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, or, 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 nonlinear groups
Alex Furman. Nicolas Monod. "Product groups acting on manifolds." Duke Math. J. 148 (1) 1 - 39, 15 May 2009. https://doi.org/10.1215/00127094-2009-018