Coxeter polytopes with a unique pair of non-intersecting facets |
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Authors: | Anna Felikson Pavel Tumarkin |
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Institution: | a Independent University of Moscow, B. Vlassievskii 11, 119002 Moscow, Russia b Department of Mathematics, University of Fribourg, Pérolles, Chemin du Musée 23, CH-1700 Fribourg, Switzerland |
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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. |
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Keywords: | Coxeter polytope Missing face Simple polytope Coxeter diagram |
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