W${}^{\ast }$-superrigidity for coinduced actions

Abstract

We prove W${}^{\ast }$-superrigidity for a large class of coinduced actions. We prove that if $\Sigma$ is an amenable almost-malnormal subgroup of an infinite conjugagy class (icc) property (T) countable group $\Gamma$, the coinduced action $\Gamma \curvearrowright X$ from an arbitrary probability measure preserving action $\Sigma \curvearrowright X_0$ is W${}^{\ast }$-superrigid. We also prove a similar statement if $\Gamma$ is an icc non-amenable group which is measure equivalent to a product of two infinite groups. In particular, we obtain that any Bernoulli action of such a group $\Gamma$ is W${}^{\ast }$-superrigid.