First Cohomology and Local Rigidity of Group Actions

Given a topological group G, and a finitely generated group Γ, a homomorphism π : Γ→G is locally rigid if any nearby by homomorphism π′ is conjugate to π by a small element of G. In 1964, Weil gave a criterion for local rigidity of a homomorphism from a finitely generated group Γ to a finite dimensional Lie group G in terms of cohomology of Γ with coefficients in the Lie algebra of G. Here we generalize Weil’s result to a class of homomorphisms into certain infinite dimensional Lie groups, namely groups of diffeomorphism compact manifolds. This gives a criterion for local rigidity of group actions which implies local rigidity of: (1) all isometric actions of groups with property (T ), (2) all isometric actions of irreducible lattices in products of simple Lie groups and certain more general locally compact groups and (3) a certain class of isometric actions of a certain class of cocompact lattices in SU(1, n). 1. A cohomological criterion for local rigidity and