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Orbit equivalence rigidity for product actions

Let $Γ_1,\dots,Γ_n$ be hyperbolic, property (T) groups, for some $n\ge 1$. We prove that if a product $Γ_1\times\dots\timesΓ_n \curvearrowright X_1\times\dots\times X_n$ of measure preserving actions is stably orbit equivalent to a measure preserving action $Λ\curvearrowright Y$, then $Λ\curvearrowright Y$ is induced from an action $Λ_0\curvearrowright Y_0$ such that there exists a direct product decomposition $Λ_0=Λ_1\times\dots\timesΛ_n$ into $n$ infinite groups. Moreover, there exists a measure preserving action $Λ_i\curvearrowright Y_i$ that is stably orbit equivalent to $Γ_i\curvearrowright X_i$, for any $1\leq i\leq n$, and the product action $Λ_1\times\dots\timesΛ_n\curvearrowright Y_1\times\dots\times Y_n$ is isomorphic to $Λ_0\curvearrowright Y_0$.