Solid Ergodicity and Orbit Equivalence Rigidity for Coinduced Actions

Abstract

Abstract We prove 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 measure preserving 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. We also obtain orbit equivalence rigidity for such actions by showing that the orbit equivalence relation of a rigid or compact measure preserving action $\Sigma \curvearrowright X_0$ of a property (T) group is “remembered” by the orbit equivalence relation of $\Gamma \curvearrowright X$.