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

TL;DR

This paper investigates a class of groups defined by specific presentations, showing they are either virtually soluble or contain a free subgroup, supporting Rosenberger's conjecture.

Contribution

It proves a dichotomy for generalized triangle groups of type (3,3,2), advancing understanding of their algebraic structure.

Findings

Groups are either virtually soluble or contain a free subgroup of rank 2.

Provides evidence supporting Rosenberger's conjecture.

Clarifies the structure of groups with presentation <x,y|x^3=y^3=W(x,y)^2=1>.

Abstract

If G is a group with a presentation of the form < x,y|x^3=y^3=W(x,y)^2=1 >, 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.