Duke Mathematical Journal

Product groups acting on manifolds

Alex Furman and Nicolas Monod

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Abstract

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

Article information

Source
Duke Math. J. Volume 148, Number 1 (2009), 1-39.

Dates
First available in Project Euclid: 22 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1240432189

Digital Object Identifier
doi:10.1215/00127094-2009-018

Mathematical Reviews number (MathSciNet)
MR2515098

Zentralblatt MATH identifier
1167.22015

Subjects
Primary: 22F10: Measurable group actions [See also 22D40, 28Dxx, 37Axx] 57S30: Discontinuous groups of transformations 58F11
Secondary: 22D40: Ergodic theory on groups [See also 28Dxx] 37A15: General groups of measure-preserving transformations [See mainly 22Fxx]

Citation

Furman, Alex; Monod, Nicolas. Product groups acting on manifolds. Duke Math. J. 148 (2009), no. 1, 1--39. doi:10.1215/00127094-2009-018. https://projecteuclid.org/euclid.dmj/1240432189


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