The Leech lattice and complex hyperbolic reflections
- Cite this article as:
- Allcock, D. Invent. math. (2000) 140: 283. doi:10.1007/s002220050363
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We construct a natural sequence of finite-covolume reflection groups acting on the complex hyperbolic spaces ℂH13, ℂH9 and ℂH5, and show that the 9-dimensional example coincides with the largest of the groups of Mostow . 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 . We also construct finite-covolume reflection groups on the quaternionic hyperbolic spaces ?H7, ?H5 and ?H3, again using the Leech lattice, and apply results of Borcherds  to obtain automorphic forms for our groups.