A complex euclidean reflection group with an elegant complement complex

Abstract

The complement of a hyperplane arrangement in $\mathbb{C}^n$ deformation retracts onto an $n$-dimensional cell complex, but the known procedures only apply to complexifications of real arrangements (Salvetti) or the cell complex produced depends on an initial choice of coordinates (Bj\"orner-Ziegler). In this article we consider the unique complex euclidean reflection group acting cocompactly by isometries on $\mathbb{C}^2$ whose linear part is the finite complex reflection group known as $G_4$ in the Shephard-Todd classification and we construct a choice-free deformation retraction from its hyperplane complement onto an elegant $2$-dimensional complex $K$ where every $2$-cell is a euclidean equilateral triangle and every vertex link is a M\"obius-Kantor graph. Since $K$ is non-positively curved, the corresponding braid group is a CAT(0) group, despite the fact that there are non-regular points in the hyperplane complement, the action of the reflection group on $K$ is not free, and the braid group is not torsion-free.