Generalised triangle groups of type (3,5,2)

TL;DR

This paper investigates a class of groups defined by specific presentations, showing they are either virtually soluble or contain a free subgroup of rank 2, supporting Rosenberger's conjecture.

Contribution

It proves a dichotomy for generalized triangle groups of type (3,5,2), 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

Classifies a specific family of generalized triangle groups

Abstract

If G is a group with a presentation of the form < x,y|x^3=y^5=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.