Daniel Allcock

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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)
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 [5], 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.