The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.

In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D.Mostow. In the case of buildings...

On utilise l'équivalence due à M. Gromov entre l'hyperbolicité d'un espace métrique géodésique et le fait que ses cônes asymptotiques sont des arbres réels. Ce résultat permet tout d'abord de donner une nouvelle preuve du fait que l'inégalité isopérimétrique sous-quadratique implique l'hyperbolicité. Les avantages de cette preuve sont qu'elle est très courte et qu'elle utilise une seule propriété de la fonction aire de remplissage des courbes fermées, l'inégalité du quadrilatère....

We introduce the notion of a critical constant $c_{rt}$ for recurrence of random walks on $G$-spaces. For a subgroup $H$ of a finitely generated group $G$ the critical constant is an asymptotic invariant of the quotient $G$-space $G/H$. We show that for any infinite $G$-space $c_{rt}\ge 1/2$. We say that $G/H$ is very small if $c_{rt}\<1$. For a normal subgroup $H$ the quotient space $G/H$ is very small if and only if it is finite. However, we give examples of infinite very small $G$-spaces. We show also that critical constants for recurrence can be used...