All hyperbolic cyclically presented groups with positive length three relators

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 $x_i{x}_{i+1}{x}_{i+m-1}$. 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.