Cubulation of a group

A cubulation of a group is a metrically proper action by cubical automorphisms on a CAT(0) cube complex. It is cocompact when the action is also a cocompact group action.

An isometric group action on a metric space is metrically proper when, for some and hence every point $x$, the set $\{g:d(x,gx)\le R\}$ is finite for every finite $R$.

If $G$ acts on $(S,\mathcal{W})$ and $x\in S$, then its action on the dual cube complex is metrically proper wheneveras $g$ leaves every finite subset of $G$.

The action of $G$ on the dual cube complex is cocompact exactly when there are finitely many $G$-orbits of finite transverse wall collections. Equivalently, there is a uniform bound on their sizes and, for every size, only finitely many orbits.