Local Rigidity and Group Cohomology I: Stowe’s Theorem for

Abstract. Stowe’s Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove if a φ is Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields onM. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group. Let Γ be a finitely generated group. Given a topological group G, the set of representations of Γ in G will be denoted by R(Γ, G). This can be given the structure of a topological space by taking the topology to be the compact-open topology. The compact-open topology may also be shown to be the topology o