Local rigidity and group cohomology I: Stowe's theorem for Banach manifolds

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 that if φ is a C^r action on a smooth, closed, manifold M satisfying H¹(Γ, D^r-1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and D^k(M) is the space of C^k tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.