### Abstract

A generalized triangle group is a group that can be presented in the form G = p = y^{q} = w(x,y)^{r} = 1> where p,q,r = 2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product Z_{p} * Z_{q} = p = y^{q} = 1>. 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.

Original language | English |
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Pages (from-to) | 181-197 |

Number of pages | 17 |

Journal | Geometriae Dedicata |

Volume | 119 |

Issue number | 1 |

DOIs | |

Publication status | Published - Apr 2006 |

### Keywords

- Free subgroup
- Generalised triangle group
- Tits alternative

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## Cite this

Howie, J., & Williams, G. (2006). Free subgroups in certain generalized triangle groups of type (2, m, 2).

*Geometriae Dedicata*,*119*(1), 181-197. https://doi.org/10.1007/s10711-006-9068-x