Compact Median Algebras are $\mu$-Boundaries in a unique way

TL;DR

This paper demonstrates that under certain conditions, group actions on compact median algebras can be uniquely realized as $$-boundaries, linking median algebra structures with boundary theory.

Contribution

It introduces a novel connection between median algebra actions and $$-boundaries, providing structural insights and uniqueness results.

Findings

Unique realization of median algebra actions as $$-boundaries

Structural results on median algebra group actions

Conditions ensuring boundary realization

Abstract

We consider group actions on compact median algebras. We show that, given a generating probability measure $\mu$ on the acting group and under suitable conditions on the median algebra, it could be realized in a unique way as a $\mu$-boundary in the sense of Furstenberg. Along the way, we prove some structural results.