The main topic of the book is amenable groups, i.e., groups on which there exist invariant finitely additive measures. It was discovered that the existence or non-existence of amenability is responsible for many interesting phenomena such as, e.g., the Banach-Tarski Paradox about breaking a sphere into two spheres of the same radius. Since then, amenability has been actively studied and a number of different approaches resulted in many examples of amenable and non-amenable groups. In the book, the author puts together main approaches to study amenability. A novel feature of the book is that the exposition of the material starts with examples which introduce a method rather than illustrating it. This allows the reader to quickly move on to meaningful material without learning and remembering a lot of additional definitions and preparatory results; those are presented after analyzing the main examples. The techniques that are used for proving amenability in this book are mainly a combination of analytic and probabilistic tools with geometric group theory.
Author(s): Kate Juschenko
Series: Mathematical Surveys and Monographs, 266
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 178
City: Providence
Cover
Title page
Contents
Preface
Chapter 1. Introduction
Chapter 2. The first germs of amenability: Paradoxical decompositions
1. Paradoxical actions
2. Hausdorff paradox
3. Banach-Tarski paradox
4. Pea to Sun paradox
5. First definitions: invariant mean, Følner condition
6. First examples
7. Operations that preserve amenability
8. Amenable actions: first definitions
9. Paradoxical decompositions and amenable actions
10. Faithful amenable actions of non-amenable groups
Chapter 3. Elementary amenable groups
1. Simplification of the class of elementary amenable groups
2. Elementary classes of amenable groups
3. Torsion groups
4. Dichotomy of the growth of elementary amenable groups
5. Groups of intermediate growth
6. Groups acting on rooted trees
7. Non-elemetary amenable groups acting on rooted trees
8. The first Grigorchuk’s group
9. Basilica group
10. Finitely generated branch groups
Chapter 4. The topological full group of a Cantor minimal system
1. The topological full group of Cantor minimal systems. Definition and basic facts
2. Basic facts on dynamics on the Cantor space
3. Simplicity of the commutator subgroup
4. Finite generation of the commutator subgroup of a minimal subshift
5. Spatial isomorphism of topological full groups
6. Boyle’s flip conjugacy theorem
7. Examples of non-isomorphic topological full groups
Unique ergodicity
Chapter 5. Lamplighter actions and extensive amenability
1. Extensive amenability: basic definition and properties
2. Recurrent actions: definition and basic properties
3. Recurrent actions are extensively amenable
4. Extensions of actions
5. Functorial properties of extensive amenability
6. Extensively amenable and amenable actions
7. Inverted orbits of random walks
Proof of “recurrent implies extensive amenability” using inverted orbits
Chapter 6. Amenability of topological full groups
1. Amenability of topological full groups of Cantor minimal system: first proof
2. The interval exchange transformation group
3. Realization of finitely generated subgroups of IET as subgroups of the full topological group of ℤ^{?}
4. Amenability of subgroups of IET of low ranks
Chapter 7. Subgroups of topological full groups of intermediate growth
1. Multisections
2. Adding two multisections together
3. Bisections
4. Finitely generated actions
5. An introduction to periodicity and intermediate growth
6. Examples
Chapter 8. An amenability criterion via actions
1. Group theoretical version of the criteria
2. Actions by homeomorphisms. Preliminary definitions
3. Topological version of criterion
Chapter 9. Groups acting on Bratteli diagrams
1. Homeomorphisms of Bratteli of bounded type
2. Vershik transformations
3. Realization of Cantor minimal systems as homeomorphisms of Bratteli-Vershik diagrams
4. One-dimensional tilings
Chapter 10. Groups acting on rooted trees
1. Finite automata of bounded activity
2. Groups of Neumann-Segal type
3. Iterated monodromy groups of polynomial iterations
4. Penrose tilings
5. Linearly and quadratically growing automata
Appendix A. Definitions of amenability and basic facts
1. Means and measures
2. Almost invariant vectors of the left regular representation
3. Weakening of the Følner condition
4. Kesten’s criterion
5. Hulanicki’s criterion
6. Weak containment of representations
7. Day’s fixed point theorem
8. Tarski theorem and Tarski numbers
9. Gromov’s doubling condition. Grasshopper criteria
Appendix B. Related open problems
Is this group amenable?
Topological full groups
Bibliography
Subject Index
Notation Index
Back Cover
