Groups with arbitrary cubical dimension gap

TL;DR

This paper demonstrates that groups acting on CAT(0) cube complexes can have a strictly larger minimal dimension for cube complex actions compared to general CAT(0) spaces, revealing a dimension gap.

Contribution

It proves a splitting theorem for groups acting on CAT(0) cube complexes and provides examples of groups with a dimension gap between cube complex and CAT(0) space actions.

Findings

Groups acting on CAT(0) cube complexes split as products.

Existence of groups with larger minimal cube complex dimension than CAT(0) space dimension.

Illustration of the dimension gap phenomenon in group actions.

Abstract

We prove that if $G = G_1\times\dots\times G_n$ 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.