We deduce from Sageev’s results that whenever a group acts locally elliptically on a finite-dimensional CAT 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 such that does not have property (T), but and all its finitely generated subgroups can not act without a fixed point on a finite-dimensional CAT cube complex, answering a question by Barnhill and Chatterji.
"A note on locally elliptic actions on cube complexes." Innov. Incidence Geom. Algebr. Topol. Comb. 18 (1) 1 - 6, 2020. https://doi.org/10.2140/iig.2020.18.1