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Kazhdan groups acting on compact manifolds. (English) Zbl 0576.22014
This is another paper out of the series studying rigidity properties of actions of semisimple Lie groups and their discrete subgroups on compact manifolds. In earlier papers it was assumed that every simple factor of the Lie group in question has \({\mathbb{R}}\)-rank at least two. This hypothesis was used for two results: the author’s superrigidity theorem for cocycles (a generalization of Margulis’ superrigidity theorem) and Kazhdan’s property T.
In this paper the author shows that in certain special situations Kazhdan’s property T itself can replace superrigidity. The main ingredient of the proof of the several theorems about actions is a theorem (theorem 10) stating that for a Kazhdan group (i.e. one satisfying Kazhdan’s property T) every cocycle with values in a real algebraic group is equivalent to one with values in a Kazhdan subgroup.
Reviewer: H.Abels

MSC:
22E40 Discrete subgroups of Lie groups
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
57S20 Noncompact Lie groups of transformations
37A99 Ergodic theory
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