Roller boundaries for median spaces and algebras

TL;DR

This paper develops a general framework for compactifying median spaces with compact intervals, extending the concept of Roller boundaries from ${\rm CAT}(0)$ cube complexes to a broader class of median spaces and algebras.

Contribution

It introduces new compactifications for median spaces and algebras, generalizing existing boundaries and recovering known completions, with new duality and property results.

Findings

Constructed compactifications for median spaces with compact intervals.

Extended Roller boundary concepts to general median spaces and algebras.

Proved properties of halfspaces and duality in median spaces.

Abstract

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of ${\rm CAT}(0)$ cube complexes. Examples of median spaces with compact intervals include all finite rank median spaces and all proper median spaces of infinite rank. Our methods also work for general median algebras, where we recover the zero-completions of Bandelt and Meletiou. Along the way, we prove various properties of halfspaces in finite rank median spaces and a duality result for locally convex median spaces.