The Tits alternative for finite rank median spaces

Abstract

We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way, we extend to median spaces the Caprace-Sageev machinery and part of Hagen's theory of unidirectional boundary sets.

Final version, incorporating referee comments; 33 pages, 2 figures

References in corpus (3)

• Superrigidity of actions on finite rank median spaces
• Group actions on median spaces
• On groups whose actions on finite-dimensional CAT(0) spaces have global fixed points

Cited by in corpus (7)

• Cross ratios on $\mathrm{CAT}(0)$ cube complexes and marked length-spectrum rigidity
• Roller boundaries for median spaces and algebras
• Coarse-median preserving automorphisms
• Superrigidity of actions on finite rank median spaces
• A note on locally elliptic actions on cube complexes
• Convex cores for actions on finite-rank median algebras
• Groups acting on hyperbolic spaces with a locally finite orbit