On One-relator products induced by generalised triangle groups

TL;DR

This paper investigates the structure of certain one-relator products formed from free products of groups, using properties of generalized triangle groups to establish foundational results like Freiheitssatz and Mayer-Vietoris sequences.

Contribution

It introduces weaker hypotheses for analyzing these groups, extending previous results and providing new theoretical insights into their algebraic properties.

Findings

Proved a Freiheitssatz for a broader class of groups.

Established the existence of Mayer-Vietoris sequences under weaker conditions.

Generalized earlier results to more inclusive group settings.

Abstract

In this paper we study a group G which is the quotient of a free product of groups by the normal closure of a word that is contained in a in a subgroup which has the form of a generalised triangle group. We use known properties of generalised triangle groups, together with detailed analyses of pictures and of words in free monoids, to prove a number of results such as a Freiheitssatz and the existence of Mayer-Vietoris sequences for such groups under suitable hypotheses. The hypotheses are weaker than those in an earlier article of Howie and Shwartz, yielding generalisations in two directions of the results in that article.