Let $G$ be a countable discrete group.

Is there a way to characterise the actions of $G$ that are orbit-equivalent to profinite ones?

Ozawa and Popa introduced the concept of weakly compact actions. Weakly compact actions are stable under orbit equivalence and profinite actions are weakly compact.

Is it possible to find a weakly compact action that is not orbit equivalent to any profinite action?

Let $G$ be a countable discrete residually finite group.

Is there a way to characterise the actions of $G$ that are orbit-equivalent to profinite ones?

Ozawa and Popa introduced the concept of weakly compact actions. Weakly compact actions are stable under orbit equivalence and profinite actions are weakly compact.

Is it possible to find a weakly compact action that is not orbit equivalent to any profinite action?