An investigation into the cyclically presented groups with length three positive relators

TL;DR

This paper explores properties of cyclically presented groups with length three positive relators, focusing on small cancellation conditions, Betti numbers, and isomorphism classifications, combining theoretical analysis with computational experiments.

Contribution

It advances understanding of these groups by analyzing invariants and isomorphism classes, and evaluates the effectiveness of abelianisation as a distinguishing tool.

Findings

Determined Betti numbers of the groups' abelianisations

Calculated orders of some abelianisations

Assessed abelianisation's effectiveness in distinguishing non-isomorphic groups

Abstract

We continue research into the cyclically presented groups with length three positive relators. We study small cancellation conditions and SQ-universality, we obtain the Betti numbers of the groups' abelianisations, we calculate the orders of the abelianisations of some groups, and we study isomorphism classes of the groups. Through computational experiments we assess how effective the abelianisation is as an invariant for distinguishing non-isomorphic groups.