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Coxeter polytopes with a unique pair of non-intersecting facets

Anna Felikson, Pavel Tumarkin
Arxiv ID: 0706.3964Last updated: 9/13/2022
We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results of Lann\'er, Kaplinskaja, Esselmann, and the second author, this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.

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