The Tits alternative for finite rank median spaces

TL;DR

This paper establishes a version of the Tits alternative for groups acting on complete, finite rank median spaces, revealing restrictions on their actions compared to general median spaces.

Contribution

It extends the Caprace-Sageev machinery and Hagen's theory to median spaces, providing new structural insights into group actions.

Findings

Groups acting on finite rank median spaces satisfy the Tits alternative.

Actions on finite rank median spaces are more constrained than on general median spaces.

Extension of boundary set theory to 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.