Summary: The Leech Lattice and Complex Hyperbolic Reflections
19 November 1997
web page: http://www.math.utah.edu/łallcock
Department of Mathematics
University of Utah
Salt Lake City, UT 84112.
We construct a natural sequence of finite≠covolume reflection groups acting on the complex hy≠
perbolic spaces C H 13 , C H 9 and C H 5 , and show that the 9≠dimensional example coincides with
the largest of the groups of Mostow . 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 finite≠covolume 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  to obtain automorphic forms for our groups.
In  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.