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SIGMA, 2020, Volume 16, 012, 17 pp. (Mi sigma1549)  

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

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

DOI: https://doi.org/10.3842/SIGMA.2020.012

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Full text: https://www.imath.kiev.ua/~sigma/2020/012/
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Bibliographic databases:

ArXiv: 1908.02365
MSC: 22E40; 20F65; 11F06
Received: August 16, 2019; in final form March 5, 2020; Published online March 11, 2020
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Citation: Dave Witte Morris, “Quasi-Isometric Bounded Generation by ${\mathbb Q}$-Rank-One Subgroups”, SIGMA, 16 (2020), 012, 17 pp.

Citation in format AMSBIB
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\by Dave~Witte~Morris
\paper Quasi-Isometric Bounded Generation by ${\mathbb Q}$-Rank-One Subgroups
\jour SIGMA
\yr 2020
\vol 16
\papernumber 012
\totalpages 17
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\crossref{https://doi.org/10.3842/SIGMA.2020.012}
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