2026/07/17 by Zhongzi Wang, Xiaoyu Xu
#math.GT #math.GR
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∈Γ has the same possible images as a given simple closed curve γ∈ Γ under epimorphisms from Γ to every finite group, then g belongs to the 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.