Profinite rigidity of simple closed curves in surface groups

This paper establishes a new characterization of simple closed curves on a closed orientable surface.

Let $\Gamma$ be the fundamental group of a closed orientable surface.

We prove that if an element $g\in\Gamma$ has the same possible images as a given simple closed curve $\gamma\in \Gamma$ under epimorphisms from $\Gamma$ to every finite group, then $g$ belongs to the $\mathrm{Aut}(\Gamma)$-orbit of $\gamma$, i.e. $g$ is itself a simple closed curve with the same topological type as $\gamma$.

Consequently, the set of simple closed curves in $\Gamma$ is closed in the profinite topology of $\Gamma$; and we obtain a new algorithm to decide whether a given element in $\Gamma$ can be represented by a simple closed curve.

Proper powers of simple closed curves and the pro-$p$ cases are also discussed.