Groups and Topological Dynamics

This book is devoted to group-theoretic aspects of topological dynamics such as studying groups using their actions on topological spaces, using group theory to study symbolic dynamics, and other connections between group theory and dynamical systems. One of the main applications of this approach to group theory is the study of asymptotic properties of groups such as growth and amenability. The book presents recently developed techniques of studying groups of dynamical origin using the structure of their orbits and associated groupoids of germs, applications of the iterated monodromy groups to hyperbolic dynamical systems, topological full groups and their properties, amenable groups, groups of intermediate growth, and other topics. The book is suitable for graduate students and researchers interested in group theory, transformations defined by automata, topological and holomorphic dynamics, and theory of topological groupoids. Each chapter is supplemented by exercises of various levels of complexity. - Publisher Contents Preface Chapter 1. Dynamical systems 1.1. Introduction by examples 1.2. Subshifts 1.3. Minimal Cantor systems 1.4. Hyperbolic dynamics 1.5. Holomorphic dynamics Exercises Chapter 2. Group actions 2.1. Structure of orbits 2.2. Micro-supported actions and Rubin’s theorem 2.3. Automata 2.4. Groups acting on rooted trees Exercises Chapter 3. Groupoids 3.1. Basic definitions 3.2. Actions and correspondences 3.3. Fundamental groups 3.4. Complexes of groups and orbispaces 3.5. Compactly generated groupoids 3.6. Hyperbolic groupoids Exercises Chapter 4. Iterated monodromy groups 4.1. Iterated monodromy groups of self-coverings 4.2. Self-similar groups 4.3. Expanding maps and contracting groups 4.4. Iterated monodromy groups of correspondences 4.5. Hyperbolicity 4.6. Iterations of polynomials 4.7. Dynamics on the sphere 4.8. Other applications Exercises Chapter 5. Groups from groupoids 5.1. Full groups 5.2. AF groupoids and bounded type 5.3. Torsion groups 5.4. Homology of totally disconnected étale groupoids 5.5. Almost finite groupoids 5.6. Purely infinite groupoids Exercises Chapter 6. Growth and amenability 6.1. Growth of groups 6.2. Groups of intermediate growth 6.3. Inverted orbits 6.4. Linearly repetitive actions 6.5. Families of groups and non-uniform exponential growth 6.6. Amenability Exercises Bibliography Index Focuses on group-theoretic aspects of topological dynamics such as studying groups using their actions on topological spaces, using group theory to study symbolic dynamics, and other connections between group theory and dynamical systems.