Heriot-Watt Mathematics Report Series
HWM05-11, 25 May 2005
Free subgroups in certain generalised triangle groups of type (2,m,2)
J Howie and A G Williams
Abstract
A generalized triangle group is a group that can be presented in
the form
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\( G = < x,y | x^p=y^q=w(x,y)^r=1 > \)
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where $p,q,r\geq 2$ and $w(x,y)$ is an element of the free product
$< x,y | x^p=y^q=1 >$ involving both $x$ and $y$.
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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
$m=6,10,12,15,20,30,60$.
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