On quasi-isometric embeddings of Lamplighter groups

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 ${\mathcal Q}I(\Gamma_{{\mathbb{Z}}_n})$.