On quasi-isometric embeddings of Lamplighter groups

Abstract

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