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

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.