Free ergodic probability measure-preserving actions of the free group

Question

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 $\mathbb{T}^2 = \widehat{\mathbb{Z}^2}$.

Answer 1

Given any positive definite function $\psi:\mathbb{F}_2\to \mathbb{C}$, there exists a stationary Gaussian process $\left(X_g\right)_{g\in\mathbb{F}_2}$ such that $\mathbb{E}\left(X_gX_h\right)=\psi\left(h^{-1}g\right)$. The shift (also called Gaussian) action of $\mathbb{F}_2$ on $\mathbb{R}^{\mathbb{F}_2}$ preserves the distribution of the Gaussian process. These actions are used extensively in various places, for example Connes and Weiss have used such constructions for characterising groups without property T.

Answer 2

Take the Bernoulli action action on the space of configurations on $\mathbb F_2$ with i.i.d. values. By the way, it's completely wrong to attribute the result on non-amenability of free probability measure preserving actions of non-amenable groups to Gaboriau.

EDIT: The construction of a Bernoulli action is the same as that of a Bernoulli shift, the only difference being that one takes an arbitrary countable group $G$ instead of the group of integers $\mathbb Z$ (look at the wiki article and the references therein). Concerning the non-amenability claim I am pretty sure it must have already been known to Zimmer. Carrière and Ghys in their 1985 note explcitily mention this fact:

Si, par exemple, une relation d'équivalence discrète mesurée $R$ est engendrée par l'action d'un groupe dénombrable $\Gamma$, alors la moyennabilité du groupe $\Gamma$ entraîne la moyennabilité de la relation R. La réciproque est fausse, même si l'on suppose l'action de $\Gamma$ essentiellement libre, c'est-à-dire si la mesure de l'ensemble des points fixes de l'action est nulle. Cette réciproque est cependant valable dans le cas d'une action essentiellement libre qui préserve une mesure de probabilité (voir [8]).

Here [8] is Zimmer's book, but I could not locate this statement there in an explicit form.

Answer 3

The class of all ergodic probability measure preserving actions of the free group up to measure-theoretic isomorphism is extraordinarily complicated and intuitively the isomorphism classes of actions with some kind of reasonable description make up a negligible proportion of the total. The isomorphism relation for ergodic actions of the integers is already not Borel and is unclassifiable in terms of countable structures. The isomorphism relation for ergodic actions of the free group is vastly more complicated still.

It is possible, however, to construct a kind of coherent parameterization that quantifies the nonamenability of actions of the free group. This is the notion of weak equivalence. It is trivial for actions of an amenable group but seems to be the right approach to the global structure of actions of the free group.

These issues are the main topic of the book Global Aspects of Ergodic Group Actions by Alexander Kechris.