A Spectral Strong Approximation Theorem for Measure Preserving Actions

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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≤Γ 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.