Hyperbolic Coxeter 𝑛-polytopes with 𝑛+3 facets

Tumarkin, P.

doi:10.1090/S0077-1554-04-00146-3

Hyperbolic Coxeter $n$-polytopes with $n+3$ facets
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by P. V. Tumarkin
Translated by: James Wiegold

Trans. Moscow Math. Soc. 2004, 235-250

DOI: https://doi.org/10.1090/S0077-1554-04-00146-3

Published electronically: October 1, 2004

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Abstract:

Noncompact hyperbolic Coxeter $n$-polytopes of finite volume and having $n+3$ facets are studied in this paper.

Unlike the spherical and parabolic cases, no complete classification exists as yet for hyperbolic Coxeter polytopes of finite volume. It has been shown that the dimension of a bounded Coxeter polytope is at most 29 (Vinberg, 1984), while an upper estimate in the unbounded case is 995 (Prokhorov, 1986). There is a complete classification of simplexes and of Coxeter $n$-polytopes of finite volume with $n+2$ facets via the complexity of the combinatorial type.

In 1994, Esselman proved that compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets can only exist when $n\le 8$. In dimension 8 there is just one such polytope; it was found by Bugaenko in 1992.

Here we obtain an analogous result for noncompact polytopes of finite volume. There are none when $n>16$. We prove that there is just one when $n=16$, and obtain its Coxeter diagram.

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