A class of digraph groups defined by balanced presentations

TL;DR

This paper introduces a new class of groups defined by balanced presentations with relators of a specific form, associating directed graphs to analyze their properties, and establishes conditions under which these groups are non-trivial and infinite.

Contribution

It defines a novel class of digraph groups based on balanced presentations and proves non-triviality and infiniteness under girth constraints, extending understanding of their algebraic structure.

Findings

Groups are non-trivial when the girth of the associated graph is at least 4.

Such groups cannot be finite of rank 3 or higher under the girth condition.

Without girth constraints, trivial and finite rank 3 groups can occur.

Abstract

We consider groups defined by non-empty balanced presentations with the property that each relator is of the form R(x,y), where x and y are distinct generators and R(.,.) is determined by some fixed cyclically reduced word R(a,b) that involves both a and b. To every such presentation we associate a directed graph whose vertices correspond to the generators and whose arcs correspond to the relators. Under the hypothesis that the girth of the underlying undirected graph is at least 4, we show that the resulting groups are non-trivial and cannot be finite of rank 3 or higher. Without the hypothesis on the girth it is well known that both the trivial group and finite groups of rank 3 can arise.