Non-triviality of some one-relator products of three groups

TL;DR

This paper investigates the non-triviality of certain one-relator products of three groups, demonstrating that such groups are non-trivial when the relator length is at most eight, with stronger results for cyclic factors.

Contribution

It proves non-triviality of specific one-relator products of three groups for relators up to length eight and shows embedding of factors in the cyclic case.

Findings

Groups are non-trivial if relator length ≤ 8

In cyclic factors, at least one factor embeds in the quotient

Provides new insights into the structure of one-relator products

Abstract

In this paper we study a group G which is the quotient of a free product of three non-trivial groups by the normal closure of a single element. In particular we show that if the relator has length at most eight, then G is non-trivial. In the case where the factors are cyclic, we prove the stronger result that at least one of the factors embeds in G.