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