# Groups with many non-conjugate but orbit equivalent actions

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?

• Many. Are you expecting the answer to provide a classification (don't) or an ilustrative example? – Uri Bader May 9 '17 at 9:22
• @Uri I would particularly like to see an example with Property (T). – hetairoi22 May 10 '17 at 11:55
• This desire will not be fulfilled, by a theorem of Hjorth. I will write an answer when I will have the time. – Uri Bader May 10 '17 at 16:25
• @Uri What if I just want two non-conjugate free ergodic actions that are orbit equivalent? Or infinitely many? – hetairoi22 May 11 '17 at 17:02

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 $X\times Y_\alpha$.