On systoles and ortho spectrum rigidity

TL;DR

This paper investigates the relationship between the ortho spectrum and the geometric properties of hyperbolic surfaces with geodesic boundary, revealing finiteness results and limitations in spectral determination.

Contribution

It demonstrates that the ortho spectrum does not uniquely determine the systolic length but implies finiteness of hyperbolic structures sharing the same spectrum.

Findings

Finitely many possibilities for systolic length given the ortho spectrum

Finiteness of hyperbolic structures with a shared ortho spectrum

The ortho spectrum does not fully determine the surface geometry

Abstract

We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we show that, up to isometry, there are only finitely many hyperbolic structures on a surface that share a given ortho spectrum.