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--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.

arXiv commentAn error in the proof of Proposition 3.2 (which is split between the proofs of Lemma 3.1 and 3.2) in the case that q is even has been corrected. The changes involve a slight modification of Definition 3.1 and a short paragraph at the end of each of the proofs of Lemmas 3.1 and 3.2