Proof of the Noferini-Williams conjecture for Gilbert-Howie groups

Abstract

The Gilbert-Howie groups $H(n,m)$ form a notable subclass within the broader family of Fibonacci-type cyclically presented groups $G_n(m,k)$. Noferini and Williams conjectured that the abelianization $H(n,m{)}^{ab}$ is torsion-free with $\mathbb{Z}$-rank $2$ if and only if $n\equiv 0(\mathrm{mod}6)$ and $m\equiv 2(\mathrm{mod}n)$. We confirm this conjecture by proving that $\mathrm{Res}(F,G)=\pm 1$, where $F=(1+t^m-t)/{\Phi }_6$ and $G=(t^n-1)/{\Phi }_6$, with ${\Phi }_6$ denoting the sixth cyclotomic polynomial. The proof uses a minimality argument, reducing the general problem to three cases: $m=2+n/3$, $m=2+n/2$, and $m=2+2n/3$. These cases are handled using polynomial resultant analysis and field-theoretic methods. As a consequence, we complete the classification of all $G_n(m,k)$ that arise as labelled oriented graph groups.

arXiv comment12 pages