Abstract:
Any probability measure preserving (pmp) action $\Gamma\curvearrowright (X,\mu)$ of a countable group $\Gamma$ gives rise to the orbit equivalence (OE) relation $\mathcal R_{\Gamma\curvearrowright X}$ of belonging to the same $\Gamma$-orbit. A main theme in measured group theory and orbit equivalence is to understand how much information about $\Gamma\curvearrowright (X,\mu)$ can be recovered from $\mathcal R_{\Gamma\curvearrowright X}$. In this talk I will present some structural and rigidity results for OE relations of coinduced actions.
     Firstly, we show that the solid ergodicity property is stable with respect to taking coinduction for a fairly large class of coinduced action. More precisely, assume that $\Sigma<\Gamma$ are countable groups such that $g\Sigma g^{-1}\cap \Sigma$ is finite for any $g\in\Gamma\setminus\Sigma$. Then any pmp action $\Sigma\curvearrowright X_0$ gives rise to a solidly ergodic equivalence relation if and only if the equivalence relation of the associated coinduced action $\Gamma\curvearrowright X$ is solidly ergodic. This result generalizes Chifan and Ioana's solid ergodicity for Bernoulli actions.
     Secondly, we obtain orbit equivalence rigidity for such actions by showing that the orbit equivalence relation of a rigid or compact pmp action $\Sigma\car X_0$ of a property (T) group is ``remembered'' by the orbit equivalence relation of $\Gamma\car X$. This provides a converse to a well known result of Bowen.