A Spectral Strong Approximation Theorem for Measure Preserving Actions

TL;DR

This paper establishes a spectral approximation theorem for measure-preserving group actions, showing how subgroup spectral properties influence ergodic components and spectral gaps, extending previous results to non-normal subgroups.

Contribution

It generalizes Shalom's result by proving a spectral approximation theorem for arbitrary subgroups, not just normal ones, in measure-preserving group actions.

Findings

Subgroups with higher relative spectral radius induce spectral gaps.

Subgroups with sufficiently large spectral radius have finitely many ergodic components.

The theorem extends spectral gap results beyond normal subgroups.

Abstract

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.