A complex euclidean reflection group with an elegant complement complex

TL;DR

This paper constructs a choice-free, non-positively curved 2-dimensional complex for a specific complex Euclidean reflection group's hyperplane complement, revealing new geometric and group-theoretic properties.

Contribution

It introduces a novel, choice-free deformation retraction onto an elegant 2D complex for a complex Euclidean reflection group, with implications for braid group properties.

Findings

The complex is non-positively curved and CAT(0).

The braid group acts on the complex with non-free action and contains torsion.

Every 2-cell is an equilateral triangle, and vertices form a M"obius-Kantor graph.

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.