The coarse classification of countable abelian groups

We prove that two countable locally finite-by-abelian groups $G,H$ endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely generated or both are infinitely generated. On the other hand, we show that each countable group $G$ that coarsely embeds into a countable abelian group is locally nilpotent-by-finite. Moreover, the group $G$ is locally abelian-by-finite if and only if $G$ is undistorted in the sense that $G$ can be written as the union $G=\bigcup_{n\in\omega}G_n$ of countably many finitely generated subgroups such that each $G_n$ is undistorted in $G_{n+1}$ (which means that the identity inclusion $G_n\to G_{n+1}$ is a quasi-isometric embedding with respect to word metrics on $G_n$ and $G_{n+1}$).