Inventiones mathematicae

, Volume 140, Issue 2, pp 283–301

The Leech lattice and complex hyperbolic reflections

  • Daniel Allcock

DOI: 10.1007/s002220050363

Cite this article as:
Allcock, D. Invent. math. (2000) 140: 283. doi:10.1007/s002220050363


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 [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 ?H7, ?H5 and ?H3, again using the Leech lattice, and apply results of Borcherds [4] to obtain automorphic forms for our groups.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Daniel Allcock
    • 1
  1. 1.Department of Mathematics, Harvard University, Cambridge, MA 02138, USAUS