Geometriae Dedicata

, Volume 119, Issue 1, pp 181–197 | Cite as

Free Subgroups in Certain Generalized Triangle Groups of Type (2, m, 2)

  • James Howie
  • Gerald Williams
Original Paper


A generalized triangle group is a group that can be presented in the form \(G = \langle x,y | x^p = y^q = w(x,y)^{r} = 1 \rangle\) where p,q,r ≥ 2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product \(\mathbb{Z}_{p}*\mathbb{Z}_{q}=\langle x,y x^p = y^q = 1\rangle\). Rosenberger has conjectured that every generalized triangle group G satisfies the Tits alternative. It is known that the conjecture holds except possibly when the triple (p,q,r) is one of (3, 3, 2), (3, 4, 2), (3, 5, 2), or (2, m, 2) where m=3, 4, 5, 6, 10, 12, 15 , 20, 30, 60. In this paper, we show that the Tits alternative holds in the cases (p,q,r)=(2, m, 2) where m=6, 10, 12, 15, 20, 30, 60.


Generalised triangle group Free subgroup Tits alternative 

Mathematics Subject Classifications (2000)

20F05 20E05 57M07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barkovich O.A., Benyash-Krivets V.V. (2003). On Tits alternative for generalized triangular groups of (2, 6, 2) type (Russian). Dokl. Nat. Akad. Nauk. Belarusi 48(3):28–33MathSciNetGoogle Scholar
  2. 2.
    Baumslag G., Morgan J.W., Shalen P.B. (1987). Generalized triangle groups. Math. Proc. Cambridge Philos. Soc. 102(1):25–31MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Benyash-Krivets V.V. (2003). On free subgroups of certain generalised triangle groups (Russian). Dokl. Nat. Akad. Nauk. Belarusi 47(3):14–17MathSciNetGoogle Scholar
  4. 4.
    Benyash-Krivets, V.V.: On Rosenberger’s conjecture for generalized triangle groups of types (2, 10, 2) and (2, 20, 2). In: Kalla, S. L. et al., (eds.), Proceedings of the International Conference on Mathematics and Its Applications, pp. 59–74. Kuwait Foundation for the Advancement of Sciences (2005)Google Scholar
  5. 5.
    Benyash-Krivets V.V., Barkovich O.A. (2004). On the Tits alternative for some generalized triangle groups. Algebra Discrete Math. 2004(2):23–43MathSciNetGoogle Scholar
  6. 6.
    Bieri R., Strebel R. (1980). Valuations and finitely presented metabelian groups. Proc. London Math. Soc. (3) 41(3):439–464MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Coxeter H.S.M., Moser W.O.J. (1972). Generators and relations for discrete groups. Ergeb. Math. Grenzgebiete. Springer-Verlag, Berlin–Heidelberg–New YorkGoogle Scholar
  8. 8.
    Fine B., Levin F., Rosenberger G. (1988). Free subgroups and decompositions of one-relator products of cyclics I The Tits alternative. Arch. Math. (Basel) 50(2):97–109MATHMathSciNetGoogle Scholar
  9. 9.
    Fine B., Roehl F., Rosenberger G. (2000). The Tits alternative for generalized triangle groups. In: Baik Y.G. et al. (eds) Groups—Korea’98 Proceedings of the 4th International Conference, Pusan, Korea, August 10–16, 1998. Walter de Gruyter, Berlin, pp. 95–131Google Scholar
  10. 10.
    The GAP Group. GAP—Groups, Algorithms, and Programming, Version 4.4 (2004). (http://www. Scholar
  11. 11.
    Gradshteyn I.S., Ryzhik I.M. (1994). Table of Integrals, Series, and Products Fifth edition. Academic Press Inc., Boston, MA, Translation edited and with a preface by Alan JeffreyMATHGoogle Scholar
  12. 12.
    Horowitz R.D. (1972). Characters of free groups represented in the two-dimensional special linear group. Comm. Pure Appl. Math. 25:635–649CrossRefMathSciNetGoogle Scholar
  13. 13.
    Howie J. (1998). Free subgroups in groups of small deficiency. J. Group Theory 1(1):95–112MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Levin, F., Rosenberger, G.: On free subgroups of generalized triangle groups. II. In: Group theory (Granville, OH, 1992), pp. 206–228. World Sci. Publishing, River Edge, NJ (1993)Google Scholar
  15. 15.
    Rosenberger G. (1989). On free subgroups of generalized triangle groups. Algebra i Logika 28(2):227–240MathSciNetGoogle Scholar
  16. 16.
    Williams, A.G.T.: Generalised triangle groups of type (2,m,2). In: Atkinson, M. et al., (eds.), Computational and Geometric Aspects of Modern Algebra, LMS Lecture Note Series 275, pp. 265–279. Cambridge University Press (2000)Google Scholar

Copyright information

© Springer Science+Business Media B.V 2006

Authors and Affiliations

  1. 1.School of Mathematical and Computer SciencesHeriot-Watt UniversityEdinburghUK
  2. 2.Institute of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK

Personalised recommendations