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?



Actions orbit equivalent to profinite ones
If a group has a free profinite action then in particular it must be residually finite. On the other hand, any measure preserving action of an amenable group is weakly compact. Thus, any free measure preserving action of infinite amenable group $G$ which is not residually finite cannot be orbit equivalent to a profinite action of $G$.


Of course by Ornstein and Weiss' theorem all such orbit equivalence relations are hyperfinte and so this doesn't work if you allow yourself to change the group $G$. 


**Update:**


For residually finite groups an example will have to be more sophisticated since this will not work with amenable groups. However, there are enough orbit equivalence rigidity theorems around that this is still possible. The relevant theorem here is Ioana's rigidity theorem for profinite actions of property (T) groups (<http://arxiv.org/abs/0805.2998>):


**Theorem**[Adrian Ioana]: Let $\Gamma$, and $\Lambda$ be countable discrete groups and suppose that $\Gamma$ has property (T). Let $\Gamma \curvearrowright X$ be a free, ergodic, profinite, probability measure preserving action and suppose that $\Lambda \curvearrowright Y$ is a free probability measure preserving action which is orbit equivalent to the action $\Gamma \curvearrowright X$. Then there exists finite index subgroups $\Gamma\_1 < \Gamma$ and $\Lambda\_1 < \Lambda$ such that restricting to an $\Gamma\_1$ (resp. $\Lambda\_1$) ergodic component $X\_1$ (resp. $Y\_1$) we have that the actions $\Gamma\_1 \curvearrowright X\_1$ and $\Lambda\_1 \curvearrowright Y\_1$ are conjugate.


In particular, this shows that for orbit equivalent actions of property (T) groups if one action is profinite then so is the other. This is because a measure preserving action $\Gamma \curvearrowright X$ is profinite if and only if the weak closure $\overline \Gamma$ of $\Gamma$ in Aut$(X)$ is a profinite group, and $\overline \Gamma$ is invariant under conjugation. Thus one just needs to construct a weakly compact action of a residually finite property (T) group which is not profinite. 


One such example is given by Dennis Sullivan (<http://www.ams.org/mathscinet-getitem?mr=590825>) who showed that for $n > 3$ there is a countable dense subgroup $\Gamma\_n$ of $O(n + 1)$ which has property (T). $\Gamma\_n$ will be residually finite by a classical result of Malcev since it is finitely generated and linear. Also, the Haar measure preserving action $\Gamma\_n \curvearrowright O(n + 1)$ given by left multiplication will be free (since $\Gamma\_n$ is a subgroup), ergodic (since $\Gamma\_n$ is dense), and compact (since $O(n + 1)$ is compact), and hence also weakly compact. However, $\Gamma\_n \curvearrowright O(n + 1)$ is not profinite since $O(n + 1) = \overline{\Gamma\_n}$ is not profinite.


Note that this gives an example if you consider orbit equivalence with other groups $\Lambda$ as well.


