The coarse classification of countable abelian groups

TL;DR

This paper classifies countable abelian groups based on coarse geometry, showing conditions for coarse equivalence and embedding related to their algebraic properties and asymptotic dimensions.

Contribution

It provides a coarse classification of countable abelian groups and characterizes groups that coarsely embed into abelian groups in terms of local nilpotency and undistorted subgroups.

Findings

Coarse equivalence depends on asymptotic dimension and generation type.

Groups coarsely embedding into abelian groups are locally nilpotent-by-finite.

Characterization of undistorted subgroups in countable groups.

Abstract

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 of countably many finitely generated subgroups G_n such that each G_n is undistorted in G_{n+1} (which means that the identity inclusion from G_n to G_{n+1} is a quasi-isometric embedding with respect to word metrics).