Orthospectrum and simple orthospectrum rigidity: finiteness and genericity

TL;DR

This paper investigates the orthospectrum and simple orthospectrum of compact hyperbolic surfaces with geodesic boundary, establishing finiteness and genericity results, and providing specific examples such as the one-holed torus.

Contribution

It proves finiteness and genericity of hyperbolic surfaces determined by their orthospectrum and simple orthospectrum, including a specific example of the one-holed torus.

Findings

Finitely many surfaces share the same orthospectrum.

Finitely many surfaces share the same simple orthospectrum.

Generic surfaces are uniquely determined by their orthospectrum.

Abstract

We study the orthospectrum and the simple orthospectrum of compact hyperbolic surfaces with geodesic boundary. We show that there are finitely many hyperbolic surfaces sharing the same simple orthospectrum and finitely many hyperbolic surfaces sharing the same orthospectrum. Then, we show that generic surfaces are determined by their orthospectrum and by their simple orthospectrum. We conclude with the example of the one-holed torus which is determined by its simple orthospectrum.