  We analyse volume-preserving actions of product groups on Riemannian
manifolds. To this end, we establish a new superrigidity theorem for ergodic
cocycles of product groups ranging in linear groups. There are no a priori
assumptions on the acting groups, except a spectral gap assumption on their
action.
  Our main application to manifolds concerns irreducible actions of Kazhdan
product groups. We prove the following dichotomy: Either the action is
infinitesimally linear, which means that the derivative cocycle arises from
unbounded linear representations of all factors. Otherwise, the action is
measurably isometric, in which case there are at most two factors in the
product group.
  As a first application, this provides lower bounds on the dimension of the
manifold in terms of the number of factors in the acting group. Another
application is a strong restriction for actions of non-linear groups.
