We extend Gromov's notion of asymptotic dimension of finitely generated groups to all discrete groups. In particular, we extend the Hurewicz type theorem proven in [B-D2] to general groups. Then we use this extension to prove a formula for the asymptotic dimension of finitely generated solvable groups in terms of their Hirsch length.

We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov, and we give an improved estimate of that dimension in terms of the relator length. The construction is similar to one of Bell and Dranishnikov, but we produce a sharper estimate.

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...

We study infinite finitely generated groups having a finite set of conjugacy classes meeting all cyclic subgroups. The results concern growth and the ascending chain condition for such groups.

We show that there exists a finitely generated group of growth $\sim f$ for all functions $f:{ℝ}_{+}\to {ℝ}_{+}$ satisfying $f\left(2R\right)\le f{\left(R\right)}^2\le f\left({\eta }_{+}R\right)$ for all $R$ large enough and ${\eta }_{+}\approx 2.4675$ the positive root of $X^3-X^2-2X-4$. Set ${\alpha }_{-}=log2/log{\eta }_{+}\approx 0.7674$; then all functions that grow uniformly faster than $exp\left(R^{{\alpha }_{-}}\right)$ are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth $\sim exp\left(R^{\alpha }\right)$ for a dense set of $\alpha \in [{\alpha }_{-},1]$.

Hass, Rubinstein, and Scott showed that every closed aspherical (irreducible) 3-manifold whose fundamental group contains the fundamental group of a closed aspherical surface, is covered by Euclidean space. This theorem does not generalize to higher dimensions. However, we provide geometric tools with which variations of this theorem can be proved in all dimensions.

We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.

We give a construction of homomorphisms from a group into the reals using random walks on the group. The construction is an alternative to an earlier construction that works in more general situations. Applications include an estimate on the drift of random walks on groups of subexponential growth admitting no nontrivial homomorphism to the integers and inequalities between the asymptotic drift and the asymptotic entropy. Some of the entropy estimates obtained have applications independent of the...

We prove an exact formula for the asymptotic dimension (asdim) of a free product. Our main theorem states that if A and B are finitely generated groups with asdim A = n and asdim B ≤ n, then asdim (A*B) = max n,1.