Profinite rigidity of simple closed curves in surface groups

Abstract

This paper establishes a new characterization of simple closed curves on a closed orientable surface. Let $\hat{I}“$ be the fundamental group of a closed orientable surface. We prove that if an element $g\in \hat{I}“$ has the same possible images as a given simple closed curve ${\hat{I}}^3\in \hat{I}“$ under epimorphisms from $\hat{I}“$ to every finite group, then $g$ belongs to the $\mathrm{Aut}(\hat{I}“)$-orbit of ${\hat{I}}^3$, i.e. $g$ is itself a simple closed curve with the same topological type as ${\hat{I}}^3$. Consequently, the set of simple closed curves in $\hat{I}“$ is closed in the profinite topology of $\hat{I}“$; and we obtain a new algorithm to decide whether a given element in $\hat{I}“$ can be represented by a simple closed curve. Proper powers of simple closed curves and the pro-$p$ cases are also discussed.

24 pages, 2 figures