The unit ball Bn in the complex vector space Cn has a unique up to scaling U(n,1)-invariant metric. The ball Bn with this negative curvature metric is called the complex hyperbolic space. We shall study a discrete subgroup M of U(13,1) acting on the complex hyperbolic space B13. Let X be the space obtained by removing the hypersurfaces in B13 that have nontrivial stabilizer in M and then quotienting the restby M. The fundamental group G of the ball quotient X is a complex hyperbolic analog of the braid group. We shall state a conjecture that relates this fundamental group G and the monster simple group and describe our results towards this conjecture. This is joint work with Daniel Allcock. The Leech lattice plays a central role in our work.