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We prove that two countable locally finite-by-abelian groups G,H endowed withproper left-invariant metrics are coarsely equivalent if and only if theirasymptotic dimensions coincide and the groups are either bothfinitely-generated or both are infinitely generated. On the other hand, we showthat each countable group G that coarsely embeds into a countable abelian groupis locally nilpotent-by-finite. Moreover, the group G is locallyabelian-by-finite if and only if G is undistorted in the sense that G can bewritten as the union of countably many finitely generated subgroups G_n suchthat each G_n is undistorted in G_{n+1} (which means that the identityinclusion from G_n to G_{n+1} is a quasi-isometric embedding with respect toword metrics).

The coarse classification of countable abelian groups