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Generators for a complex hyperbolic braid group

Daniel Allcock and Tathagata Basak

Geometry & Topology 22 (2018) 3435–3500

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.

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
Received: 18 February 2017
Revised: 14 September 2017
Accepted: 20 November 2017
Published: 23 September 2018
Proposed: Walter Neumann
Seconded: Anna Wienhard, Benson Farb
Daniel Allcock
Department of Mathematics
University of Texas at Austin
Austin, TX
United States
Tathagata Basak
Department of Mathematics
Iowa State University
Ames, IA
United States