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Orbit Equivalence Rigidity

Alex Furman
Annals of Mathematics
Second Series, Vol. 150, No. 3 (Nov., 1999), pp. 1083-1108
Published by: Annals of Mathematics
DOI: 10.2307/121063
Stable URL: http://www.jstor.org/stable/121063
Page Count: 26
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Orbit Equivalence Rigidity
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Abstract

Consider a countable group Γ acting ergodically by measure preserving transformations on a probability space (X, μ ), and let RΓ be the corresponding orbit equivalence relation on X. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation RΓ on X determines the group Γ and the action (X, μ,Γ ) uniquely, up to finite groups. The natural action of SLn( Z) on the n-torus Rn/ Zn, for n > 2, is one of such examples. The interpretation of these results in the context of von Neumann algebras provides some support to the conjecture of Connes on rigidity of group algebras for groups with property T. Our rigidity results also give examples of countable equivalence relations of type II1, which cannot be generated (mod 0) by a free action of any group. This gives a negative answer to a long standing problem of Feldman and Moore.

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