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 $\operatorname{CAT}(0)$ group.

中文翻译：

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具有非正弯曲补复复数的复欧几里得反射群

$\mathbb{C}^n$ 变形中的超平面排列的补充收缩到 $n$ 维细胞复合体上，但已知程序仅适用于实际排列 (Salvetti) 的复杂化或产生的细胞复合体取决于坐标的初始选择（Björner-Ziegler）。在这篇文章中，我们考虑了唯一的复欧几里得反射群通过 $\mathbb{C}^2$ 上的等距协同作用，其线性部分是 Shephard-Todd 分类中称为 $G_4$ 的有限复反射群，我们构造了一个选择-自由变形从其超平面补体收缩到 $2$-维复合 $K$，其中每个 $2$-cell 是一个欧几里得等边三角形，每个顶点链接都是一个 Möbius-Kantor 图。超平面补包含非常规点，反射群对$K$的作用不是自由的，辫群不是无扭转的。尽管如此，由于 $K$ 是非正弯曲的，因此相应的编织群是 $\operatorname{CAT}(0)$ 群。