 
Summary: The Leech Lattice and Complex Hyperbolic Reflections
Daniel Allcock
19 November 1997
allcock@math.utah.edu
web page: http://www.math.utah.edu/¸allcock
Department of Mathematics
University of Utah
Salt Lake City, UT 84112.
Abstract.
We construct a natural sequence of finitecovolume reflection groups acting on the complex hy
perbolic spaces C H 13 , C H 9 and C H 5 , and show that the 9dimensional example coincides with
the largest of the groups of Mostow [10]. These 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. We also construct finitecovolume reflection groups on the quater
nionic hyperbolic spaces H H 7 , H H 5 and H H 3 , again using the Leech lattice, and apply results of
Borcherds [3] to obtain automorphic forms for our groups.
1 Introduction
In [2] we constructed a large number of reflection groups acting on complex and quaternionic hy
perbolic spaces. For the most part, the groups appeared as symmetry groups of selfdual Lorentzian
lattices (i.e., those of signature (1; n)) over the rings of Eisenstein, Gaussian and Hurwitz integers.
