Roller boundaries for median spaces and algebras

Abstract

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of $\mathrm{CAT}(0)$ 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.

39 pages, 3 figures. Compared to published version, corrected Figure 1 (needed in an example). Many thanks to Mark Hagen for spotting this

References in corpus (3)

• Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups
• The Tits alternative for finite rank median spaces
• Superrigidity of actions on finite rank median spaces

Cited by in corpus (7)

• Cross ratios and cubulations of hyperbolic groups
• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• Coarse-median preserving automorphisms
• Convex cores for actions on finite-rank median algebras
• On automorphisms and splittings of special groups
• Homotopy equivalent boundaries of cube complexes
• CAT(0) Polygonal Complexes are 2-Median