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We say that a subset $X$ quasi-isometrically boundedly generates a finitely generated group $Gamma$ if each element $gamma$ of a finite-index subgroup of $Gamma$ can be written as a product $gamma = x_1 x_2 cdots x_r$ of a bounded number of elements of $X$, such that the word length of each $x_i$ is bounded by a constant times the word length of $gamma$. A. Lubotzky, S. Mozes, and M.S. Raghunathan observed in 1993 that ${rm SL}(n,{mathbb Z})$ is quasi-isometrically boundedly generated by the elements of its natural ${rm SL}(2,{mathbb Z})$ subgroups. We generalize (a slightly weakened version of) this by showing that every $S$-arithmetic subgroup of an isotropic, almost-simple ${mathbb Q}$-group is quasi-isometrically boundedly generated by standard ${mathbb Q}$-rank-1 subgroups.
We describe a family of finitely presented groups which are quasi-isometric but not bilipschitz equivalent. The first such examples were described by the first author and are the lamplighter groups $F wr mathbb{Z}$ where $F$ is a finite group; these
We use basic tools of descriptive set theory to prove that a closed set $mathcal S$ of marked groups has $2^{aleph_0}$ quasi-isometry classes provided every non-empty open subset of $mathcal S$ contains at least two non-quasi-isometric groups. It fol
For a positive integer $g$, let $mathrm{Sp}_{2g}(R)$ denote the group of $2g times 2g$ symplectic matrices over a ring $R$. Assume $g ge 2$. For a prime number $ell$, we give a self-contained proof that any closed subgroup of $mathrm{Sp}_{2g}(mathbb{
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group $G$ (e.g., a non-elementary hyperbolic or relatively hyperbolic group, non-exceptional mapping class group, ${rm Out}(F_n)$, dots) embeds
We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically