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.
Keywords
cube complexes, locally elliptic actions, global fixed
points