Description:
<jats:p>Let<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718000639_inline1.png" /><jats:tex-math>$\unicode[STIX]{x1D6E4}$</jats:tex-math></jats:alternatives></jats:inline-formula>be a finitely generated group acting by probability measure-preserving maps on the standard Borel space<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718000639_inline2.png" /><jats:tex-math>$(X,\unicode[STIX]{x1D707})$</jats:tex-math></jats:alternatives></jats:inline-formula>. We show that if<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718000639_inline3.png" /><jats:tex-math>$H\leq \unicode[STIX]{x1D6E4}$</jats:tex-math></jats:alternatives></jats:inline-formula>is a subgroup with relative spectral radius greater than the global spectral radius of the action, then<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718000639_inline4.png" /><jats:tex-math>$H$</jats:tex-math></jats:alternatives></jats:inline-formula>acts with finitely many ergodic components and spectral gap on<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0143385718000639_inline5.png" /><jats:tex-math>$(X,\unicode[STIX]{x1D707})$</jats:tex-math></jats:alternatives></jats:inline-formula>. This answers a question of Shalom who proved this for normal subgroups.</jats:p>