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