2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129418We show that an automorphism of an arbitrary CAT(0) cube complex either has a fixed point or preserves some combinatorial axis. It follows that when a group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls.Group Theory20F65, 20F67, (53C21, 20F18)text