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 ${ℂ}^n$, 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 $P\Gamma$. 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 $ℂ{ℍ}^{13}$. 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 $\left(M\times M\right):2$. We also suggest some other applications of our machinery.