Characterizing covers via simple closed curves

Abstract

Given two finite covers p : X → S and q : Y → S of a connected, oriented, closed surface S of genus at least 2, we attempt to characterize the equivalence of p and q in terms of which curves lift to simple curves. Using Teichmüller theory and the complex of curves, we show that two regular covers p and q are equivalent if for any closed curve γ ⊂ S, γ lifts to a simple closed curve on X if and only if it does to Y . When the covers are abelian, we also give a characterization of equivalence in terms of which powers of simple closed curves lift to closed curves.

Unmarked simple length spectral rigidity for covers

“…If X and Y are generically simple length isospectral over S, are the covers p and q isomorphic? Our strategy for answering Question 1.2 is discussed in prior work of Aougab-Lahn-Loving-Xiao [Aou+20] and at the beginning of Section 5. We obtain the following strengthening of [Aou+20, Theorem 1.1], using entirely new proof techniques in order to remove the regularity hypothesis, thus resolving [Aou+20,Conjecture 1.4].…”

“…Our strategy for answering Question 1.2 is discussed in prior work of Aougab-Lahn-Loving-Xiao [Aou+20] and at the beginning of Section 5. We obtain the following strengthening of [Aou+20, Theorem 1.1], using entirely new proof techniques in order to remove the regularity hypothesis, thus resolving [Aou+20,Conjecture 1.4]. Given a finite cover p : X − → S, a curve α on X is an elevation of a curve γ on S along p if p(α) is a nonzero power of γ.…”