We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented groups we show that the linear sci is preserved under amalgamated products over finitely generated one-ended groups. Eventually one proves that most non-uniform lattices have linear sci.

* The authors thank the “Swiss National Science Foundation” for its support.We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras...

We establish the lower bound $p_{2t}(e,e)\succsim exp\left(-t^{1/3}\right)$, for the large times asymptotic behaviours of the probabilities $p_{2t}(e,e)$ of return to the origin at even times $2t$, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer $r$, such that any of its finitely generated subgroup admits a generating set of cardinality less or equal to $r$.)

Nous précisons le comportement exponentiel de la fonction orbitale d'un quelconque groupe discret d'isométries en courbure négative.

We describe a sufficient condition for a finitely generated group to have infinite asymptotic dimension. As an application, we conclude that the first Grigorchuk group has infinite asymptotic dimension.

Let $G$ be a group, $\beta G$ be the Stone-Čech compactification of $G$ endowed with the structure of a right topological semigroup and $G^{*}=\beta G\setminus G$. Given any subset $A$ of $G$ and $p\in G^{*}$, we define the $p$-companion ${\Delta }_p\left(A\right)=A^{*}\cap Gp$ of $A$, and characterize the subsets with finite and discrete ultracompanions.