Actions of Product Groups on Manifolds

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.