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Register a new userBilipschitz versus quasi-isometric equivalence for higher rank lamplighter groups
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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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Let $Gamma_d(q)$ denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph $DL_d(q)$, as described by Bartholdi, Neuhauser and Woess. We compute both $Aut(Gamma_d(q))$ and $Out(Gamma_d(q))$ for $d geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d geq 3$. Specifically, we show that when $d geq 3$, the groups $Gamma_d(q)$ have property $R_{infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $Gamma_2(q)=L_q = {mathbb Z}_q wr {mathbb Z}$ have property $R_{infty}$ if and only if $(q,6) eq 1$.
We show that the higher rank lamplighter groups, or Diestel-Leader groups $Gamma_d(q)$ for $d geq 3$, are graph automatic. This introduces a new family of graph automatic groups which are not automatic.
We use basic tools of descriptive set theory to prove that a closed set $mathcal S$ of marked groups has $2^{aleph_0}$ quasi-isometry classes provided every non-empty open subset of $mathcal S$ contains at least two non-quasi-isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains $2^{aleph_0}$ quasi-isometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. They can also be used to prove the existence of $2^{aleph_0}$ quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties.
In this paper, which is the continuation of [EFW2], we complete the proof of the quasi-isometric rigidity of Sol and the lamplighter groups. The results were announced in [EFW1].
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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