Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 134 / 2

A group is cubulated when it admits a cubulation of a group, namely a metrically proper group action by cubical automorphisms on a CAT(0) cube complex. It is cocompactly cubulated when that action is also a cocompact group action. Thus a cocompact cubulation is a proper, cocompact cubical action.

Fix $x\in S$ and let ${\sigma }_x$ be its principal vertex of a dual cube complex. The combinatorial distance in the dual complex is the wall metric:Consequently the wall-metric properness criterion says that the action on $C$ is metrically proper provided this number tends to infinity as $g$ leaves every finite subset of $G$. Equivalently, for every $R$, only finitely many $g$ separate $x$ from $gx$ by at most $R$ walls.

The necessary and sufficient condition is the transverse-wall cocompactness criterion: there must be finitely many $G$-orbits of finite transverse wall collections. Equivalently, their cardinalities must be uniformly bounded and, for every cardinality, there must be only finitely many orbits.

Indeed, an $n$-cube of the dual cube complex of a wallspace is dual to an $n$-element collection of pairwise crossing walls, and this correspondence respects the $G$-action and passage to faces. If there are finitely many orbits of transverse collections, there are finitely many cube orbits, so the quotient $G\backslash C$ is a finite cube complex and is compact. Conversely, if the action is cocompact, a compact fundamental set meets only finitely many open unit cubes. Hence there are finitely many cube orbits and therefore finitely many orbits of their dual transverse wall collections.

The walls occur in three families of parallel lines, and a choice of halfspaces in one family is determined by an integer cut. Lines from different families cross, so the three cuts can be chosen independently. It follows directly from the dual cube complex of a wallspace construction thatand that $C$ is the standard cubulation of ${\mathbb{R}}^3$.

Choose affine coordinates $x_1,x_2,x_3$ for the three wall families so that the original Euclidean plane is $x_1+x_2+x_3=0$ and the walls are the integer level sets. On the cut coordinates $(n_1,n_2,n_3)\in {\mathbb{Z}}^3$, the translation subgroup of the Affine Coxeter group $W$ adds vectors $(m_1,m_2,m_3)$ satisfying $m_1+m_2+m_3=0$, while its finite reflection subgroup permutes the three coordinates. Thereforeis constant on every $W$-orbit. Since $h(n,0,0)=n$ is unbounded, there are infinitely many vertex orbits. A cocompact cubical action on this locally finite cube complex would have only finitely many cube orbits, so the action of $W$ on $C$ is not cocompact.