# Free ergodic probability measure-preserving actions of the free group

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

• Just to clarify: Is your question about the known actions modulo some equivalence (for instance orbit equivalence) or about the explicit actions that have appeared so far in the literature? Nov 17, 2018 at 14:58
• @AdriánGonzález-Pérez: Perhaps I should remove the word "known", I put it just to avoid the comments like "too broad". Now, sure, if you think to an unknown action, it is on-topic. Explicit or modulo equivalence, it is ok. Nov 17, 2018 at 17:20

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.

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.

• Could you write this action explicitly or cite a reference? For Gaboriau you are certainly right, which paper should be credited first? Nov 17, 2018 at 17:30
• It's a bit too long for a comment - I have added an edit
– R W
Nov 17, 2018 at 18:38
• It should be Proposition 4.3.3 in Zimmer's book. Nov 17, 2018 at 22:23
• Indeed - there was a little contamination on my part here. You were imposing the freeness condition, so that I implicitly deduced that you were asking about amenability of the orbit equivalence relation, and it is this latter result that I could not find in Zimmer's book.
– R W
Nov 17, 2018 at 22:47
• Amenability of an action is equivalent to amenability of the orbit equivalence relation and amenability of the stabilizers if this is what you are asking about.
– R W
Nov 18, 2018 at 0:17

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.