# Orbit equivalence rigidity

### Abstract

Consider a countable group $\Gamma$ acting ergodically by measure preserving transformations on a probability space $(X,\mu)$, and let $\mathcal{R}_\Gamma$ be the corresponding orbit equivalence relation on $X$. The following rigidity phenomenon is shown: there exist group actions such that the equivalence relation $\mathcal{R}_\Gamma$ on $X$ determines the group $\Gamma$ and the action $(X,\mu,\Gamma)$ uniquely, up to finite groups. The natural action of $\mathrm{SL}_n(\mathbb{Z})$ on the $n$-torus $\mathbb{R}^n/\mathbb{Z}^n$, 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 of countable equivalence relations of type $\mathrm{II}_1$, 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.

Alex Furman