  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 (respectively, solvable,
satisfies a non-trivial identity, elementary amenable, of finite decomposition
complexity, etc.) whenever $H$ is. We also discuss some applications to
compression functions of Lipschitz embeddings into uniformly convex Banach
spaces, F{\o}lner functions, and elementary classes of amenable groups.
