Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.

Dans cet article, on montre que l’espace des groupes marqués est un sous-espace fermé d’un ensemble de Cantor dont la dimension de Hausdorff est infinie. On prouve que la dimension de Minkowski de cet espace est infinie en exhibant des sous-ensembles de groupes marqués à petite simplification dont les dimensions de Minkowski sont arbitrairement grandes. On donne une estimation des dimensions de Minkowski de sous-espaces de groupes à un relateur. On démontre enfin que les dimensions de Minkowski...

Let $S$ be a fixed symmetric finite subset of $S{L}_d\left({𝒪}_K\right)$ that generates a Zariski dense subgroup of $S{L}_d\left({𝒪}_K\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({𝒪}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.

Complexes of groups $G\left(X\right)$ over ordered simplicial complexes $X$ are generalizations to higher dimensions of graphs of groups. We first relate them to complexes of spaces by considering their classifying space $BG\left(X\right)$. Then we develop their homological algebra aspects. We define the notions of homology and cohomology of a complex of groups $G\left(X\right)$ with coefficients in a $G\left(X\right)$-module and show the existence of free resolutions. We apply those notions to study extensions of complexes of groups with constant or abelian kernel....