A Complex Euclidean Reflection Group and Its Braid Group

Book excerpt: There is an interesting correspondence between groups generated by reflections, the arrangement of hyperplanes fixed by their reflections, and the braid groups arising from the complement of said hyperplanes. The spherical and euclidean Coxeter groups, their corresponding reflection arrangements and Artin groups have been previously investigated. More recent advances have been made in understanding complex spherical reflection groups and finite complex hyperplane arrangements. Little is known about the complex euclidean reflection groups, the infinite hyperplane arrangements provided by their mirrors, and their corresponding braid groups. For the affine extension of the complex reflection group Refl(G4), we construct a piecewise euclidean complex onto which the affine hyperplane complement deformation retracts, and show this is a CAT(0) space with nonregular points. Visualization techniques for the group and its geometry are described.