Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 012, 17 pages      arXiv:1908.02365      https://doi.org/10.3842/SIGMA.2020.012

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

Dave Witte Morris
Department of Mathematics and Computer Science, University of Lethbridge,Lethbridge, Alberta, T1K 3M4, Canada

Received August 16, 2019, in final form March 05, 2020; Published online March 11, 2020

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 ${\rm SL}(n,{\mathbb Z})$ 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.

Key words: arithmetic group; quasi-isometric; bounded generation; discrete subgroup.

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