Group cocycles and the ring of affiliated operators

In this article we study cocycles of discrete countable groups with values in ℓ ² G and the ring of affiliated operators UG\mathcal{U}G. We clarify properties of the first cohomology of a group G with coefficients in ℓ ² G and answer several questions from De Cornulier et al. (Transform. Groups 13(1):125–147, 2008). 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.