A note on locally elliptic actions on cube complexes

Abstract

We deduce from Sageev’s results that whenever a group acts locally elliptically on a finite-dimensional CAT$\left(0\right)$ cube complex, then it must fix a point. As an application, we partially prove a conjecture by Marquis concerning actions on buildings and we give an example of a group $G$ such that $G$ does not have property (T), but $G$ and all its finitely generated subgroups can not act without a fixed point on a finite-dimensional CAT$\left(0\right)$ cube complex, answering a question by Barnhill and Chatterji.