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, F?lner functions, and elementary classes of amenable groups.
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