On cohomology rings of infinite groups

Let R be any ring (with 1), G a torsion free group and RG the corresponding group ring. Let image be the cohomology ring associated with the RG-module M. Let H be a subgroup of finite index of G. The following is a special version of our main Theorem: Assume the profinite completion of G is torsion free. Then an element image is nilpotent (under Yoneda’s product) if and only if its restriction to image is nilpotent. In particular this holds for the Thompson group. There are torsion free groups for which the analogous statement is false.