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.
A collection ∆ of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number α, β is equal to k in absolute value for every α, β ∈ ∆ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is 2g + 1 when g ≥ 3 or k is odd, and 2g otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as g 2 .
We prove that every closed orientable surface S of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over S but generically not simple length isospectral over S. To do this, we first characterize when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations. We also prove an effective version of this result, weakening the hypotheses to consider curves with explicitly bounded self-intersection number. We further show that for sufficiently large N , the set of unmarked traces associated to simple closed curves in a generically chosen representation to SLN (R) distinguishes between pairs of non-isomorphic covers. Along the way, we construct hyperbolic surfaces X and Y with the same full unmarked length spectrum but so that for each k, the set of lengths associated to curves with at most k self-intersections differ.
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