Skip to main content
SpringerLink
Log in

Hyperbolic Coxeter N-Polytopes with n+2 Facets

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In this paper, we classify all the hyperbolic noncompact Coxeter polytopes of finite volume whose combinatorial type is either that of a pyramid over a product of two simplices or a product of two simplices of dimension greater than one. Combined with the results of Kaplinskaja (1974) and Esselmann (1996), this completes the classification of hyperbolic Coxeter N-polytopes of finite volume with n+2 facets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. E. M. Andreev, “On convex polyhedra in Lobachevskii spaces,” Mat. Sb. [Math. USSR-Sb.], 81 (1970), no. 3, 445–478.

    Google Scholar 

  2. E. M. Andreev, “On convex polyhedra of finite volume in Lobachevskii spaces,” Mat. Sb. [Math. USSR-Sb.], 83 (1970), no. 2, 256–260.

    Google Scholar 

  3. F. Lannér, “On complexes with transitive groups of automorphisms,” Comm. Sem. Math. Univ. Lund, 11 (1950), 1–71.

    Google Scholar 

  4. É. B. Vinberg, “Discrete groups generated by reflections in Lobachevskii spaces,” Mat. Sb. [Math. USSR-Sb.], 72 (1967), no. 4, 471–488.

    Google Scholar 

  5. I. M. Kaplinskaya, “Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces,” Mat. Zametki [Math. Notes], 15 (1974), no. 1, 159–164.

    Google Scholar 

  6. F. Esselmann, “The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets,” Comm. Math. Helv., 71 (1996), 229–242.

    Google Scholar 

  7. B. Grünbaum, Convex Polytopes, John Wiley & Sons, New York, 1967.

    Google Scholar 

  8. V. A. Emelichev, M. M. Kovalev, and M. K. Kravtsov, Polytopes, Graphs, Optimization [in Russian], Nauka, Moscow, 1981.

    Google Scholar 

  9. É. B. Vinberg, “Hyperbolic reflection groups.,” Uspekhi Mat. Nauk [Russian Math. Surveys], 40 (1985), no. 1, 29–64.

    Google Scholar 

  10. É. B. Vinberg, “The absence of crystallographic groups of reflections in Lobachevsky spaces of large dimensions,” Trudy Moskov. Mat. Obshch. [Trans. Moscow Math. Soc.], 47 (1984), 68–102.

    Google Scholar 

  11. P. Tumarkin, “Hyperbolic Coxeter n-polytopes with n + 2 facets,” in: E-print math.MG/0301133, 2003.

Download references

Author information

Authors and Affiliations

  1. Independent Moscow University, Russia

    P. V. Tumarkin

Authors
  1. P. V. Tumarkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tumarkin, P.V. Hyperbolic Coxeter N-Polytopes with n+2 Facets. Mathematical Notes 75, 848–854 (2004). https://doi.org/10.1023/B:MATN.0000030993.74338.dd

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:MATN.0000030993.74338.dd

Search

Navigation