Coxeter polytopes with a unique pair of non-intersecting facets
arXiv:0706.3964 · doi:10.1016/j.jcta.2008.10.008
Abstract
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 Lannér, 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.
28 pages, lots of figures; v2: Lemma 2.2.1 corrected (thanks to Ruth Kellerhals for the correction!), further minor changes