Rigidity of ergodic volume-preserving actions of semisimple groups of higher rank on compact manifolds

Abstract

Let M be a compact manifold, H a semisimple Lie group of higher rank (e.g., H = SL(n, R) with n > 3) and a E' (H, M) an ergodic H-action on M which preserves a volume v . Such an H-action is conjectured to be locally rigid: if a' is a sufficiently Cl -small perturbation of a, then there should exist a diffeomorphism D of the manifold M which conjugates a' to a. This conjecture would imply that if co is an a-invariant geometrical structure on M, then there should exist an a'-invariant geometrical structure ct on M of the same type. Using Kazhdan property, superrigidity for cocycles, and Sobolev spaces techniques we prove, under suitable conditions, two such results with co = v and with co a Riemannian metric along the leaves of a foliation of M.