Intrinsic uniform structure on median algebras

Abstract

We introduce the median uniformity ${\mathcal{U}}_{\mathrm{m}}$, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology ${\tau }_{\mathrm{m}}$ provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When the intrinsic topology ${\tau }_{\mathrm{m}}$ is Hausdorff and all intervals are compact, the MMC is the unique proper median compactification and is canonically isomorphic to the Roller--Fioravanti compactification. We apply this uniform framework to continuous actions of a topological group $G$ by median automorphisms. We show that the MMC is a median $G$-compactification. In the finite-rank case, the resulting compact $G$-system is Rosenthal representable and dynamically tame.

arXiv comment30 pages, 1 figure