Coxeter polytopes with a unique pair of non-intersecting facets

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 by Kaplinskaja [I.M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Math. Notes 15 (1974) 88–91] and the second author [P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n + 3 facets, Electron. J. Combin. 14 (2007), R69, 36 pp.], 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.