Characterizing covers via simple closed curves (Q2681863)

Using Teichmüller theory and the complex of curves, the authors prove that two finite regular covers \(p:X \to S\) and \(q:Y \to S\) of a closed surface \(S\) of genus at least two are equivalent if the following holds: a closed curve on \(S\) lifts to a simple closed curve on \(X\) if and only if it does to \(Y\) (they conjecture that this holds also for non-regular covers). For abelian covers, they give a characterization of equivalence of coverings by considering which powers of simple closed curves lift to closed curves. The results are motivated also by \textit{T. Sunada}'s [Ann. Math. (2) 121, 169--186 (1985; Zbl 0585.58047)] construction of isospectral hyperbolic surfaces which are not isometric; an interesting open question here is the existence of non-isometric surfaces with the same simple length spectrum (i.e., considering only the lengths of simple closed curves).