Appears in collections : Combinatorics, automata and number theory / Combinatoire, automates et théorie des nombres, Ecoles de recherche

I shall discuss old and new results on amenability of groups, and more generally G-sets. This notion traces back to von Neumann in his study of the Hausdorff-Banach-Tarski paradox, and grew into one of the fundamental properties a group may / may not have -- each time with important consequences. Lecture 1. I will present the classical notions and equivalent definitions of amenability, with emphasis on group actions and on combinatorial aspects: Means, Folner sets, random walks, and paradoxical decompositions. Lecture 2. I will describe recent work by de la Salle et al. leading to a quite general criterion for amenability, as well as some still open problems. In particular, I will show that full topological groups of minimal Z-shifts are amenable. Lecture 3. I will explain links between amenability and cellular automata, in particular the "Garden of Eden" properties by Moore and Myhill: there is a characterization of amenable groups in terms of whether these classical theorems still hold.