 
Summary: New complex and quaternionhyperbolic reflection groups
Daniel Allcock*
9 September 1997
allcock@math.utah.edu
web page: http://www.math.utah.edu/¸allcock
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
1991 mathematics subject classification: 22E40 (11F06, 11E39, 53C35)
Dedicated to my father John Allcock, 19401991
Abstract.
We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian,
and Hurwitz integers, and in some of them we find reflection groups of finite index. These provide
explicit constructions of new finitecovolume reflection groups acting on complex and quaternionic
hyperbolic spaces of high dimensions. Specifically, we provide groups acting on C H n for all n ! 6
and n = 7, and on H H n for n = 1; 2; 3; and 5. We compare our groups with those discovered by
Deligne and Mostow and by Thurston, and show that our most interesting examples are new. For
many of these Lorentzian lattices we show that the entire symmetry group is generated by reflec
tions, and obtain a description of the group in terms of the combinatorics of a lowerdimensional
positivedefinite lattice. The techniques needed for our lowerdimensional examples are elementary;
to construct our best examples we also use certain facts about the Leech lattice. We conjecture
that Lorentzian lattices provide examples of hyperbolic reflection groups in dimensions even higher
