| Abstract: A long studied problem in hyperbolic (or more generally, negatively curved) geometry is the extent to which a manifold M is determined by its collection of lengths of closed geodesics on M. For instance, Otal showed that any negatively curved metric on a surface is determined up to isometry by its marked length spectrum, that is, by the function which associates to each closed curve the length of the unique geodesic in its free homotopy class. By classic work of Fricke, the similar result is true if one restricts this function only to simple closed curves. By celebrated constructions of Vigneras and Sunada, we now know that the corresponding statement is false when one forgets the marking, that is, there exist non-isometric surfaces which have the same collections of lengths of closed geodesics. In this talk, we will explore the extent to which surfaces arising from Sunada's construction can have the same collection of lengths of simple closed curves. Along the way we will also discuss some new general results about how simple lifts of curves can determine equivalence of covers. This represents joint work with Tarik Aougab, Max Lahn, and Marissa Loving. |