Some unique group-measure space decomposition results

Abstract

Using an approach emerging from the theory of closable derivations on von Neumann algebras, we exhibit a class of groups $\mathcal{CR}$ satisfying the following property: given any groups ${\Gamma }_1,{\Gamma }_2\in \mathcal{CR}$, then any free, ergodic, measure-preserving action on a probability space ${\Gamma }_1\times {\Gamma }_2\curvearrowright X$ gives rise to a von Neumann algebra with a unique group-measure space Cartan subalgebra. Pairing this result with Popa’s orbit equivalence superrigidity theorem we obtain new examples of $W^{\ast }$-superrigid actions.