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Algebra and Discrete Mathematics, 2008, Issue 4, Pages 40–48
(Mi adm177)
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This article is cited in 1 scientific paper (total in 1 paper)
RESEARCH ARTICLE
The Tits alternative for generalized triangle groups of type $(3,4,2)$
James Howiea, Gerald Williamsb a Maxwell Institute of Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
b Department of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ
United Kingdom
Abstract:
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 $p,q,r\geq 2$ 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 $(2,3,2)$, $(2,4,2)$, $(2,5,2)$, $(3,3,2)$, $(3,4,2)$ or $(3,5,2)$. Building on a result of Benyash–Krivets and Barkovich from this journal, we show that the Tits alternative holds in the case $(p,q,r)=(3,4,2)$.
Keywords:
Generalized triangle group, Tits alternative, free subgroup.
Received: 15.05.2007 Revised: 16.10.2007
Citation:
James Howie, Gerald Williams, “The Tits alternative for generalized triangle groups of type $(3,4,2)$”, Algebra Discrete Math., 2008, no. 4, 40–48
Linking options:
https://www.mathnet.ru/eng/adm177 https://www.mathnet.ru/eng/adm/y2008/i4/p40
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