Horowitz–Randol pairs of curves in $q$–differential metrics

Abstract

The Euclidean cone metrics coming from $q$–differentials on a closed surface of genus $g\ge 2$ define an equivalence relation on homotopy classes of closed curves, where two classes are equivalent if they have the equal length in every such metric. We prove an analogue of the result of Randol for hyperbolic metrics (building on the work of Horowitz): for every integer $q\ge 1$, the corresponding equivalence relation has arbitrarily large equivalence classes. In addition, we describe how these equivalence relations are related to each other.