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

TL;DR

This paper proves the Noferini-Williams conjecture for Gilbert-Howie groups by analyzing polynomial resultants and field theory, confirming conditions for torsion-free abelianizations and completing the classification of related graph groups.

Contribution

It confirms the Noferini-Williams conjecture for Gilbert-Howie groups using polynomial and field-theoretic methods, completing their classification within Fibonacci-type cyclically presented groups.

Findings

Confirmed the torsion-free abelianization condition for specific group parameters.

Reduced the problem to three key cases using minimality arguments.

Utilized polynomial resultant analysis and field theory in the proof.

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 $n\equiv 0\pmod{6}$ and $m\equiv 2\pmod{n}$. We confirm this conjecture by proving that $\mathrm{Res}(F,G)=\pm1$, 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.