# Orbit equivalence rigidity for product actions

## TL;DR

This paper establishes rigidity results for product actions of hyperbolic property (T) groups, showing that stable orbit equivalence implies a specific decomposition of the acting groups and actions, revealing deep structural invariances.

## Contribution

It proves that stable orbit equivalence of product actions of hyperbolic property (T) groups forces a product decomposition of the acting groups and actions, extending orbit equivalence rigidity to product settings.

## Key findings

- Stable orbit equivalence implies a product decomposition of the acting groups.
- Each factor action is stably orbit equivalent to the original group actions.
- The product of the decomposed actions is isomorphic to the original product action.

## Abstract

Let $\Gamma_1,\dots,\Gamma_n$ be hyperbolic, property (T) groups, for some $n\ge 1$. We prove that if a product $\Gamma_1\times\dots\times\Gamma_n \curvearrowright X_1\times\dots\times X_n$ 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 $\Lambda_0\curvearrowright Y_0$ such that 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 $\Lambda_0\curvearrowright Y_0$.