Generalised triangle groups of type \((3,q,2)\)

Abstract

If \(G\) is a group with a presentation of the form \(\langle x,y|x^3=y^q=W(x,y)^2=1\rangle\), then either \(G\) is virtually soluble or \(G\) contains a free subgroup of rank \(2\). This provides additional evidence in favour of a conjecture of Rosenberger.