  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.
