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Volume preserving actions of lattices in semisimple groups on compact manifolds. (English) Zbl 0576.22013
This paper is the first one in a series studying volume preserving actions of lattices $$\Gamma$$ in higher rank semisimple Lie groups on compact manifolds. All presently known actions are of an ”algebraic” nature. The question is whether these are essentially all the examples. One way of exhibiting new actions would be to perturb a given action. One of the main results shows that a sufficiently smooth perturbation of an isometric action is at least topologically isometric.
The proof consists in first showing the existence of a measurable $$\Gamma$$-invariant Riemannian metric - making use of the author’s superrigidity theorem for cocycles which generalizes Margulis’ superrigidity theorem - and then to construct a continuous $$\Gamma$$- invariant Riemannian metric by a Sobolev-lemma type argument.
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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