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Quasi-Isometric Bounded Generation by ${\mathbb Q}$-Rank-One Subgroups
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: arithmetic group, quasi-isometric, bounded generation, discrete subgroup. 
@article{SIGMA_2020_16_a11,
     author = {Dave Witte Morris},
     title = {Quasi-Isometric {Bounded} {Generation} by ${\mathbb Q}${-Rank-One} {Subgroups}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a11/}
}
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Dave Witte Morris. Quasi-Isometric Bounded Generation by ${\mathbb Q}$-Rank-One Subgroups. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a11/

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