Skip to main content
SpringerLink
Log in

Essential hyperbolic Coxeter polytopes

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We introduce a notion of an essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least 6 which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, realize it in the four-dimensional case, and formulate a conjecture on finiteness of the number of essential polytopes.

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. I. Agol, M. Belolipetsky, P. Storm and K. Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups, Geometry, and Dynamics 2 (2008), 481–498.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. Allcock, Infinitely many hyperbolic Coxeter groups through dimension 19, Geometry and Topology 10 (2006), 737–758.

    Article  MATH  MathSciNet  Google Scholar 

  3. E. M. Andreev, Intersection of plane boundaries of acute-angled polyhedra, Mathematical Notes 8 (1971), 761–764.

    Article  MATH  Google Scholar 

  4. E. M. Andreev, On convex polyhedra in Lobachevskii spaces, Math. USSR Sb. 10 (1970), 413–440.

    Article  Google Scholar 

  5. A. F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  6. M. Belolipetsky and V. Emery, On volumes of arithmetic quotients of PO(n, 1)(○), n odd, Proceedings of the London Mathematical Society 105 (2012), 541–570.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Borcherds, Coxeter groups, Lorentzian lattices, and K3 surfaces, International Mathematics Research Notices (1998), 1011–1031.

  8. V. O. Bugaenko, Groups of automorphisms of unimodular hyperbolic quadratic forms over the ring \(\mathbb{Z}[(\sqrt 5 + 1)/2]\), Moscow University Mathematics Bulletin 39 (1984), 6–14.

    MathSciNet  Google Scholar 

  9. V. O. Bugaenko, On reflective unimodular hyperbolic quadratic forms, Selecta Mathematica Sovietica 9 (1990), 263–271.

    MATH  MathSciNet  Google Scholar 

  10. V. O. Bugaenko, Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, in Lie Groups, their Discrete Subgroups, and Invariant Theory, Advances in Soviet Mathematics, Vol. 8, American Mathematical Society, Providence, RI, 1992, pp. 33–55.

    Google Scholar 

  11. F. Esselmann, Über kompakte hyperbolische Coxeter-Polytope mit wenigen Facetten, Universität Bielefeld, SFB 343, Preprint 94-087.

  12. F. Esselmann, The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets, Commentarii Mathematici Helvetici 71 (1996), 229–242.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Felikson, Coxeter decomposition of hyperbolic simplices, Sbornik. Mathematics 193 (2002), 1867–1888.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Felikson and P. Tumarkin, On subgroups generated by reflections in groups generated by reflections, Functional Analysis and its Applications 38 (2004), 313–314.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Felikson and P. Tumarkin, Reflection subgroups of Euclidean reflection groups, Sbornik. Mathematics 196 (2005), 1349–1369.

    Article  MATH  MathSciNet  Google Scholar 

  16. A. Felikson and P. Tumarkin, On hyperbolic Coxeter polytopes with mutually intersecting facets, Journal of Combinatorial Theory. Series A 115 (2008), 121–146.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Felikson and P. Tumarkin, On d-dimensional compact hyperbolic Coxeter polytopes with d+4 facets, Transactions of the Moscow Mathematical Society 69 (2008), 105–151.

    Article  MathSciNet  Google Scholar 

  18. A. Felikson and P. Tumarkin, Coxeter polytopes with a unique pair of non-intersecting facets, Journal of Combinatorial Theory. Series A 116 (2009), 875–902.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Felikson and P. Tumarkin, Reflection subgroups of Coxeter groups, Transactions of the American Mathematical Society 362 (2010), 847–858.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Felikson, P. Tumarkin and T. Zehrt, On hyperbolic Coxeter n-polytopes with n + 2 facets, Advances in Geometry 7 (2007), 177–189.

    Article  MATH  MathSciNet  Google Scholar 

  21. H.-C. Im Hof, Napier cycles and hyperbolic Coxeter groups, Bulletin de la Société Mathématique de Belgique. Série A 42 (1990), 523–545.

    MATH  MathSciNet  Google Scholar 

  22. N. W. Johnson, R. Kellerhals, J. G. Ratcliffe and S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups 4 (1999), 329–353.

    Article  MATH  MathSciNet  Google Scholar 

  23. N. W. Johnson, R. Kellerhals, J. G. Ratcliffe and S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, Linear Algebra and its Applications 345 (2002), 119–147.

    Article  MATH  MathSciNet  Google Scholar 

  24. I. M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Mathematical Notes 15 (1974), 88–91.

    Article  Google Scholar 

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

    Google Scholar 

  26. V. S. Makarov, The Fedorov groups of four-dimensional and five-dimensional Lobacevskii space, in Studies in General Algebra, No. 1 (Russian), Kishinev. Gos. Univ., Kishinev, 1968, pp. 120–129.

    Google Scholar 

  27. V. V. Nikulin, Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces, Izvestiya. Mathematics 71 (2007), 53–56.

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Poincaré, Théorie des groupes fuchsiens, Acta Mathematica 1 (1882), 1–62.

    Article  MATH  MathSciNet  Google Scholar 

  29. L. Potyagailo and E. Vinberg, On right-angled reflection groups in hyperbolic spaces, Commentarii Mathematici Helvetici 80 (2005), 63–73.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Schlettwein, Hyperbolische Simplexe, Diplomarbeit, Basel, 1995.

    Google Scholar 

  31. P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n + 3 facets, Electronic Journal of Combinatorics 14 (2007), #R69, 36 pp.

    Google Scholar 

  32. E. B. Vinberg, The absence of crystallographic groups of reflections in Lobachevsky spaces of large dimension, Transactions of the Moscow Mathematical Society 47 (1985), 75–112.

    Google Scholar 

  33. E. B. Vinberg, Hyperbolic reflection groups, Russian Mathematical Surveys 40 (1985), 31–75.

    Article  MATH  MathSciNet  Google Scholar 

  34. E. B. Vinberg (Ed.), Geometry II, Encyclopaedia of Mathematical Sciences, Vol. 29, Springer-Verlag, Berlin, 1993.

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Anna Felikson

    Present address: School of Engineering and Science, Jacobs University Bremen, P.O. Box 750 561, 28725, Bremen, Germany

Authors and Affiliations

  1. Independent University of Moscow, B. Vlassievskii 11, 119002, Moscow, Russia

    Anna Felikson

  2. Department of Mathematical Sciences, Durham University Science Laboratories, South Road, Durham, DH1 3LE, UK

    Pavel Tumarkin

Authors
  1. Anna Felikson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavel Tumarkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anna Felikson.

Additional information

Research was partially supported by grants RFBR 07-01-00390-a and RFBR 11-01-00289-a.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Felikson, A., Tumarkin, P. Essential hyperbolic Coxeter polytopes. Isr. J. Math. 199, 113–161 (2014). https://doi.org/10.1007/s11856-013-0046-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-013-0046-3

Keywords

Search

Navigation