A coarse classification of countable abelian groups

TL;DR

This paper classifies countable abelian groups of finite torsion free rank up to coarse equivalence using invariants like Q-cohomological dimension and torsion free rank, and relates them to Z^n plus cyclic groups.

Contribution

It provides a coarse classification of such groups and extends Gromov's rigidity theorem to this context.

Findings

Countable abelian groups of finite torsion free rank are coarsely equivalent to Z^n plus a sum of cyclic groups.

The Q-cohomological dimension and torsion free rank serve as key invariants for classification.

A partial generalization of Gromov's rigidity theorem is established for these groups.

Abstract

We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H where H is a direct sum (possibly infinite) of cyclic groups. A partial generalization to countable abelian groups of the Gromov rigidity theorem for abelian groups is shown.