Volume 20, issue 2 (2016)

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Geometric generators for braid-like groups

Daniel Allcock and Tathagata Basak

Geometry & Topology 20 (2016) 747–778
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 ${ℂ}^{n}$, or complex hyperbolic space $ℂ{ℍ}^{n}$, or the Hermitian symmetric space for $O\left(2,n\right)$, and then takes the quotient by a discrete group $P\phantom{\rule{0.3em}{0ex}}\Gamma$. 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 $P\phantom{\rule{0.3em}{0ex}}\Gamma$ 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 $ℂ{ℍ}^{13}$. 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 $\left(M×M\right):2$. We also suggest some other applications of our machinery.

Keywords
fundamental groups, infinite hyperplane arrangement, complex hyperbolic geometry, braid groups, Artin groups, Leech lattice, presentations, Monster
Mathematical Subject Classification 2010
Primary: 57M05
Secondary: 20F36, 52C35, 32S22