Let $(X,\mathcal{B},\mu)$ be a standard Borel probability space. Let $\Gamma$ be a countable group.

An action of $\Gamma$ on $X$ is:

- essentially free if for all $g \in \Gamma \setminus \{e \}$, $\mu(\{x \in X \ | \ g \cdot x = x \}) = 0$,
- ergodic if for any $Y \in \mathcal{B}$ such that $\Gamma \cdot Y \subset Y$, we have $\mu(Y) \in \{0,1 \}$,
- measure-preserving if for any $Y \in \mathcal{B}$ and any $g \in \Gamma$, $\mu(g \cdot Y) = \mu(Y)$.

**Question**: What are the (known) essentially free ergodic measure-preserving actions of the free group $\mathbb{F}_2$ on a standard Borel probability space $(X,\mathcal{B},\mu)$?

Note that by Gaboriau (here, Corollary 5.7), it cannot be an amenable action.

*Example*: the usual action of $\mathbb{F}_2 \subset {\rm SL}(2,\mathbb{Z})$ on $\mathbb{T}^2 = \widehat{\mathbb{Z}^2}$.