Geometric generators for braid-like groups

Abstract

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group $P\Gamma$. The classical example is the braid group, but there are many similar "braid-like" groups that arise in topology and algebraic geometry. Our main result is that if $P\Gamma$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then G is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for G in a particular intricate example in complex hyperbolic space of dimension 13. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group M, that gives geometric meaning to the generators and relations in the Conway-Simons presentation of $(M\times M):2$.

24 pages, 5 figures. Revised version 2: minor typos corrected, expository material added, bibliography updated

Cited by in corpus (5)

• Generators for a complex hyperbolic braid group
• Non-crossing partitions and Milnor fibers
• The Deligne-Mostow 9-ball, and the monster
• A new complex reflection group in $PU(9,1)$ and the Barnes-Wall lattice
• Locally toroidal polytopes of rank 6 and sporadic groups