College of Liberal Arts & Sciences

# Operator Theory Seminar

**Abastract: **

In this talk we provide a natural complement to Monod and Shalom's orbit equivalence superrigidity theorem for irreducible actions of product groups by providing a large class of product actions whose orbit equivalence relation remember the product structure. More precisely, we show that if a product $\Gamma_1\times\dots\times\Gamma_n \curvearrowright X_1\times\dots\times X_n$ 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 $\Lambda_0\curvearrowright Y_0$ and there exists a direct product decomposition $\Lambda_0=\Lambda_1\times\dots\times\Lambda_n$ 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 $1\leq i\leq n$, and the product action $\Lambda_1\times\dots\times\Lambda_n\curvearrowright Y_1\times\dots\times Y_n$ is isomorphic to $\Lambda_0\curvearrowright Y_0$.