Quasi-Isometric Bounded Generation by $\mathbb{Q}$-Rank-One Subgroups

Abstract

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 $\mathrm{SL}(n,\mathbb{Z})$ is quasi-isometrically boundedly generated by the elements of its natural $\mathrm{SL}(2,\mathbb{Z})$ 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.