Structure of words with short 2-length in a free product of groups

TL;DR

This paper extends the decomposition properties of words in free products of groups to include elements of order two, generalizing previous results and providing new insights into the structure of such words.

Contribution

It generalizes the decomposition results of Howie and Duncan to free products with elements of order two, broadening the understanding of word structure in these groups.

Findings

Decomposition properties hold in more general settings with elements of order two

Extension of cyclic subword decomposition to broader class of free products

Implications for understanding one-relator groups with torsion

Abstract

Howie and Duncan observed that a word in a free product with length at least two and which is not a proper power can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved various important results about one-relator product of groups. In this paper, we show that similar results hold in a more general setting where we allow elements of order two.