On quasi-isometric embeddings of Lamplighter groups

Inamdar, S.; Naolekar, Aniruddha

doi:10.1090/S0002-9939-07-08970-8

On quasi-isometric embeddings of Lamplighter groups
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by S. P. Inamdar and Aniruddha C. Naolekar PDF

Proc. Amer. Math. Soc. 135 (2007), 3789-3794 Request permission

Abstract:

We denote by $\Gamma _G$ the Lamplighter group of a finite group $G$. In this article, we show that if $G$ and $H$ are two finite groups with at least two elements, then there exists a quasi-isometric embedding from $\Gamma _G$ to $\Gamma _H$. We also prove that the quasi-isometry group ${\mathcal Q}I(\Gamma _G)$ of $\Gamma _G$ contains all finite groups. We then show that the group of automorphisms of $\Gamma _{{\mathbb Z}_n}$ has infinite index in ${\mathcal Q}I(\Gamma _{{\mathbb Z}_n})$.

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