A coarse classification of countable abelian groups

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.