Amenability of Discrete Groups by Examples Paperback / softback

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.