  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.
