Abstract:
We study amenability of affine algebras (based on the notion of almost-invariant finite-dimensional subspace), and apply it to algebras associated with finitely generated groups. We show that a group G is amenable if and only if its group ring KG is amenable for some (and therefore for any) field K.

Abstract:
We prove a converse to Moore's ``Garden-of-Eden'' theorem: a group G is amenable if and only if all cellular automata living on G that admit mutually erasable patterns also admit gardens of Eden. It had already been conjectured in that amenability could be characterized by cellular automata. We prove the first part of that conjecture.

Abstract:
We show that contracting self-similar groups satisfy the Farrell-Jones conjectures as soon as their universal contracting cover is non-positively curved. This applies in particular to bounded self-similar groups. We define, along the way, a general notion of contraction for groups acting on a rooted tree in a not necessarily self-similar manner.

Abstract:
We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive abscissa of convergence, is algebraic over the ring Q[2^{-s},...,P^{-s}] for some integer P, and show that it can be analytically continued (through root singularities) to the left half-plane. We compute the abscissa of convergence and the functional equation for some prominent examples of branch groups, such as the Grigorchuk and Gupta-Sidki groups.

Abstract:
We describe, up to degree equal to the rank, the Lie algebra associated with the automorphism group of a free group. We compute in particular the ranks of its homogeneous components, and their structure as modules over the linear group. Along the way, we infirm (but confirm a weaker form of) a conjecture by Andreadakis, and answer a question by Bryant-Gupta-Levin-Mochizuki.

Abstract:
In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth $\gamma$: $e^{\sqrt{n}}\precsim\gamma(n)\precsim e^{n^\beta},$ where $\beta=\log_{32}(31)\approx0.991$. He conjectured that the lower bound is actually tight. In this paper we improve the lower bound to $e^{n^\alpha}\precsim\gamma(n),$ where $\alpha\approx0.5157$, with the aid of a computer. This disproves the conjecture that the lower bound be tight.

Abstract:
We introduce L-presentations: group presentations given by a generating set, a set of relations and a set of substitution rules on the generating set producing more relations. We first study in full generality the structure of finitely L-presented groups, i.e. groups for which all the above sets are finite, and then show that a broad class of groups acting on rooted trees admit explicitly constructible finite L-presentations: they are the "branch" groups defined by R. Grigorchuk.

Abstract:
I answer a question from the 1993 International Mathematical Olympiads by constructing an equivalent algebraic problem, and unearth a surprising behaviour of some polynomials over the two-element field.

Abstract:
These are introductory notes on word growth of groups, and include a gentle presentation of wreath products as well as recent results on construction of groups of with given growth function. They are are an expanded version of a mini-course given at "Le Louverain", June 24-27, 2014.

Abstract:
We compute the homology groups $H_*(\operatorname{Out}(F_7);\mathbb Q)$ of the outer automorphism group of the free group of rank $7$. We produce in this manner the first rational homology classes of $\operatorname{Out}(F_n)$ that are neither constant ($*=0$) nor Morita classes ($*=2n-4$).