Generators for a complex hyperbolic braid group

Abstract

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$–space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient space. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the “monstrous proposal”, which posits a relationship between this braid group and the monster finite simple group.