Length spectra of flat metrics coming from $q$-differentials

Abstract

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves’ lengths do we really need to know? It is a result of Duchin, Leininger, and Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We generalize the notion of simple curves to that of $q$-simple curves, for any positive integer $q$, and show that the lengths of $q$-simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a $q$-differential.