Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the infinite cyclic subgroup generated by $c$; in other words, $\psi(c)$ is conjugate to either $c$ or $c^{-1}$.
Now let us consider the profinite setting. Let $\widehat{F_2}$ denote the free profinite group over two generators $a,b$, and $c=[a,b]$.
- Which automorphisms of $\widehat{F_2}$ preserve the procyclic subgroup generated by $c$? In other words, for which $\psi\in \mathrm{Aut}(\widehat{F_2})$, $\psi(c)$ is conjugate to $c^\mu$ for some $\mu\in\widehat{\mathbb{Z}}^{\times}$?
- Which units $\mu\in\widehat{\mathbb{Z}}^{\times}$ can appear in the above question?
Some extras:
I found it not easy to "compute" elements in $\mathrm{Aut}(\widehat{F_2})$ or $\mathrm{Out}(\widehat{F_2})$. It is widely known that $\mathrm{Out}(F_2)\cong \mathrm{GL}_2(\mathbb{Z})$, and an element $\varphi$ in $\mathrm{Out}(F_2)\cong \mathrm{GL}_2(\mathbb{Z})$ sends $c$ to (a conjugate of) $c$ or $c^{-1}$ if and only if the determinant of $\varphi$ is $1$ or $-1$. However, in the profinite case, the map $\mathrm{Out}(\widehat{F_2})\to \mathrm{GL}_2(\widehat{\mathbb{Z}})$ has non-trivial kernel (see for example this question). So is there a straightforward way to express elements in $\mathrm{Out}(\widehat{F_2})$, and can we deduce the image of $c=[a,b]$ (and the unit $\mu$ in the above questions) easily just like we examine the determinant in the discrete case?
