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New complex- and quaternion-hyperbolic reflection groups. (English) Zbl 0962.22007
E. B. Vinberg researched a number of reflection groups that are subgroups of the symmetry groups of the Lorentzian lattice \(I_{n,1}\) over \(\mathbb{R}\) [Mat. Sbornik (N.S.) 87, 18-36 (1972); English translation in Math. USSR-Sb. 87, 17-35 (1972; Zbl 0244.20058)]. These groups act on the real hyperbolic space \(\mathbb{R} H^n\). In the paper under review the author carries out complex and quaternionic analogues of Vinberg’s study. He researches the symmetry groups of Lorentzian lattices over each of the rings \(G,E,H\) where \(G, E,H\) denote the rings of Gaussian, Eisenstein, Hurwitz integers, i.e. \(G=\mathbb{Z} [i]\), \(E=\mathbb{Z} [-1+{1\over 2}\sqrt{-3}]\), \(H\) is the integral span of the units \(\pm 1,\pm i,\pm j.\pm k\) and \({1\over 2}(\pm 1,\pm i,\pm j,\pm k)\) in the field \(H\) of quaternions. A Lorentzian lattice is a free \(R\)-module equipped with a Hermitian form of signature \([-1,+1, \dots,+1]\). The symmetry groups provide a large number of discrete groups generated by reflections and acting with finite volume quotient on the hyperbolic spaces \(\mathbb{C} H^n\), \(HH^n\). The author constructs a total of 19 such groups. Some of them are known, some other ones are new. It should be pointed out that there are some applications on the moduli space of complex cubic surfaces.

MSC:
22E40 Discrete subgroups of Lie groups
11H06 Lattices and convex bodies (number-theoretic aspects)
11F55 Other groups and their modular and automorphic forms (several variables)
11E39 Bilinear and Hermitian forms
53C35 Differential geometry of symmetric spaces
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