Kazhdan groups acting on compact manifolds.

*(English)*Zbl 0576.22014This 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.

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 |

##### Keywords:

compact Lorentz manifold; rigidity properties; semisimple Lie groups; discrete subgroups; compact manifolds; superrigidity theorem for cocycles; Kazhdan’s property T##### Citations:

Zbl 0576.22013**OpenURL**

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