New complex and quaternion-hyperbolic re ection groups

We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian, and Hurwitz integers, and in some of them we nd reeection groups of nite index. These provide explicit constructions of new nite-covolume reeection groups acting on complex and quaternionic hyperbolic spaces of high dimensions. Speciically, 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 reeec-tions, and obtain a description of the group in terms of the combinatorics of a lower-dimensional positive-deenite lattice. The techniques needed for our lower-dimensional 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 reeection groups in dimensions even higher than those considered here, and mention connections to moduli of cubic surfaces. By studying orbits of norm 0 vectors in certain selfdual Lorentzian lattices we provide a new and geometric proof of the classiications of selfdual Eisenstein lattices of dimension 6 and of selfdual Hurwitz lattices of dimension 4. 1. Introduction In this paper we carry out complex and quaternionic analogues of some of Vinberg's extensive study of reeection groups on real hyperbolic space. In 24] and 25] he investigated the symmetry groups of the integral quadratic forms diag ?1; +1; : : : ; +1], or equivalently the Lorentzian lattices I n;1. He was able to describe these groups quite concretely for n 17, and extensions of his work by Vinberg and Kaplinskaja 26] and Borcherds 6] provide concrete descriptions for all n 23. In particular, the subgroup of Aut I n;1 generated by reeections has nite index just when n 19. In our work, we study the symmetry groups of Lorentzian lattices over the rings G and E of Gaussian and Eisenstein integers and the ring H of Hurwitz integers (a discrete subring of the skew eld H of quaternions). Most of the paper is devoted to the most natural of such lattices, the selfdual ones. The symmetry groups of these lattices provide a large number of discrete groups generated by reeections and acting with …