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 <= 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.