We study the problem of finding generators for the fundamental group of a space of the following sort: one removes a family of complex hyperplanes from , or complex hyperbolic space , or the Hermitian symmetric space for , and then takes the quotient by a discrete group . 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 contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for in a particular intricate example in . The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group , that gives geometric meaning to the generators and relations in the Conway–Simons presentation of . We also suggest some other applications of our machinery.
Daniel Allcock. Tathagata Basak. "Geometric generators for braid-like groups." Geom. Topol. 20 (2) 747 - 778, 2016. https://doi.org/10.2140/gt.2016.20.747