On cohomology rings of infinite groups

Let R be any ring (with 1), G a torsion free group and R G the corresponding group ring. Let Ext_{R G}^* (M, M) be the cohomology ring associated with the R G-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 ζ ∈ Ext_{R G}^* (M, M) is nilpotent (under Yoneda's product) if and only if its restriction to Ext_{R H}^* (M, M) is nilpotent. In particular this holds for the Thompson group. There are torsion free groups for which the analogous statement is false.