All hyperbolic cyclically presented groups with positive length three relators

TL;DR

This paper classifies all hyperbolic cyclically presented groups with positive length three relators, showing they are hyperbolic only for specific values of m, thus completing the classification for this class.

Contribution

It provides a complete classification of hyperbolic cyclically presented groups with positive length three relators based on the parameter m.

Findings

Groups are hyperbolic if and only if m is in {1,2,3,6,9}.

Completes the classification of these groups.

Identifies the exact conditions for hyperbolicity.

Abstract

We consider the cyclically presented groups defined by cyclic presentations with $2m$ generators $x_i$ whose relators are the $2m$ positive length three relators $x_ix_{i+1}x_{i+m-1}$. We show that they are hyperbolic if and only if $m\in \{1,2,3,6,9\}$. This completes the classification of the hyperbolic cyclically presented groups with positive length three relators.