  In this article we study cocycles of discrete countable groups with values in
l^2(G) and the ring of affiliated operators UG. We clarify properties of the
first cohomology of a group G with coefficients in l^2(G) and answer several
questions from [CTV]. Moreover, we obtain strong results about the existence of
free subgroups and the subgroup structure, provided the group has a positive
first l^2-Betti number. We give numerous applications and examples of groups
which satisfy our assumptions.
