Given two finite degree regular covers Y, Z of an orientable surface S (not necessarily of the same degree), suppose that for any closed curve gamma on S, some iterate of gamma lifts to a

*simple*closed curve on Y if and only if some (perhaps distinct) iterate of gamma lifts to a simple closed curve on Z. Then we prove that Y and Z are equivalent covers. We’ll discuss connections to Sunada’s construction of isospectral hyperbolic surfaces. This represents joint work with Max Lahn, Marissa Loving, and Sunny Yang Xiao.