### 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 |

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### Keywords

- Free subgroup
- Generalised triangle group
- Tits alternative

### Cite this

*Geometriae Dedicata*,

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

}

*Geometriae Dedicata*, vol. 119, no. 1, pp. 181-197. https://doi.org/10.1007/s10711-006-9068-x

**Free subgroups in certain generalized triangle groups of type (2, m, 2).** / Howie, James; Williams, Gerald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Free subgroups in certain generalized triangle groups of type (2, m, 2)

AU - Howie, James

AU - Williams, Gerald

PY - 2006/4

Y1 - 2006/4

N2 - A generalized triangle group is a group that can be presented in the form G = p = yq = 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 Zp * Zq = p = yq = 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.

AB - A generalized triangle group is a group that can be presented in the form G = p = yq = 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 Zp * Zq = p = yq = 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.

KW - Free subgroup

KW - Generalised triangle group

KW - Tits alternative

UR - http://www.scopus.com/inward/record.url?scp=33745075034&partnerID=8YFLogxK

U2 - 10.1007/s10711-006-9068-x

DO - 10.1007/s10711-006-9068-x

M3 - Article

VL - 119

SP - 181

EP - 197

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

IS - 1

ER -