Weakly hyperbolic actions of Kazhdan groups on tori

In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov's classical theorem on ergodicity of Anosov diffeomorphisms to the wider class of weakly hyperbolic group actions. To state the second main result, suppose that A denotes a weakly hyperbolic action of a higher rank lattice (such as SL(n,Z) for n>2) by C^2 volume preserving diffeomorphisms on the torus. If A is covered by an action of the universal covering space, R^n, of the torus, then A induces a weakly hyperbolic representation on the first homology of the torus. This result may be viewed as an analogue of Manning's contribution to the Franks/Manning classification of Anosov diffeos. on tori. We draw conclusions related to Zimmer's program of classifying ergodic volume preserving actions of higher rank lattices. Both results rely on a new regularity result of Rauch and Talyor relating Sobolev classes of functions measured tangentially with respect to leaves of absolutely continuous foliations to global Sobolev classes of functions.