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 |xp=yq=w(x,y)r=1⟩ where p,q,r≥2 and w(x,y) is a cyclically reduced word of length at least 2 in the free product Zp∗Zq=⟨x,y |xp=yq=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). 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).