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Ergodic theory, group representations, and rigidity. (English) Zbl 0532.22009
In a recent paper the author proved [Ann. Math., II. Ser. 112, 511-529 (1980; Zbl 0468.22011)] (see also H. Furstenberg [Lect. Notes Math. 842, 273-292 (1981; Zbl 0471.22007)] a remarkable result which in particular asserts that no two free ergodic finite-measure-preserving actions of two connected noncompact centerfree simple Lie groups of R- ran$$k\geq 2$$ are orbit-equivalent, unless the groups are isomorphic and the actions correspond to each other under appropriate isomorphisms of the groups and the measure spaces respectively. (We recall that two actions are said to be orbit-equivalent if modulo null-sets there exists an isomorphism of the measure spaces which maps orbits of one of the actions onto orbits of the other.) A similar assertion holds for actions of lattices in these Lie groups and also in a more general set-up that we shall not go into. This kind of rigidity of ergodic actions is sharp contrast with the situation for actions of amenable groups; indeed any two free properly ergodic finite-measure-preserving actions of two (possibly different) continuous (respectively, discrete) amenable unimodular locally compact second countable groups are orbit equivalent.
The paper under review gives an account of various notions and results about ergodic actions which, though vaguely leading to the rigidity theorem noted above, covers much more than what is directly involved in the proof of the theorem. It is written in a relaxed style and illustrates various ideas in the area. Proofs of many (but not all) well- chosen assertions and also many examples are included.
It may be noted that the proof of rigidity of ergodic actions is motivated by Margulis’ proof of ”superrigidity” of lattices in the same class of groups, which in turn was a major ingredient of his proof of arithmeticity of these lattices. In the later sections of the paper, the present author gives an exposition of the arithmeticity of these lattices. In the later sections of the paper, the present author gives an expostion of the arithmeticity theorem and also certain other more recent results about lattices. The author expects some of these to have appropriate analogues for ergodic actions of the Lie groups in question.

##### MSC:
 22D40 Ergodic theory on groups 28D05 Measure-preserving transformations 22E46 Semisimple Lie groups and their representations 43A05 Measures on groups and semigroups, etc. 22E40 Discrete subgroups of Lie groups
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