Explicit generators for the relation module in the example of Gruenberg-Linnell

TL;DR

This paper explicitly constructs a minimal generating set for the relation module of a free product of groups, demonstrating a potential relation gap in group theory.

Contribution

It provides an explicit set of generators for the relation module in the Gruenberg-Linnell example, confirming the existence of a relation gap.

Findings

Explicit generators for the relation module are constructed.

The result confirms the possibility of a relation gap in this context.

Provides insight into the structure of relation modules in free products.

Abstract

Gruenberg and Linnell showed that the standard relation module of a free product of $n$ groups of the form $C_r \times \mathbb{Z}$ could be generated by just $n+1$ generators, raising the possibility of a relation gap. We explicitly give such a set of generators.