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

Abstract:

We establish some conditions for an abstract finitely generated group Γ to be SS-rigid, i.e. to have only finitely many inequivalent completely reducible representations in each dimension. One of the conditions requires that be of bounded dimension for all irreducible representatons ρ of Γ, which is reminiscent of A. Weil's criterion for 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 the groups , n≥ 3, and by purely combinatorial computations.