  A group G has homological dimension less or equal to 1 if it is locally free.
We prove the converse provided that G satisfies the Atiyah Conjecture about
L^2-Betti numbers. We also show that a finitely generated elementary amenable
group G of cohomological dimension less or equal to 2 possesses a finite
2-dimensional model for BG and in particular that G is finitely presented and
the trivial ZG-module Z has a 2-dimensional resolution by finitely generated
free ZG-modules.
