Which countable discrete groups (apart from the infinite amenable ones) admit uncountably many mutually non-conjugate free ergodic probability measure preserving actions that are all mutually orbit equivalent?



Groups with many non-conjugate but orbit equivalent actions
We consider infinite countable groups and ask about the *this property*: admitting uncountably many mutually non-isomorphis free ergodic probability measure preserving (pmp) actions that are all mutually orbit equivalent.


You ask which groups satisfy *this property*. I do not know exactly. I think no one knows. But it is a big class of groups, maybe all groups which do not satisfy property (T).


In Corollary 2.6 of [*A converse to Dye’s theorem*](http://www.ams.org/journals/tran/2005-357-08/S0002-9947-04-03672-4/S0002-9947-04-03672-4.pdf) Hjorth proves that property (T) groups do not satisfy *this property*. 


On the other hand, amenable groups do satisfy *this property*. Indeed, any amenable group has uncountably many mutually non-isomorphic pmp actions (say, by entropy theory), but these are all orbit equivalent, by the celebrated Ornstein-Weiss theorem.


To see that there are many non-amenable groups satisfying *this property*, fix a pmp action of a non-amenable group $\Gamma\_1$ on $X$ and uncountably many pmp actions of an amenable group $\Gamma\_2$ on $Y\_\alpha$. Then the non-amenable group $\Gamma=\Gamma\_1\times \Gamma\_2$ will satisfy *this property*, as witness by its actions on $X\times Y\_\alpha$.


If you want to study *this property* further, I suggest you to start with the following two things: try to show that if a group has a quotient that satisfies *this property* then the group satisfies *this property*, and try to show that *this property* is a measure equivalence invariant. 


