Local spectral gap in simple Lie groups and applications

TL;DR

This paper introduces the concept of local spectral gap for measure-preserving actions and proves it for dense subgroups generated by algebraic elements in simple Lie groups, with applications to measure uniqueness, rigidity, and expanders.

Contribution

It establishes local spectral gap for dense algebraic subgroups in simple Lie groups, extending previous results to non-compact settings and diverse applications.

Findings

Proves local spectral gap for dense subgroups in simple Lie groups.

Shows Haar measure is essentially unique among invariant finitely additive measures.

Provides applications to rigidity, expanders, and random walks.

Abstract

We introduce a novel notion of {\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $\Gamma\curvearrowright G$, whenever $\Gamma$ is a dense subgroup generated by algebraic elements of an arbitrary connected simple Lie group $G$. This extends to the non-compact setting recent works of Bourgain and Gamburd \cite{BG06,BG10}, and Benoist and de Saxc\'{e} \cite{BdS14}. We present several applications to the Banach-Ruziewicz problem, orbit equivalence rigidity, continuous and monotone expanders, and bounded random walks on $G$. In particular, we prove that, up to a multiplicative constant, the Haar measure is the unique $\Gamma$-invariant finitely additive measure defined on all bounded measurable subsets of $G$.