On one-relator products induced by generalized triangle groups I

ABSTRACT

In this paper and a sequel, we study a group which is the quotient of a free product of groups by the normal closure of a single word that is contained in a subgroup which has the form of a free product of two cyclic groups. We use known properties of generalized triangle groups, together with detailed analysis 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 results generalize those in an earlier article of the second author and Shwartz.

KEYWORDS:

• Generalized triangle groups
• one-relator group
• pictures

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20F05
• Secondary: 20E06
• 20F06
• 20F10

Acknowledgments

We are grateful to an anonymous referee for useful and insightful comments on an earlier version of the paper, which have significantly improved the exposition. The first author was supported in this work by a Maxwell Institute Scholarship from Heriot-Watt University.