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. [-]

Quadratic polynomials have been investigated since the beginnings of complex dynamics, and are often approached through combinatorial theories such as laminations or Hubbard trees. I will explain how both of these approaches fit in a more algebraic framework: that of iterated monodromy groups. The invariant associated with a quadratic polynomial is a group acting on the infinite binary tree, these groups are interesting in their own right, and provide insight and structure to complex dynamics: I will explain in particular how the conversion between Hubbard trees and external angles amounts to a change of basis, how the limbs and wakes may be defined in the language of group theory, and present a model of the Mandelbrot set consisting of groups. This is joint work with Dzmitry Dudko and Volodymyr Nekrashevych.[-]

I will present a general notion of automatic action, based on Büchi automata, and show how it unifies a large number of subclasses, in particular the automatic groups by Cannon, Thurston et al., the transducer groups by Aleshin, Grigorchuk, Sushchansky, Sidki et al., and substitutional subshifts. I will present some algorithms for these groups, and in particular show under an extra condition (boundedness) that their orbit relation is computable. This will have strong decidability consequences, such as that the order problem, aperiodicity, minimality, etc. for automatic transformations is decidable.[-]