Furman, AlexMonod, Nicolas2008-10-292008-10-292008-10-29200910.1215/00127094-2009-018https://infoscience.epfl.ch/handle/20.500.14299/30509WOS:000265672400001We 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.Kac-Moody GroupsSemisimple Lie-GroupsBounded CohomologyIrreducible LatticesAutomorphism-GroupsFundamental-GroupsOperator MethodsKazhdan GroupsRigiditySuperrigiditytext::journal::journal article::research article