David Fisher (Yale University)

Local rigidity of partially hyperbolic and isometric lattice actions.

Abstract. Let G be semisimple Lie group with all simple factors noncompact and of real rank at least 2. Let $\Gamma$ be a lattice in G. I will discuss the proof of the following:

THEOREM: Any standard affine action of G or $\Gamma$ on a compact manifold is locally rigid.

By a standard affine action we mean any affine action on a compact homogeneous space of the form $H/{\Lambda}$ where $\Lambda$ is discrete and cocompact. By local rigidity of an action $\rho$ we mean that any other action $\rho'$ that is close to $\rho$ on a compact generating set (in an appropriate topology on diffeomorphisms) is conjugate to the original action.

We also prove the theorem for some more general actions. Let M be a compact manifold with an isometric $\Gamma$ action and let $\Gamma$ act affinely on $H/{\Lambda}$ as above. Then the diagonal action of $\Gamma$ on $M{\times}H/{\Lambda}$ is also locally rigid.

The results are primarily new in the case where the action is partially hyperbolic. The only previous results in the partially hyperbolic setting require strong assumptions on the central leaves of the action where we need none.

This is joint work with G.A. Margulis.