Infinite stationary measures of co-compact group actions

TL;DR

This paper proves that co-compact actions of finitely generated groups on locally compact spaces always admit nonzero stationary measures, extending classical results through a stationary version of Tarski's theorem.

Contribution

It introduces a stationary analogue of Tarski's theorem and demonstrates the existence of nonzero stationary measures for co-compact group actions.

Findings

Every co-compact group action admits a nonzero stationary measure.

Established a stationary version of Tarski's theorem for finitely supported measures.

Provided a method to construct finitely additive stationary measures on groups.

Abstract

Let $\Gamma$ be a finitely generated group, and let $\mu$ be a nondegenerate, finitely supported probability measure on $\Gamma$. We show that every co-compact $\Gamma$ action on a locally compact Hausdorff space admits a nonzero $\mu$-stationary Radon measure. The main ingredient of the proof is a stationary analogue of Tarski's theorem: we show that for every nonempty subset $A \subseteq \Gamma$ there is a $\mu$-stationary, finitely additive measure on $\Gamma$ that assigns unit mass to $A$.