Proof of the Noferini-Williams conjecture for Gilbert-Howie groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908818371248128 |
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| author | Chinyere, Ihechukwu |
| author_facet | Chinyere, Ihechukwu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07171 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proof of the Noferini-Williams conjecture for Gilbert-Howie groups Chinyere, Ihechukwu Group Theory 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. |
| title | Proof of the Noferini-Williams conjecture for Gilbert-Howie groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2602.07171 |