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Transactions of the Moscow Mathematical Society

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Hyperbolic Coxeter $n$-polytopes with $n+3$ facets

Author: P. V. Tumarkin
Translated by: James Wiegold
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 65 (2004).
Journal: Trans. Moscow Math. Soc. 2004, 235-250
MSC (2000): Primary 52B11; Secondary 20F55, 22E40
Published electronically: October 1, 2004
MathSciNet review: 2193442
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Abstract: Noncompact hyperbolic Coxeter $n$-polytopes of finite volume and having $n+3$ facets are studied in this paper.

Unlike the spherical and parabolic cases, no complete classification exists as yet for hyperbolic Coxeter polytopes of finite volume. It has been shown that the dimension of a bounded Coxeter polytope is at most 29 (Vinberg, 1984), while an upper estimate in the unbounded case is 995 (Prokhorov, 1986). There is a complete classification of simplexes and of Coxeter $n$-polytopes of finite volume with $n+2$ facets via the complexity of the combinatorial type.

In 1994, Esselman proved that compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets can only exist when $n\le 8$. In dimension 8 there is just one such polytope; it was found by Bugaenko in 1992.

Here we obtain an analogous result for noncompact polytopes of finite volume. There are none when $n>16$. We prove that there is just one when $ n=16$, and obtain its Coxeter diagram.

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  • 1. V. V. Ryzhkov (ed.), \cyr Geometriya. 2, \cyr Itogi Nauki i Tekhniki. [Progress in Science and Technology], Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988 (Russian). \cyr Sovremennye Problemy Matematiki. Fundamental′nye Napravleniya [Current Problems in Mathematics. Fundamental Directions], 29. MR 1315326
  • 2. E. M. Andreev, On convex polytopes in Lobachevskii spaces, Mat. Sb. 81 (1970), 445-478. English transl. in Math. USSR-Sbornik 12 (1970). MR 259734 (41:4367)
  • 3. E. M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space, Mat. Sb. (N.S.) 83 (125) (1970), 256–260 (Russian). MR 0273510
  • 4. V. O. Bugaenko, On groups of automorphisms of unimodular hyperbolic quadratic forms over the ring $\mathbb{Z}\left[\frac{\sqrt{5}+1}{2}\right]$, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1984), 6-12. English transl. in Moscow Univ. Math. Bull. 39 (1984). MR 0764026 (86d:11030)
  • 5. V. O. Bugaenko, Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, Lie groups, their discrete subgroups, and invariant theory, Adv. Soviet Math., vol. 8, Amer. Math. Soc., Providence, RI, 1992, pp. 33–55. MR 1155663
  • 6. È. B. Vinberg, Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sb. (N.S.) 72 (114) (1967), 471–488; correction, ibid. 73 (115) (1967), 303 (Russian). MR 0207853
  • 7. E. B. Vinberg, The absence of crystallographic reflection groups in Lobachevskii spaces of large dimensions, Trudy. Moscov. Mat. Obshch. 47 (1984), 65-102. English transl. in Trans. Moscow Math. Soc. 47 (1984). MR 0774946 (86i:22020)
  • 8. E. B. Vinberg, Hyperbolic reflection groups, Uspekhi Mat. Nauk 40 (1985), 29-64. English transl. in Russian Math. Surveys, 40 (1985). MR 0783604 (86m:53059)
  • 9. È. B. Vinberg and I. M. Kaplinskaja, The groups 𝑂_{18,1}(𝑍) and 𝑂_{19,1}(𝑍), Dokl. Akad. Nauk SSSR 238 (1978), no. 6, 1273–1275 (Russian). MR 0476640
  • 10. È. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., vol. 29, Springer, Berlin, 1993, pp. 139–248. MR 1254933,
  • 11. V. A. Emelichev, M. M. Kovalev, and M. K. Kravtsov, Polytopes, graphs and optimisation, Nauka, Moscow, 1981. MR 0656517 (83h:52014)
  • 12. I. M. Kaplinskaja, The discrete groups that are generated by reflections in the faces of simplicial prisms in Lobačevskiĭ spaces, Mat. Zametki 15 (1974), 159–164 (Russian). MR 0360858
  • 13. M. N. Prokhorov, The absence of discrete reflection groups with a noncompact fundamental polytope of finite volume in Lobachevskii spaces, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 413-424. English transl. in Math. USSR-Izv. 21 (1986). MR 0842588 (87k:22016)
  • 14. R. Borcherds, Automorphism groups of Lorentzian lattices, Journal of Algebra 111 (1987), 133-153. MR 0913200 (89b:20018)
  • 15. Frank Esselmann, The classification of compact hyperbolic Coxeter 𝑑-polytopes with 𝑑+2 facets, Comment. Math. Helv. 71 (1996), no. 2, 229–242. MR 1396674,
  • 16. F. Esselmann, Über kompakte hyperbolische Coxeter-Polytope mit wenigen Facetten, Universität Bielefeld. SFB 343. Preprint No. 94-087.
  • 17. Branko Grünbaum, Convex polytopes, With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. MR 0226496
  • 18. Folke Lannér, On complexes with transitive groups of automorphisms, Comm. Sém., Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 11 (1950), 71. MR 0042129
  • 19. P. Tumarkin, Hyperbolic Coxeter $n$-polytopes with $n+2$facets, math.MG/0301133.

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Additional Information

P. V. Tumarkin
Affiliation: Moscow Independent University, B. Vlas’evskii Per. 11, Moscow 119002, Russia

Published electronically: October 1, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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