Geometriae Dedicata

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

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

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 


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© 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

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