## Duke Mathematical Journal

### Cocycle superrigidity for profinite actions of property (T) Groups

#### Abstract

Consider a free ergodic measure-preserving profinite action $\Gamma\curvearrowright X$ (i.e., an inverse limit of actions $\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally, of a group $\Gamma$ which admits an infinite normal subgroup $\Gamma_0$ such that the inclusion $\Gamma_0\subset\Gamma$ has relative property (T) and $\Gamma/\Gamma_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\Gamma\times X\rightarrow \Lambda$ is a measurable cocycle with values in a countable group $\Lambda$, then $w$ is cohomologous to a cocycle $w\prime$ which factors through the map $\Gamma\times X\rightarrow \Gamma\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $\Gamma\curvearrowright X$ with any free ergodic measure-preserving action $\Lambda\curvearrowright Y$ comes from a (virtual) conjugacy of actions.

#### Article information

Source
Duke Math. J., Volume 157, Number 2 (2011), 337-367.

Dates
First available in Project Euclid: 25 March 2011

https://projecteuclid.org/euclid.dmj/1301059111

Digital Object Identifier
doi:10.1215/00127094-2011-008

Mathematical Reviews number (MathSciNet)
MR2783933

Zentralblatt MATH identifier
1235.37005

#### Citation

Ioana, Adrian. Cocycle superrigidity for profinite actions of property (T) Groups. Duke Math. J. 157 (2011), no. 2, 337--367. doi:10.1215/00127094-2011-008. https://projecteuclid.org/euclid.dmj/1301059111

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