Geometric Generators for Braid-like Groups

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 Cn, or complex hyperbolic space CHn, or the Hermitian symmetric space for O(2, n), and then takes the quotient by a discrete group PΓ. 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 PΓ 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 CH. 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 ×M) : 2.