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Register a new userQuasi-Isometric Bounded Generation by ${mathbb Q}$-Rank-One Subgroups
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Authors
Dave Witte Morris
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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.
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We describe a family of finitely presented groups which are quasi-isometric but not bilipschitz equivalent. The first such examples were described by the first author and are the lamplighter groups $F wr mathbb{Z}$ where $F$ is a finite group; these groups are finitely generated but not finitely presented. The examples presented in this paper are higher rank generalizations of these lamplighter groups and include groups that are of type $F_n$ for any $n$.
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