On cohomology rings of infinite groups

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