A Spectral Strong Approximation Theorem for Measure Preserving Actions

Let $\Gamma$ be a finitely generated group acting by probability measure preserving maps on the standard Borel space $(X,\mu)$. We show that if $H\leq\Gamma$ 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 $(X,\mu)$. This answers a question of Shalom who proved this for normal subgroups.