A quasi-isometric embedding theorem for groups

Abstract

We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (resp., solvable, satisfies a nontrivial identity, elementary amenable, of finite decomposition complexity) whenever $H$ also shares those conditions. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.