We show that every group of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group such that is amenable (resp., solvable, satisfies a nontrivial identity, elementary amenable, of finite decomposition complexity) whenever 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.
Alexander Yu. Olshanskii. Denis V. Osin. "A quasi-isometric embedding theorem for groups." Duke Math. J. 162 (9) 1621 - 1648, 15 June 2013. https://doi.org/10.1215/00127094-2266251