#### Volume 22, issue 6 (2018)

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Generators for a complex hyperbolic braid group

### Daniel Allcock and Tathagata Basak

Geometry & Topology 22 (2018) 3435–3500
##### 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.

##### Keywords
fundamental group, presentations, Artin groups, braid group, hyperplane arrangements, lattices in PU(1,n), Leech lattice, monster
##### Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F36, 32S22, 52C35