Horowitz-Randol pairs of curves in q-differential metrics

TL;DR

This paper investigates the equivalence classes of closed curves in q-differential metrics on surfaces, showing that for each q > 1, these classes can be arbitrarily large, extending classical results from hyperbolic geometry.

Contribution

It generalizes Randol's result to q-differential metrics, demonstrating the existence of arbitrarily large equivalence classes of curves and exploring their interrelations.

Findings

Equivalence classes can be arbitrarily large for q > 1

Extension of Randol's hyperbolic result to q-differential metrics

Description of relationships between different equivalence relations

Abstract

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