Orbit equivalence rigidity

Alex Furman

Review from Zentralblatt MATH:

Author's abstract: ``Consider a countable group $\Gamma$ acting ergodically by measure preserving transformations on a probability space $(X,\mu)$, and let $R_\Gamma$ be the corresponding orbit equivalence relation on $X$. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation $R_\Gamma$ on $X$ determines the group $\Gamma$ and the action $(X,\mu, \Gamma)$ uniquely, up to finite groups. The natural action of $SL_n(\bbfZ)$ on the $n$-torus $\bbfR^n/ \bbfZ^n$, 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 and countable equivalence relations of type $\text{II}_1$, which {\it cannot} be generated $\pmod 0$ by a {\it free action} of any group. This gives a negative answer to a long standing problem of Feldman and Moore''.

Reviewed by S.K.Kaul

Keywords: ergodicity; measure preserving transformations; probability space; rigidity; group actions; von Neumann algebras; group algebras

Classification (MSC2000): 22E40 37A05

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