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., $H = SL(n,{\mathbf {R}})$ with $n \geq 3$) and $a \in \mathcal {A}(H,M)$ an ergodic H-action on M which preserves a volume v. Such an H-action is conjectured to be "locally rigid": if $a \prime$ is a sufficiently ${C^1}$-small perturbation of a, then there should exist a diffeomorphism $\Phi$ of the manifold M which conjugates $a \prime$ to a. This conjecture would imply that if $\omega$ is an a-invariant geometrical structure on M, then there should exist an. $a \prime$-invariant geometrical structure $\omega \prime$ 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 $\omega = v$ and with $\omega$ a Riemannian metric along the leaves of a foliation of M.