Generalised triangle groups of type (2,3,2) with no cyclic essential representations

TL;DR

This paper verifies Roseberger's conjecture for a specific class of generalized triangle groups, showing they are either virtually soluble or contain non-abelian free subgroups, except for two special cases.

Contribution

It proves the conjecture for (2,3,2) type groups without essential representations onto cyclic groups of order 6, extending understanding of their algebraic structure.

Findings

Most (2,3,2) groups are either virtually soluble or contain free subgroups.

Two exceptional cases do not fit the general pattern.

Supports Roseberger's conjecture in a significant class of groups.

Abstract

A conjecture of Roseberger asserts that every generalised triangle group either is virtually soluble or contains a non-abelian free subgroup. Modulo two exceptional cases, we verify this conjecture for generalised triangle groups of type (2,3,2) which do not admit essential representations onto the cyclic group of order 6.