Congruence subgroups and the Atiyah conjecture

Let (Q) over bar denote the algebraic closure of Q in C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra of G. We prove that there exists a division ring D(G) such that (Q) over bar [G] subset of D(G) subset of U(G). This establishes some versions of the Atiyah conjecture for the group G.