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Báo cáo toán học: "Compact hyperbolic Coxeter n-polytopes with n + 3 facets"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Compact hyperbolic Coxeter n-polytopes with n + 3 facets. | Compact hyperbolic Coxeter n-polytopes with n 3 facets Pavel Tumarkin Independent University of Moscow B. Vlassievskii 11 119002 Moscow Russia pasha@mccme.ru Submitted Apr 23 2007 Accepted Sep 30 2007 Published Oct 5 2007 Mathematics Subject Classifications 51M20 51F15 20F55 Abstract We use methods of combinatorics of polytopes together with geometrical and computational ones to obtain the complete list of compact hyperbolic Coxeter n-polytopes with n 3 facets 4 n 7. Combined with results of Esselmann this gives the classification of all compact hyperbolic Coxeter n-polytopes with n 3 facets n 4. Polytopes in dimensions 2 and 3 were classified by Poincaré and Andreev. 1 Introduction A polytope in the hyperbolic space Hn is called a Coxeter polytope if its dihedral angles are all integer submultiples of K. Any Coxeter polytope P is a fundamental domain of the discrete group generated by reflections in the facets of P. There is no complete classification of compact hyperbolic Coxeter polytopes. Vin-berg V1 proved there are no such polytopes in H n 30. Examples are known only for n 8 see B1 B2 . In dimensions 2 and 3 compact Coxeter polytopes were completely classified by Poincare P and Andreev A . Compact polytopes of the simplest combinatorial type the simplices were classified by Lannér L . Kaplinskaja K see also V2 listed simplicial prisms Esselmann E2 classified the remaining compact n-polytopes with n 2 facets. In the paper ImH Im Hof classified polytopes that can be described by Napier cycles. These polytopes have at most n 3 facets. Concerning polytopes with n 3 facets Esselmann proved the following theorem E1 Th. 5.1 Partially supported by grants MK-6290.2006.1 NSh-5666.2006.1 INTAS grant YSF-06-10000014-5916 and RFBR grant 07-01-00390-a THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R69 1 Let P be a compact hyperbolic Coxeter n-polytope bounded by n 3 facets. Then n 8 if n 8 then P is the polytope found by Bugaenko in B2 . This polytope has the following .