Abstract: It is well known that given any homotopically non-trivial closed curve gamma on a surface S, there is some finite cover Y of S to which gamma admits an embedded lift. in the other direction, one can fix a cover Y and analyze the collection of closed curves on S that lift to an embedding on Y. We prove that Y is determined uniquely (up to the natural equivalence for covers) by this collection. The proof uses Teichmuller theory, and we’ll explain a connection to constructing isospectral hyperbolic surfaces. This represents joint work with Maxie Lahn, Marissa Loving, and Nick Miller.