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Number Theory Seminar

Arithmetic and quasi-arithmetic hyperbolic reflection groups

Nikolay Bogachev (Skoltech & MIPT)

Location:  Zoom Link:
Date & time: Tuesday, 22 September 2020 at 2:00PM - 3:00PM

Abstract: In 1967, Vinberg started a systematic study of hyperbolic reflection groups. In particular, he showed that Coxeter polytopes are natural fundamental domains of hyperbolic reflection groups and developed practically efficient methods that allow to determine compactness or volume finiteness of a given Coxeter polytope by looking at its Coxeter diagram. He also proved a (quasi-)arithmeticity criterion for hyperbolic lattices generated by reflections. In 1981, Vinberg showed that there are no compact Coxeter polytopes in hyperbolic spaces \(H^n\) for \(n>29\). Also, he showed that there are no arithmetic hyperbolic reflection groups \(H^n\) for \(n>29\), either. Due to the results of Nikulin (2007) and Agol, Belolipetsky, Storm, and Whyte (2008) it became known that there are only finitely many maximal arithmetic hyperbolic reflection groups in all dimensions. These results give hope that maximal arithmetic hyperbolic reflection groups can be classified.

A very interesting moment is that compact Coxeter polytopes are known only up to \(H^8\), and in \(H^7\) and \(H^8\) all the known examples are arithmetic. Thus, besides the problem of classification of arithmetic hyperbolic reflection groups (which remains open since 1970-80s) we have another very natural question (which is again open since 1980s): do there exist compact (both arithmetic and non-arithmetic) hyperbolic Coxeter polytopes in \(H^n\) for \(n>8\) ?

The talk will be devoted to the discussion of these two related problems. One part of the talk is based on the recent preprint where some new geometric classification method is described. The second part is based on a joint work with Alexander Kolpakov where we prove that each lower-dimensional face of a quasi-arithmetic Coxeter polytope, which happens to be itself a Coxeter polytope, is also quasi-arithmetic. We also provide a few illustrative examples.

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