A restricted Magnus property for profinite surface groups

Abstract:

Magnus proved in 1930 that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle ^F=\langle y\rangle ^F$, $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property, called the Magnus property, has been generalized to oriented surface groups.

In this paper, we consider an analogue property for profinite surface groups. While the Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for $\mathscr {S}$ a class of finite groups, we prove that if $x$ and $y$ are algebraically simple elements of the pro-$\mathscr {S}$ completion $\widehat {\Pi }^{\mathscr {S}}$ of an orientable surface group $\Pi$ such that, for all $n\in \mathbb {N}$, there holds $\langle x^n\rangle ^{\widehat {\Pi }^{\mathscr {S}}}=\langle y^n\rangle ^{\widehat {\Pi }^{\mathscr {S}}}$, then $x$ is conjugated to $y^s$ for some $s\in (\widehat {\mathbb Z}^{\mathscr {S}})^\ast$. As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions.

The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists given in [Marco Boggi, Trans. Amer. Math. Soc. 366 (2014), 5185–5221] to profinite Dehn multitwists.