A restricted Magnus property for profinite surface groups

Abstract

Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ 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 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 \emph{algebraically simple} elements of the pro-${\mathscr S}$ completion $\hat{\Pi}^{\mathscr S}$ of an orientable surface group $\Pi$, such that, for all $n\in{\mathbb N}$, there holds $\langle x^n\rangle^{\hat{\Pi}^{\mathscr S}}=\langle y^n\rangle^{\hat{\Pi}^{\mathscr S}}$, then $x$ is conjugated to $y^s$ for some $s\in(\hat{\mathbb Z}^{\mathscr S})^\ast$. 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 to profinite Dehn multitwists.