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

Abstract

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