On SS-rigid groups and A. Weil's criterion for local rigidity. I

We establish some conditions for an abstract finitely generated group Gamma to be SS-rigid, i.e. to have only finitely many inequivalent completely re ducible representations in each dimension. One of the conditions requires t hat H-1(Gamma, Ad omicron rho) be of bounded dimension for all irreducible representatons rho of Gamma, which is reminiscent of A. Well's criterion fo r local rigidity. We also link these new conditions to the previous results on the SS-rigidity of groups with bounded generation and verify them for t he groups SLn(Z), n greater than or equal to 3, and SL2(Z[1/p]) by purely c ombinatorial computations.