Orbit equivalence rigidity for product actions

Abstract

Let ${\Gamma }_1,\dots ,{\Gamma }_n$ be hyperbolic, property (T) groups, for some $n\ge 1$. We prove that if a product ${\Gamma }_1\times \cdots \times {\Gamma }_n\curvearrowright X_1\times \cdots \times X_n$ of measure preserving actions is stably orbit equivalent to a measure preserving action $\Lambda \curvearrowright Y$, then $\Lambda \curvearrowright Y$ is induced from an action ${\Lambda }_0\curvearrowright Y_0$ such that there exists a direct product decomposition ${\Lambda }_0={\Lambda }_1\times \cdots \times {\Lambda }_n$ into $n$ infinite groups. Moreover, there exists a measure preserving action ${\Lambda }_i\curvearrowright Y_i$ that is stably orbit equivalent to ${\Gamma }_i\curvearrowright X_i$, for any $1\le i\le n$, and the product action ${\Lambda }_1\times \cdots \times {\Lambda }_n\curvearrowright Y_1\times \cdots \times Y_n$ is isomorphic to ${\Lambda }_0\curvearrowright Y_0$.

To appear in Communications in Mathematical Physics