**Mathematical Surveys and Monographs**

Volume: 117;
2005;
231 pp;
Hardcover

MSC: Primary 20; 37;
Secondary 22

Print ISBN: 978-0-8218-3831-0

Product Code: SURV/117

List Price: $69.00

Individual Member Price: $55.20

**Electronic ISBN: 978-1-4704-1344-6
Product Code: SURV/117.E**

List Price: $69.00

Individual Member Price: $55.20

#### Supplemental Materials

# Self-Similar Groups

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*Volodymyr Nekrashevych*

Self-similar groups (groups generated by automata) initially appeared as examples of groups that are easy to define but have exotic properties like nontrivial torsion, intermediate growth, etc. This book studies the self-similarity phenomenon in group theory and shows its intimate relationship with dynamical systems and more classical self-similar structures, such as fractals, Julia sets, and self-affine tilings. This connection is established through the central topics of the book, which are the notions of the iterated monodromy group and limit space.

A wide variety of examples and different applications of self-similar groups to dynamical systems and vice versa are discussed. In particular, it is shown that Julia sets can be reconstructed from the respective iterated monodromy groups and that groups with exotic properties can appear not just as isolated examples, but as naturally defined iterated monodromy groups of rational functions.

The book offers important, new mathematics that will open new avenues of research in group theory and dynamical systems. It is intended to be accessible to a wide readership of professional mathematicians.

#### Table of Contents

# Table of Contents

## Self-Similar Groups

- Contents v6 free
- Preface vii8 free
- Chapter 1. Basic Definitions and Examples 114 free
- 1.1. Rooted tree X* and its boundary X[sup(ω)] 114
- 1.2. Groups acting on rooted trees 215
- 1.3. Automata 316
- 1.4. Wreath products 922
- 1.5. Self-similar actions 1023
- 1.6. The Grigorchuk group 1225
- 1.7. The adding machine and self-similar actions of Z[sup(n)] 1629
- 1.8. Branch groups 1730
- 1.9. Other examples 2134
- 1.10. Bi-reversible automata and free groups 2336

- Chapter 2. Algebraic Theory 3144
- 2.1. Permutational bimodules 3144
- 2.2. Bases of a covering bimodule and wreath recursions 3245
- 2.3. Tensor products and self-similar actions 3346
- 2.4. The left G-space M[sup(⊗ω)] 3649
- 2.5. Virtual endomorphisms 3750
- 2.6. The linear recursion 4053
- 2.7. Invariant subgroups and the kernel of a self-similar action 4154
- 2.8. Recurrent actions 4457
- 2.9. Example: free abelian groups 4659
- 2.10. Rigidity 5063
- 2.11. Contracting actions 5770
- 2.12. Finite-state actions of Z[sup(n)] 6275
- 2.13. Defining relations and word problem 6477

- Chapter 3. Limit Spaces 7184
- 3.1. Introduction 7184
- 3.2. The limit G-space χ[sub(G)] 7386
- 3.3. Digit tiles 7891
- 3.4. Axiomatic description of χ[sub(G)] 8295
- 3.5. Connectedness of χ[sub(G)] 91104
- 3.6. The limit space J[sub(G)] 92105
- 3.7. Limit spaces of self-similar subgroups 96109
- 3.8. The limit space J[sub(G)] as a hyperbolic boundary 97110
- 3.9. Groups of bounded automata 102115
- 3.10. One-dimensional subdivision rules 110123
- 3.11. Uniqueness of the limit space 113126

- Chapter 4. Orbispaces 117130
- Chapter 5. Iterated Monodromy Groups 137150
- 5.1. Definition of iterated monodromy groups 137150
- 5.2. Standard self-similar actions of IMG(p) on X* 142155
- 5.3. Iterated monodromy groups of limit dynamical systems 146159
- 5.4. Length structures and expanding maps 148161
- 5.5. Limit spaces of iterated monodromy groups 150163
- 5.6. Iterated monodromy group of a pull-back 154167
- 5.7. The limit solenoid and inverse limits of self-coverings 156169

- Chapter 6. Examples and Applications 161174
- 6.1. Expanding self-coverings of orbifolds 161174
- 6.2. Limit spaces of free Abelian groups 165178
- 6.3. Examples of self-coverings of orbifolds 169182
- 6.4. Rational functions 174187
- 6.5. Combinatorial equivalence and Thurston's Theorem 176189
- 6.6. "Twisted rabbit" question of J. Hubbard 179192
- 6.7. Abstract kneading automata 185198
- 6.8. Topological polynomials and critical portraits 190203
- 6.9. Iterated monodromy groups of complex polynomials 193206
- 6.10. Polynomials from kneading automata 196209
- 6.11. Quadratic polynomials 203216
- 6.12. Examples of iterated monodromy groups of polynomials 208221
- 6.13. Matings 215228

- Bibliography 223236
- Index 229242

#### Readership

Graduate students and research mathematicians interested in group theory and dynamical systems.