The near Frattini subgroup of a knot group is trivial
We prove that the near Frattini subgroup of every knot group is trivial. The argument first identifies the lower and upper near Frattini subgroups for finitely generated groups, and then shows that every nontrivial element of a finitely generated torsion-free virtually RFRS group is a near generator. Virtual specialness of knot groups then gives the result. This answers both parts of Problem 19.1 of the Kourovka Notebook.
Lemma 2.1
The set $\lambda(G)$ is a characteristic subgroup of $G$.
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- External referenceI. Agol, Criteria for virtual fibering, J. Topol. 1 (2008), 269--284.LaTeX source\bibitem{agol} I. Agol, Criteria for virtual fibering, \emph{J. Topol.} 1 (2008), 269--284.
- External referenceF. Haglund and D. T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), 1551--1620.LaTeX source\bibitem{haglundwise} F. Haglund and D. T. Wise, Special cube complexes, \emph{Geom. Funct. Anal.} 17 (2008), 1551--1620.
- External referenceE. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Novosibirsk, 2026; arXiv:1401.0300.LaTeX source\bibitem{kourovka} E. I. Khukhro and V. D. Mazurov (eds.), \emph{Unsolved Problems in Group Theory: The Kourovka Notebook}, No. 21, Novosibirsk, 2026; arXiv:1401.0300.
- External referenceP. Przytycki and D. T. Wise, Graph manifolds with boundary are virtually special, J. Topol. 7 (2014), 419--435.LaTeX source\bibitem{pwgraph} P. Przytycki and D. T. Wise, Graph manifolds with boundary are virtually special, \emph{J. Topol.} 7 (2014), 419--435.
- External referenceP. Przytycki and D. T. Wise, Mixed $3$-manifolds are virtually special, J. Amer. Math. Soc. 31 (2018), 319--347.LaTeX source\bibitem{pwmixed} P. Przytycki and D. T. Wise, Mixed $3$-manifolds are virtually special, \emph{J. Amer. Math. Soc.} 31 (2018), 319--347.
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Machine-readable theorem index · 5 statements
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- PF:2026.000005/v1/LEM-2.1
The set $\lambda(G)$ is a characteristic subgroup of $G$.
- PF:2026.000005/v1/THM-2.2
If $G$ is finitely generated, then $\lambda(G)=\mu(G)$.
- PF:2026.000005/v1/THM-3.1
If $G$ is finitely generated, torsion-free, and virtually RFRS, then $\psi(G)=1$.
- PF:2026.000005/v1/THM-4.1
Let $K\subset S^3$ be a knot and let $G_K=\pi_1(S^3\setminus\nu K)$. Then \[ \psi(G_K)=1. \]
- PF:2026.000005/v1/COR-4.2
As a consequence of Theorem 4.1, the answer to both parts of Kourovka Problem 19.1 is affirmative. In fact the conclusion holds for all knot groups, not only groups of composite or cable knots.