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Vinberg’s theory of hyperbolic reflection groups

Nikolay Bogachev (Skoltech & MIPT)

Location:  Zoom
Date & time: Wednesday, 21 April 2021 at 3:30PM - 4:30PM

Abstract: This talk is devoted to the memory of my teacher, Professor Ernest Borisovich Vinberg: 1937 -- 2020. All of us are familiar with the kaleidoscope, in which multicolored glasses form an attractive and amazing reflection pattern. Such pictures are obtained by a system of reflections with respect to certain mirrors. Generalizing this concept, we may consider discrete reflection groups of finite covolume in Euclidean spaces E^n and on the spheres S^n. Such were studied by many mathematicians and finally classified completely by Coxeter in 1933 via Coxeter diagrams. They exist in all dimensions n for both E^n and S^n.

The story of reflection groups in hyperbolic Lobachevsky spaces H^n (and to the phenomenal impact of Vinberg in this theory) goes back to the 19th century, to the works of Poincare and Dyck about the classification of Fuchsian groups. However only low-dimensional examples of such groups were known. In 1967, Vinberg initiated his fundamental theory of hyperbolic reflection groups. In 1972, he suggested an algorithm for constructing the fundamental Coxeter polytope of an arbitrary hyperbolic reflection group. The Vinberg algorithm is now widely used by many people in different branches of mathematics. In 1981, Vinberg obtained the following celebrated and surprising result: there are no compact hyperbolic Coxeter polytopes and no arithmetic finite volume Coxeter polytopes in H^n with n>29. With these results, Vinberg was selected as an Invited Speaker at the ICM 1983. In 2014, he constructed the first examples of higher-dimensional non-arithmetic non-compact hyperbolic Coxeter polytopes.

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