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Geometriae Dedicata

, Volume 136, Issue 1, pp 57–77 | Cite as

Chordal Coxeter groups

  • John G. RatcliffeEmail author
  • Steven T. Tschantz
Original Paper

Abstract

A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every cycle of length at least four has a chord. As an application, we construct counterexamples to two conjectures concerning the isomorphism problem for Coxeter groups.

Keywords

Coxeter groups Diagram twisting Isomorphism problem Chordal graphs 

Mathematics Subject Classification (2000)

20F55 

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References

  1. 1.
    Brady N., McCammond J.P., Mühlherr B., Neumann W.D.: Rigidity of Coxeter groups and Artin groups. Geom. Dedicata 94, 91–109 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brink B., Howlett R.B.: Normalizers of parabolic subgroups in Coxeter groups. Invent. Math. 136, 323–351 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Caprace P-E., Mühlherr B.: Reflection rigidity of 2-spherical Coxeter groups. Proc. Lond. Math. Soc 94, 520–542 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Coxeter H.S.M.: Regular Polytopes. Dover, New York (1973)Google Scholar
  5. 5.
    Dirac G.A.: On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25, 71–76 (1961)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Franzsen, W.N.: Automorphisms of Coxeter groups. PhD thesis, University of Sydney (2001)Google Scholar
  7. 7.
    Franzsen W.N., Howlett R.B., Mühlherr B.: Reflections in abstract Coxeter groups. Comment. Math. Helv 81, 665–697 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Franzsen W.N., Howlett R.B.: Automorphisms of nearly finite Coxeter groups. Adv. Geom. 3, 301–338 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Grassi, M.: The Isomorphism problem for a class of finitely generated Coxeter groups. PhD thesis, University of Sydney (2007)Google Scholar
  10. 10.
    Howlett, R.B., Mühlherr, B.: Isomorphisms of Coxeter groups which do not preserve reflections. 18 p (2004, Preprint)Google Scholar
  11. 11.
    Kloks T., Kratsch D.: Listing all minimal separators of a graph. SIAM J. Comput. 27, 605–613 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mihalik M., Ratcliffe J., Tschantz S.: Matching theorems for systems of a finitely generated Coxeter group. Algebr. Geom. Topol. 7, 919–956 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Mihalik, M., Tschantz, S.: Visual decompositions of Coxeter groups, arXiv: math.GR/0703439v1 14 Mar 2007.Google Scholar
  14. 14.
    Mühlherr, B.: The isomorphism problem for Coxeter groups. In: The Coxeter legacy: reflections and projections. Davis C., Ellers E.W. (eds.) Amer. Math. Soc. pp. 1–15 (2006)Google Scholar
  15. 15.
    Ratcliffe, J., Tschantz, S.: Chordal Coxeter groups. arXiv:math.GR/0607301v2 12 Jul 2006.Google Scholar
  16. 16.
    Richardson R.W.: Conjugacy classes of involutions in Coxeter groups. Bull. Austral. Math. Soc. 26, 1–15 (1982)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentVanderbilt UniversityNashvilleUSA

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