Can You Hear the Shape of a Hyperbolic Surface? Now for Real

Abstract

We associate a musical instrument, a "hyperbolic marimba", to every pair $(X,\Gamma)$ where $X$ is a hyperbolic surface and $\Gamma\subset X$ a simple multicurve labeled with musical keys. It works as follows: take a geodesic and every time it hits $\Gamma$, play the corresponding note. In this paper we investigate to which extent the so-produced melodies characterize $(X,\Gamma)$ up to isometry. In the accompanying website "HyperMarimba" (available at https://ludox73.github.io/HyperMarimba/story.html ), the reader can actually listen to the produced melodies. They can also visualize some of the phenomena we investigate.