The coarse classification of countable Abelian groups

This is joint work with T. Banakh and J. Higes.

We will consider abelian groups endowed with proper left-invariant metrics. We prove that such two groups G, H are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both large scele connected or are both not large scale connected. Also we prove that two groups G, H are coarsely equivalent if and only if their torsion free rang coincide and the groups are either both finitely-generated or both are infinitely generated.

As a result we have that matric space X of asymptotic dimension n=asdim(X) is coarsely equivalent to a countable abelian group, then X is coarsely equivalent to Zⁿ iff X is large scale connected or X is coarsely equivalent to Zⁿ ⊕ Q/Z iff X is not large scale connected.