# Amenability of groups and $G$-sets

## TL;DR

This survey provides a comprehensive, self-contained overview of classical and recent results on the amenability of groups and G-sets, emphasizing combinatorial methods and including recent examples like groups acting on Cantor sets.

## Contribution

It offers an accessible introduction to amenability, highlighting combinatorial tools and extending the discussion to amenable actions and recent classes of examples.

## Key findings

- Overview of classical and recent results in amenability
- Introduction of combinatorial methods in the study of groups
- Discussion of new examples like groups acting on Cantor sets

## Abstract

This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups.