The Leech lattice and complex hyperbolic reflections

Abstract.

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.