Combinatorial and Geometric Group Theory (eBook)

Title page 3
Copyright page 4
Table of contents 5
Preface 7
Subgroups of Small Index in Aut(Fn) and Kazhdan’s Property (T) 9
1. Definitions, problems and motivations 9
2. A sketch of the proof of F. Grunewald and A. Lubotzky that Aut(F3) has no Kazhdan’s property (T) 13
3. Some notations and useful automorphisms 15
4. Finite index subgroups of Aut(Fn) containing IA(Fn) 15
5. Congruence subgroups SCong(n, k) in SAut(Fn) 16
6. A subgroup K(n) of index 2 in SCong(n, 2) 18
7. K(3) and some its overgroups with infinite abelianization 21
8. The group K(n) for n 4 23
References 24
Dynamics of Free Group Automorphisms 26
Introduction 26
1. Improved relative train track maps 28
2. More on train tracks 32
3. Terminology and examples 36
4. Strata of superlinear growth 40
5. Polynomially growing automorphisms 44
6. Proof of the main result 49
Glossary 59
References 59
Geodesic Rewriting Systems and Pregroups 61
1. Introduction 61
2. Rewriting techniques 65
2.1. Basics 65
2.2. Rewriting in monoids 65
2.3. Convergent rewriting systems 66
2.4. Computing with infinite systems 69
3. Length-reducing and Dehn systems 70
3.1. Finite length-reducing systems 70
3.2. Infinite length-reducing systems 71
3.3. Weight-reducing systems 72
4. Preperfect systems 72
4.1. General results 72
5. Geodesically perfect rewriting systems 75
5.1. Geodesic systems 75
5.2. Geodesically perfect systems 79
6. Knuth-Bendix completion for geodesically perfect systems 81
7. Examples of preperfect systems in groups 83
7.1. Graph groups 83
7.2. Coxeter groups 84
7.3. HNN-extensions 85
7.4. Free products with amalgamation 86
8. Stallings’ pregroups and their universal groups 87
8.1. Rewriting systems for universal groups 89
8.2. Characterisation of pregroups in terms of geodesic systems 92
References 94
Regular Sets and Counting in Free Groups 98
1. Introduction 98
2. Preliminaries 100
2.1. Asymptotic densities 100
2.2. Generating random elements and multiplicative measures 101
2.3. The frequency measure 103
2.4. Asymptotic classification of subsets 104
2.5. Context-free and regular languages as a measuring tool 104
3. Schreier systems of representatives 105
3.1. Subgroup and coset graphs 105
3.2. Schreier transversals 108
4. Measuring subsets of F 109
5. Comparing sets at infinity 113
5.1. Comparing Schreier representatives 113
5.2. Comparing regular sets 114
References 120
Twisted Conjugacy for Virtually Cyclic Groups and Crystallographic Groups 124
1. Preliminaries 124
2. Elementary groups 126
3. Crystallographic groups 129
4. Cases 1–5 131
4.1. Case 1. 131
4.2. Case 4. 131
4.3. Case 2. 131
4.4. Case 3. 132
4.5. Case 5. 132
5. Cases 6–9 133
5.1. Case 7. 133
5.2. Case 8. 135
5.3. Case 6. 137
5.4. Case 9. 138
6. Cases 10, 13, 16 142
6.1. Case 13. 142
6.2. Case 10. 142
6.3. Case 16. 144
7. Cases 11, 12, 14, 15, and 17 146
7.1. Case 14. 146
7.2. Case 15. 148
7.3. Case 11. 149
7.4. Case 12. 149
7.5. Case 17. 150
8. Concluding remarks 151
References 151
Solving Random Equations in Garside Groups Using Length Functions 153
1. Solving random equations 153
1.1. Making the problems meaningful 154
1.2. The probabilistic model 155
1.3. Decision problems 155
2. The memory-length approach 156
2.1. The memory-length algorithm 157
2.2. Sufficiency for the general problem 158
2.3. Improvements 158
3. Excursion: Garside groups 158
3.1. Garside monoids and groups 158
3.2. Greedy normal form 159
3.3. Rational normal form 159
4. Several length functions on Garside groups 161
4.1. Quasi-geodesics in Garside groups 163
4.2. Quasi-geodesics in embedded Garside groups 165
4.3. The case of the braid group 166
5. Experimental results 167
5.1. Initial experiments 167
5.2. A detailed comparison 167
5.3. When the sentence length varies 168
5.4. When the word length varies 168
5.5. When the number of generators varies 168
5.6. When the number of strings varies 171
6. Concluding remarks and proposed future research 171
References 172
An Application of Word Combinatorics to Decision Problems in Group Theory 174
Introduction 174
1. Preliminaries on small cancellation theory 178
1.1. Diagrams 178
1.2. Diagrams with small cancellation conditions 180
1.3. Transversals in diagrams with Small Cancellation Conditions 184
1.4. The Main Theorem for almost s-complete presentations 186
2. Word combinatorics 189
2.1. Words 189
3. Piece configurations of 1-corner regions and 2-corner regions 193
3.1. 1-corner regions 193
3.2. 2-corner regions 194
3.3. Proof of the Main Theorem 197
Appendix 197
Acknowledgement 205
References 205
Equations and Fully Residually Free Groups 206
1. Introduction 206
1.1. Motivation 206
1.2. Milestones of the theory of equations in free groups 207
1.3. New age 208
2. Basic notions of algebraic geometry over groups 209
3. Fully residually free groups 211
3.1. Definitions and elementary properties 211
3.2. Lyndon’s completion FZ[t] 214
4. Main results in [38] 216
4.1. Structure and embeddings 216
4.2. Triangular quasi-quadratic systems 217
5. Elimination process 221
5.1. Generalized equations 222
5.2. Elementary transformations 224
5.3. Derived transformations and auxiliary transformations 226
5.4. Rewriting process for O 231
5.4.1. Tietze cleaning and entire transformation. 231
5.4.2. Solution tree. 232
5.4.3. Quadratic case. 232
5.4.4. Entire transformation goes infinitely. 233
6. Elementary free groups 240
7. Stallings foldings and algorithmic problems 240
8. Residually free groups 242
References 242
The FN-action on the Product of the Two Limit Trees for an Iwip Automorphism 246
1. Introduction 246
2. The set-up 247
3. The proof of Proposition 1.2 249
4. A little history and some references 251
References 252
Mather Invariants in Groups of Piecewise-linear Homeomorphisms 254
1. Introduction 254
2. The stair algorithm for functions in PL< +(I)
3. Mather invariants for functions in PL> +(I)
4. Equivalence of the two points of view 259
5. Applications: centralizers and generalizations 261
References 262
Algebraic Geometry over the Additive Monoid of Natural Numbers: Systems of Coefficient Free Equations 264
1. Introduction 264
2. A-monoids 266
2.1. Logical preliminaries 266
3. Introduction to algebraic geometry 267
3.1. Systems of equations 267
3.2. Algebraic sets 268
3.3. Radicals 268
3.4. Coordinate monoids 269
3.5. Equationally Noetherian monoids 270
4. Commutative monoids with cancellation 270
5. Coefficient free algebraic geometry over N 272
5.1. Properties of finitely generated commutative positive monoids with cancellation 273
5.2. Ordering of submonoids of Zn 277
5.3. Proof of Theorem A 278
6. Geometric and universal equivalence 278
7. Dimension theory 280
References 281
Some Graphs Related to Thompson’s Group F 282
Introduction 282
1. Thompson’s group 283
2. The Schreier graph of the action of F on the set of dyadic rational numbers 284
3. Coamenability of stabilizers of several dyadic rationals 287
4. The Schreier graph of the action of F on L2([0, 1]) 289
5. Parts of the Cayley graph of F 292
References 298
Generating Tuples of Virtually Free Groups 300
1. Introduction 300
2. Virtually free groups 301
3. The reduced core of a graph of groups 302
4. The proof of the theorem 302
5. The rank problem 305
6. Nielsen equivalence and T-systems 305
References 307
Limits of Thompson’s Group F 309
1. Preliminaries 309
2. Free products 312
3. HNN-extensions 314
References 316