Coxeter polytopes with a unique pair of non-intersecting facets

Details Coxeter polytopes with a unique pair of non-intersecting facets Coxeter polytopes with a unique pair of non-intersecting facets

2007-06-27

arxiv.org/abs/0706.3964v1

J. Combin. Theory A 116 (2009), 875--902

Mathematics

Metric Geometry

35 pages, a lot of figures

Scientific paper

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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