To Herb Clemens on his 60th birthday
1. Introduction. In this paper, we carry out complex and quaternionic analogues of some of Vinberg's extensive study of reflection groups on real hyperbolic space. In  and , Vinberg 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… (More)
The absolute logarithmic Weil height is well defined on the quotient group Q × / Tor`Q × ´ and induces a metric topology in this group. We show that the completion of this metric space is a Banach space over the field R of real numbers. We further show that this Banach space is isometrically isomorphic to a co-dimension one subspace of L 1 (Y, B, λ), where… (More)
We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A n , B n = C n and D n and the affine diagrams˜A n , ˜ B n , ˜ C n and˜D n as subgroups of the braid groups of various simple orbifolds. The cases D n , ˜ B n ,… (More)
We construct a natural sequence of nite-covolume reeection 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 10]. These reeection groups arise as automorphism groups of certain Lorentzian lattices over the Eisenstein integers, and we obtain… (More)
The moduli space of cubic threefolds in CP 4 , with some minor birational modifications, is the Baily-Borel compacti-fication of the quotient of the complex 10-ball by a discrete group. We describe both the birational modifications and the discrete group explicitly.
We show that the Heisenberg groups H 2n+1 of dimension ve and higher, considered as Rieman-nian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length L bounds a disk of area L 2). This implies several important results about isoperimetric inequalities for discrete groups that act either on H 2n+1 or on complex… (More)
It is well-known that the isomorphism classes of complex Enriques surfaces are in 1-1 correspondence with a Zariski-open subset (D − H)/Γ of the quotient of the Hermitian symmetric space D for O(2, 10). Here H is a totally geodesic divisor in D and Γ is a certain arithmetic group. In the usual formulation of this result , Γ is described as the isometry… (More)
For any root system and any commutative ring, we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac–Moody theory, for which the Steinberg group was defined by Tits and Morita–Rehmann. In most cases, our group equals St, giving a presentation with many advantages… (More)
We describe the moduli space of cubic hypersurfaces in CP 4 in the sense of geometric invariant theory. That is, we characterize the stable and semistable hypersurfaces in terms of their singularities, and determine the equivalence classes of semistable hypersurfaces under the equivalence relation of their orbit-closures meeting.