Unmarked simple length spectral rigidity for covers

TL;DR

This paper demonstrates that for any closed orientable surface with negative Euler characteristic, there exist finite covers that are length isospectral but typically not simple length isospectral, revealing nuanced spectral distinctions.

Contribution

It characterizes when two covers are isomorphic based on simple elevations and constructs hyperbolic surfaces with identical length spectra but differing simple length spectra for curves with bounded self-intersections.

Findings

Existence of length isospectral covers not sharing simple length spectra

Characterization of isomorphic covers via simple elevations

Construction of hyperbolic surfaces with identical full length spectra but differing simple length spectra

Abstract

We prove that every closed orientable surface S of negative Euler characteristic admits a pair of finite-degree covers which are length isospectral over S but generically not simple length isospectral over S. To do this, we first characterize when two finite-degree covers of a connected, orientable surface of negative Euler characteristic are isomorphic in terms of which curves have simple elevations. We also construct hyperbolic surfaces X and Y with the same full unmarked length spectrum but so that for each k, the sets of lengths associated to curves with at most k self-intersections differ.