Convex cores for actions on finite-rank median algebras

Abstract

We show that every action of a finitely generated group on a finite-rank median algebra admits a nonempty "convex core", even when no metric or topology is given. We then use this to deduce an analogue of the flat torus theorem for actions on connected finite-rank median spaces. We also prove that isometries of connected finite-rank median spaces are either elliptic or loxodromic.

arXiv comment29 pages, no figures; to appear in Annales de l'Institut Fourier

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Cited by in corpus (4)

• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• Coarse-median preserving automorphisms
• On automorphisms and splittings of special groups
• Coarse cubical rigidity