Hyperbolic Coxeter Pyramids


Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by Tumarkin for small p. In this article we generalise Tumarkins methods and find the remaining hyperbolic Coxeter pyramids.

Share and Cite:

J. Mcleod, "Hyperbolic Coxeter Pyramids," Advances in Pure Mathematics, Vol. 3 No. 1, 2013, pp. 78-82. doi: 10.4236/apm.2013.31010.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] E. B. Vinberg, “Hyperbolic Groups of Reflections,” Uspekhi Matematicheskikh Nauk, Vol. 40, 1985, pp. 29-66.
[2] F. Lannér, “On Complexes with Transitive Groups of Auto-morphisms,” Communications du Séminaire Mathématique de l'Université de Lund, Vol. 11, 1950, p. 71.
[3] N. W. Johnson, R. Kellerhals, J. G. Ratcliffe and S. T. Tschantz, “Commensurability Classes of Hyperbolic Coxeter Groups,” Linear Algebra and Its Applications, Vol. 345, No. 1, 2002, pp. 119-147. doi:10.1016/S0024-3795(01)00477-3
[4] N. W. Johnson, R. Kellerhals, J. G. Ratcliffe and S. T. Tschantz, “The Size of a Hyperbolic Coxeter Simplex,” Transformation Groups, Vol. 4, No. 4, 1999, pp. 329-353. doi:10.1007/BF01238563
[5] M. Chein, “Recherce des Graphes des Matrices de Coxeter Hyperboliques d’Ordre ≤10,” Revue Francaise Informatique Recherche Opérationnelle, Vol. 3, No. 2, 1969, pp. 3-16.
[6] A. A. Felikson, “Coxeter Decomposition of Hyperbolic Simplexes,” Sbornik: Mathematics, Vol. 193, No. 12, 2002, pp. 11-12. doi:10.1070/SM2002v193n12ABEH000702
[7] P. V. Tumarkin, “Hyperbolic Coxeter Polytopes in Hm with n + 2 Hyperfacets,” Mathematical Notes, Vol. 75, No. 5-6, 2004, pp. 848-854. doi:10.1023/B:MATN.0000030993.74338.dd
[8] P. V. Tumarkin, “Hyperbolic n-Dimensional Coxeter Poly-topes with n + 3 Facets,” Transactions of the Moscow Mathematical Society, Vol. 58, No. 4, 2004, pp. 235-250.
[9] G. Ziegler, “Lectures on Polytopes,” Springer-Verlag, New York, 1995.
[10] E. B. Vinberg, “Geometry II,” Springer-Verlag, Berlin, 1993.
[11] A. A. Felikson and P. V. Tumarkin, “Hyperbolic Subalgebras of Hyperbolic Kac-Moody Algebras,” Transformation Groups, Vol. 17, No. 1, 2012, pp. 87-122. doi:10.1007/s00031-011-9169-y

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.