A complex Euclidean reflection group with a non-positively curved 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 (Björner–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 a $2$-dimensional complex $K$ where every $2$-cell is a Euclidean equilateral triangle and every vertex link is a Möbius–Kantor graph. The hyperplane complement contains non-regular points, the action of the reflection group on $K$ is not free, and the braid group is not torsion-free. Despite all of this, since $K$ is non-positively curved, the corresponding braid group is a $\mathrm{CAT}(0)$ group.