Proof of the Noferini-Williams conjecture for Gilbert-Howie groups
Ihechukwu Chinyere
TL;DR
The paper resolves the Noferini–Williams conjecture by showing that the abelianization of Gilbert-Howie groups $H(n,m)$ is torsion-free with $ obreak\mathbb{Z}$-rank $2$ if and only if $n\equiv0\pmod{6}$ and $m\equiv2\pmod n$. The authors recast the problem in terms of a resultant condition $\mathrm{Res}(F,G)=\pm1$ with $F=(1+t^m-t)/\Phi_6$ and $G=(t^n-1)/\Phi_6$, and employ a minimality argument to reduce to three parameter families $m=2+n/3$, $m=2+n/2$, and $m=2+2n/3$, analyzed via polynomial resultant and field-theoretic methods. Through detailed arithmetic and cyclotomic-structure arguments, they exclude all potential counterexamples, establishing the stated criterion and thereby completing the classification of $G_n(m,k)$ as LOG groups. A key corollary is the precise identification of which Fibonacci-type cyclically presented groups yield abelianizations that are free of rank two, and the result sharpens the understanding of the relationship between algebraic invariants and the combinatorial presentation data.
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 $F=(1+t^m-t)/Φ_6$ and $G=(t^n-1)/Φ_6$, with $Φ_6$ 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.