Universal lower bounds for the discrepancies of actions of a locally compact group

TL;DR

This paper establishes fundamental lower bounds on spectral gaps for measure-preserving actions of locally compact groups, extending previous results from discrete groups and highlighting cases where these bounds are tight.

Contribution

It generalizes universal lower bounds for spectral gaps from discrete groups to locally compact groups under specific conditions.

Findings

Lower bounds for spectral gaps are established for actions of locally compact groups.

The bounds generalize previous results for discrete groups.

Examples show the bounds are tight for certain Lie group actions.

Abstract

We prove universal lower bounds for discrepancies (i.e. sizes of spectral gaps of averaging operators) of measure-preserving actions of a locally compact group on probability spaces. For example, a locally compact Hausdorff unimodular group $G$, acting continuously, by measure-preserving transformations, on a compact atomless probability space $(X,\nu)$, with an orbit $Gx_0$ of measure zero, contained in the support of $\nu$, and with compact stabilizer (i.e. $G_{x_0}$ is compact) has the following property: any finite positive regular Borel measure $\mu$ on $G$ satisfies $$\|\pi_0(\mu)\|_{2\to 2}\geq \|\lambda_G(\mu)\|_{2\to 2},$$ where $\pi_0$ denotes the Koopman representation of $G$, defined by the given action, and $\lambda_G$ denotes the left-regular representation of $G$. The lower bounds we prove generalize the universal lower bounds for the discrepancies of measure-preserving actions of a discrete group. Many examples show that the generalization from discrete groups to locally compact groups requires some additional hypothesis on the action (we detail some examples of actions of amenable groups with a spectral gap, due to Margulis). Well-known examples and results of Kazhdan and Zimmer show that the discrepancies of some actions of Lie groups on homogeneous spaces match exactly the universal lower bounds we prove.