Groups with arbitrary cubical dimension gap

Abstract

We prove that if G " G1 ˆ¨¨¨ˆGn acts essentially, properly and cocompactly on a CAT(0) cube complex X, then the cube complex splits as a product. We use this theorem to give various examples of groups for which the minimal dimension of a cube complex the group acts on is strictly larger than that of the minimal dimension of a CAT(0) space upon which the group acts.