First Cohomology and Local Rigidity of Group Actions 3

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 applications In 1964, André Weil showed that a homomorphism π from a finitely generated group Γ to a Lie group G is locally rigid whenever H1(Γ, g) = 0. Here π is locally rigid if any nearby homomorphism is conjugate to π by a small element of G, g is the Lie algebra of G, and Γ acts on g by the composition of π and the adjoint representation of G. Weil’s proof also applies to G an algebraic group over a local field of characteristic zero, but his use of the implicit function theorem forced G to be finite dimensional. Here we prove the following generalization of Weil’s theorem to some cases where G is an infinite dimensional Lie group. Theorem 1.1. Let Γ be a finitely presented group, (M,g) a compact Riemannian manifold and π : Γ→ Isom(M,g)⊂Diff(M) a homomorphism. If H1(Γ,Vect(M)) = 0, the homomorphism π is locally rigid as a homomorphism into Diff(M). *Author partially supported by NSF grant DMS-0226121 and a PSC-CUNY grant.