The Tits alternative for generalized triangle groups of type (3,4,2)

Abstract

A generalized triangle group is a group that can be presented in the form $G = < x,y | x^p=y^q=w(x,y)^r=1>$, where $p,q,r\geq 2$ and $w(x,y)$ is a cyclically reduced word of length at least 2 in the free product $\Z_p*\Z_q=< x,y | x^p=y^q=1>$. 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 $(2,3,2), (2,4,2),(2,5,2),(3,3,2),(3,4,2)$, or $(3,5,2)$. In this paper we show that the Tits alternative holds in the case $(p,q,r)=(3,4,2)$.