  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 $(M \times M):2$.
