The Leech Lattice and Complex Hyperbolic Re ections

We construct a natural sequence of nite-covolume reeection groups acting on the complex hy-perbolic spaces C H 13 , C H 9 and C H 5 , and show that the 9-dimensional example coincides with the largest of the groups of Mostow 10]. These reeection groups arise as automorphism groups of certain Lorentzian lattices over the Eisenstein integers, and we obtain our largest example by using the complex Leech lattice. We also construct nite-covolume reeection groups on the quater-Borcherds 3] to obtain automorphic forms for our groups.