This
is a joint work with Nicolas Monod. We analyze volume-preserving
actions of products of Kazhdan groups on Riemannian manifolds. Under a
natural irreducibility assumption we obtain lower bounds on the
dimension of the manifold in terms of the number of factors in the
acting group, and strong restrictions for actions of non-linear groups.
We prove our results by means of a new cocycle superrigidity theorem of
independent interest, in analogy to Zimmer's programme.
