2017/07/20 by Ben Coté, Jon McCammond, Coté, Ben +1
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Mathematics and Applications #math.GR #msc:20F36 #msc:20F55 #msc:51F15 #msc:52C35
paper · pdf · doi:10.48550/arxiv.1707.06624
30 pages, 4 figures
arxiv created 2017/07/20 · arxiv updated 2017/07/21
The complement of a hyperplane arrangement in ℂn 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 ℂ2 whose linear part is the finite complex reflection group known as G4 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 Möbius-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.