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We say that a subset $X$ quasi-isometrically boundedly generates a finitelygenerated 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 boundednumber of elements of $X$, such that the word length of each $x_i$ is boundedby a constant times the word length of $\gamma$. A. Lubotzky, S. Mozes, andM.S. Raghunathan observed in 1993 that ${\rm SL}(n,{\mathbb Z})$ isquasi-isometrically boundedly generated by the elements of its natural ${\rmSL}(2,{\mathbb Z})$ 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 bystandard ${\mathbb Q}$-rank-1 subgroups.

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