  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.
