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Pro-Hp density in the free group of rank two: a Frobenian criterion on the mod-p torus
Abstract
Let F be the free group of rank two and, for a prime p, let Hp=Gp*Abp−1 be the pseudovariety of finite groups having a normal p-subgroup with abelian quotient of exponent dividing p−1. For H of rank two with abF(H)=Fab we determine the set D(H) of primes p where H is Hp-dense in F, and we show that this prime set is Frobenian in the sense of Serre: it is governed by a single Laurent polynomial g attached to H. We prove that p∈D(H) if and only if g has no zero on the torus (Fp×)2, and hence that D(H) possesses a computable natural density d(H). Exactly one of three cases holds: D(H) is the set of all primes, it is finite, or it is neither and d(H) satisfies 1/|G|≤ d(H)≤1−1/|G|, where G is the finite Galois group attached to g.
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BibTeX
@misc{wu2026prohp,
title = {{Pro-$\mathbf H_p$ density in the free group of rank two: a Frobenian criterion on the mod-$p$ torus}},
author = {Jianchun Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28941},
eprint = {2609.28941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28941},
}
RIS
TY - UNPB
TI - Pro-H_p density in the free group of rank two: a Frobenian criterion on the mod-p torus
AU - Jianchun Wu
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28941
UR - https://arxiv.org/abs/2609.28941
AB - Let F be the free group of rank two and, for a prime p, let H_p=G_p*Ab_p−1 be the pseudovariety of finite groups having a normal p-subgroup with abelian quotient of exponent dividing p−1. For H of rank two with ab_F(H)=F^ab we determine the set D(H) of primes p where H is H_p-dense in F, and we show that this prime set is Frobenian in the sense of Serre: it is governed by a single Laurent polynomial g attached to H. We prove that p∈D(H) if and only if g has no zero on the torus (F_p^×)^2, and hence that D(H) possesses a computable natural density d(H). Exactly one of three cases holds: D(H) is the set of all primes, it is finite, or it is neither and d(H) satisfies 1/|G|≤ d(H)≤1−1/|G|, where G is the finite Galois group attached to g.
ER -