A complex Euclidean reflection group with a non-positively curved complement complex

  • Ben Coté

    Western Oregon University, Monmouth, USA
  • Jon McCammond

    University of California, Santa Barbara, USA
A complex Euclidean reflection group with a non-positively curved complement complex cover
Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

The complement of a hyperplane arrangement in Cn\mathbb{C}^n deformation retracts onto an nn-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 C2\mathbb{C}^2 whose linear part is the finite complex reflection group known as G4G_4 in the Shephard-Todd classification and we construct a choice-free deformation retraction from its hyperplane complement onto a 22-dimensional complex KK where every 22-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 KK is not free, and the braid group is not torsion-free. Despite all of this, since KK is non-positively curved, the corresponding braid group is a CAT(0)\operatorname{CAT}(0) group.

Cite this article

Ben Coté, Jon McCammond, A complex Euclidean reflection group with a non-positively curved complement complex. Groups Geom. Dyn. 15 (2021), no. 3, pp. 989–1013

DOI 10.4171/GGD/620