A Spectral Strong Approximation Theorem for Measure Preserving Actions

Let $Γ$ be a finitely generated group acting by probability measure preserving maps on the standard Borel space $(X,μ)$. We show that if $H\leqΓ$ is a subgroup with relative spectral radius greater than the global spectral radius of the action, then $H$ acts with finitely many ergodic components and spectral gap on $(X,μ)$. This answers a question of Shalom who proved this for normal subgroups.

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