Product groups acting on manifolds
We analyse volume-preserving actions of product groups on Riemannian manifolds. Under a natural spectral irreducibility assumption, we prove the following dichotomy: Either the action is measurably isometric, in which case there are at most two factors; or the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors. 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. We prove our results by means of a new cocycle superrigidity theorem of independent interest, in analogy to Zimmer's programme.
Keywords: Kac-Moody Groups ; Semisimple Lie-Groups ; Bounded Cohomology ; Irreducible Lattices ; Automorphism-Groups ; Fundamental-Groups ; Operator Methods ; Kazhdan Groups ; Rigidity ; Superrigidity
Record created on 2008-10-29, modified on 2016-08-08