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 \( G = \langle {x,y}\ |{x^p=y^q=w(x,y)^r=1} \rangle \) 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 \((p,q,r)=(3,4,2)\).