All hyperbolic cyclically presented groups with positive length three relators
Ihechukwu Chinyere, Martin Edjvet, Gerald Williams
TL;DR
The paper resolves the hyperbolicity status of the family $\Gamma_{2m}$ of cyclically presented groups with positive length three relators by reducing hyperbolicity to the absence of a rank-2 free abelian subgroup in a finite extension $E_{2m}$. The authors construct $E_{2m}$ and define commuting elements $A=xt^{-3}$ and $B=t^mxt^{m+3}xt^{m-3}$, then prove a Main Lemma via a detailed curvature-distribution analysis on relative diagrams, which shows $A^\alpha B^\beta=1$ only when $\alpha=\beta=0$ for most $m$. This approach, combining relative presentations, star-graph vertex-label restrictions, and a two-stage curvature redistribution, yields a complete classification: Γ_{2m} is hyperbolic iff $m\in\{1,2,3,6,9\}$, with explicit isomorphisms for these cases. The results extend the classification program for hyperbolic cyclically presented groups with positive length three relators, complementing prior work on Fibonacci-type and other length-three relator families, and provide a rigorous geometric-combinatorial proof via curvature arguments.
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.