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 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 CAT(0) group.
Classification :
20-XX, 51-XX
Mots-clés :
Complex Euclidean reflection group, hyperplane complement, Salvetti complex, non-positive curvature, braid group of a group action
Affiliations des auteurs :
Ben Coté
1
;
Jon McCammond
2
1
Western Oregon University, Monmouth, USA
2
University of California, Santa Barbara, USA
Ben Coté; Jon McCammond. A complex Euclidean reflection group with a non-positively curved complement complex. Groups, geometry, and dynamics, Tome 15 (2021) no. 3, pp. 989-1013. doi: 10.4171/ggd/620
@article{10_4171_ggd_620,
author = {Ben Cot\'e and Jon McCammond},
title = {A complex {Euclidean} reflection group with a non-positively curved complement complex},
journal = {Groups, geometry, and dynamics},
pages = {989--1013},
year = {2021},
volume = {15},
number = {3},
doi = {10.4171/ggd/620},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/620/}
}
TY - JOUR
AU - Ben Coté
AU - Jon McCammond
TI - A complex Euclidean reflection group with a non-positively curved complement complex
JO - Groups, geometry, and dynamics
PY - 2021
SP - 989
EP - 1013
VL - 15
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/620/
DO - 10.4171/ggd/620
ID - 10_4171_ggd_620
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%A Ben Coté
%A Jon McCammond
%T A complex Euclidean reflection group with a non-positively curved complement complex
%J Groups, geometry, and dynamics
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%P 989-1013
%V 15
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/620/
%R 10.4171/ggd/620
%F 10_4171_ggd_620
