Length spectra of flat metrics coming from \(q\)-differentials (Q2035053)

Summary: When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: \textit{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 \(q\)-\textit{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.