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arXiv 2003.037082020-03math.DSmath.FAmath.OA

Solid ergodicity and orbit equivalence rigidity for coinduced actions

Daniel Drimbe

备注：: 20 pages. Some errors are corrected. To appear as such in International Mathematics Research Notices

英文摘要：

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 $Σ<Γ$ are countable groups such that $gΣg^{-1}\cap Σ$ is finite for any $g\inΓ\setminusΣ$. Then any measure preserving action $Σ\curvearrowright X_0$ gives rise to a solidly ergodic equivalence relation if and only if the equivalence relation of the associated coinduced action $Γ\curvearrowright X$ 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 $Σ\curvearrowright X_0$ of a property (T) group is "remembered" by the orbit equivalence relation of $Γ\curvearrowright X$.

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