptotic densities......Page 105
2.2. Generating random elements and multiplicative measures......Page 106
2.3. The frequency measure......Page 108
2.5. Context-free and regular languages as a measuring tool......Page 109
3.1. Subgroup and coset graphs......Page 110
3.2. Schreier transversals......Page 113
4. Measuring subsets of F......Page 114
5.1. Comparing Schreier representatives......Page 118
5.2. Comparing regular sets......Page 119
References......Page 125
1. Preliminaries......Page 129
2. Elementary groups......Page 131
3. Crystallographic groups......Page 134
4.3. Case 2.......Page 136
4.5. Case 5.......Page 137
5.1. Case 7.......Page 138
5.2. Case 8.......Page 140
5.3. Case 6.......Page 142
5.4. Case 9.......Page 143
6.2. Case 10.......Page 147
6.3. Case 16.......Page 149
7.1. Case 14.......Page 151
7.2. Case 15.......Page 153
7.4. Case 12.......Page 154
7.5. Case 17.......Page 155
References......Page 156
1. Solving random equations......Page 159
1.1. Making the problems meaningful......Page 160
1.3. Decision problems......Page 161
2. The memory-length approach......Page 162
2.1. The memory-length algorithm......Page 163
3.1. Garside monoids and groups......Page 164
3.3. Rational normal form......Page 165
4. Several length functions on Garside groups......Page 167
4.1. Quasi-geodesics in Garside groups......Page 169
4.2. Quasi-geodesics in embedded Garside groups......Page 171
4.3. The case of the braid group......Page 172
5.2. A detailed comparison......Page 173
5.5. When the number of generators varies......Page 174
6. Concluding remarks and proposed future research......Page 177
References......Page 178
Introduction......Page 181
1.1. Diagrams......Page 185
1.2. Diagrams with small cancellation conditions......Page 187
1.3. Transversals in diagrams with Small Cancellation Conditions......Page 191
1.4. The Main Theorem for almost σ-complete presentations......Page 193
2.1. Words......Page 196
3.1. 1-corner regions......Page 200
3.2. 2-corner regions......Page 201
Appendix......Page 204
References......Page 212
1.1. Motivation......Page 213
1.2. Milestones of the theory of equations in free groups......Page 214
1.3. New age......Page 215
2. Basic notions of algebraic geometry over groups......Page 216
3.1. Definitions and elementary properties......Page 218
3.2. Lyndon’s completion FZ[t]......Page 221
4.1. Structure and embeddings......Page 223
4.2. Triangular quasi-quadratic systems......Page 224
5. Elimination process......Page 228
5.1. Generalized equations......Page 229
5.2. Elementary transformations......Page 231
5.3. Derived transformations and auxiliary transformations......Page 233
5.4.1. Tietze cleaning and entire transformation.......Page 238
5.4.3. Quadratic case.......Page 239
5.4.4. Entire transformation goes infinitely.......Page 240
7. Stallings foldings and algorithmic problems......Page 247
References......Page 249
1. Introduction......Page 253
2. The set-up......Page 254
3. The proof of Proposition 1.2......Page 256
4. A little history and some references......Page 258
References......Page 259
1. Introduction......Page 261
2. The stair algorithm for functions in PL<+(I)......Page 262
3. Mather invariants for functions in PL>+(I)......Page 265
4. Equivalence of the two points of view......Page 266
5. Applications: centralizers and generalizations......Page 268
References......Page 269
1. Introduction......Page 271
2.1. Logical preliminaries......Page 273
3.1. Systems of equations......Page 274
3.3. Radicals......Page 275
3.4. Coordinate monoids......Page 276
4. Commutative monoids with cancellation......Page 277
5. Coefficient free algebraic geometry over N......Page 279
5.1. Properties of finitely generated commutative positive monoids with cancellation......Page 280
5.2. Ordering of submonoids of Zn......Page 284
6. Geometric and universal equivalence......Page 285
7. Dimension theory......Page 287
References......Page 288
Introduction......Page 289
1. Thompson’s group......Page 290
2. The Schreier graph of the action of F on the set of dyadic rational numbers......Page 291
3. Coamenability of stabilizers of several dyadic rationals......Page 294
4. The Schreier graph of the action of F on L2([0, 1])......Page 296
5. Parts of the Cayley graph of F......Page 299
References......Page 305
1. Introduction......Page 307
2. Virtually free groups......Page 308
4. The proof of the theorem......Page 309
6. Nielsen equivalence and T-systems......Page 312
References......Page 314
1. Preliminaries......Page 317
2. Free products......Page 320
3. HNN-extensions......Page 322
References......Page 324