Pro-$\mathbf H_p$ density in the free group of rank two: a Frobenian criterion on the mod-$p$ torus

Source abstract

Let $F$ be the free group of rank two and, for a prime $p$, let $\mathbf H_p=\mathbf G_p*\mathbf{Ab}_{p-1}$ 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 $\mathfrak D(H)$ of primes $p$ where $H$ is $\mathbf H_p$-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 $p\in\mathfrak D(H)$ if and only if $g$ has no zero on the torus $(\F_p^\times)^2$, and hence that $\mathfrak D(H)$ possesses a computable natural density $d(H)$. Exactly one of three cases holds: $\mathfrak D(H)$ is the set of all primes, it is finite, or it is neither and $d(H)$ satisfies $\frac1{|G|}\le d(H)\le1-\frac1{|G|}$, where $G$ is the finite Galois group attached to $g$.