#### Volume 20, issue 3 (2020)

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Roller boundaries for median spaces and algebras

### Elia Fioravanti

Algebraic & Geometric Topology 20 (2020) 1325–1370
##### Abstract

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of $CAT\left(0\right)$ 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 apply to general median algebras, where we recover the zero-completions of Bandelt and Meletiou (1993). Along the way, we prove various properties of halfspaces in finite-rank median spaces and a duality result for locally convex median spaces.

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##### Keywords
median space, median algebra, horofunction compactification, Roller boundary, spaces with walls
##### Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20F67, 22F50, 51F99, 57M99