Geometriae Dedicata

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

Chordal Coxeter groups

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


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.


Coxeter groups Diagram twisting Isomorphism problem Chordal graphs 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations