The Leech lattice and complex hyperbolic reflections

We construct a natural sequence of finite-covolume reflection groups acting on the complex hyperbolic spaces ℂH ¹³, ℂH ⁹ and ℂH ⁵, and show that the 9-dimensional example coincides with the largest of the groups of Mostow 11]. Our reflection 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 in a manner inspired by Conway 5]. We also construct finite-covolume reflection groups on the quaternionic hyperbolic spaces ?H ⁷, ?H ⁵ and ?H ³, again using the Leech lattice, and apply results of Borcherds 4] to obtain automorphic forms for our groups. Oblatum 25-III-1999 & 2-IX-1999?Published online: 21 February 2000