New Complex- and Quaternion-hyperbolic Reflection Groups

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 new finitecovolume reflection groups acting on complex and quaternionic hyperbolic spaces. Specifically, we provide groups acting on CH for all n < 6 and n = 7, and on HH for n = 1, 2, 3, and 5. We compare our groups to those discovered by Deligne and Mostow and show that our largest examples are new. For many of these Lorentzian lattices we show that the entire symmetry group is generated by reflections, and obtain a description of the group in terms of the combinatorics of a lower-dimensional positive-definite lattice. The techniques needed for our lower-dimensional examples are elementary, but to construct our best examples we also need certain facts about the Leech lattice. We give a new and geometric proof of the classifications of selfdual Eisenstein lattices of dimension ≤ 6 and of selfdual Hurwitz lattices of dimension ≤ 4.