Interrogating surface length spectra and quantifying isospectrality

TL;DR

This paper investigates how the length spectrum of hyperbolic surfaces determines their geometry, providing bounds on the number of questions needed and on the count of isospectral but non-isometric surfaces.

Contribution

It offers a quantitative analysis of inverse spectral problems, establishing bounds on questions needed for spectrum determination and on isospectral surface counts.

Findings

Quantitative bounds on questions needed to determine length spectra

Upper bounds on the number of isospectral non-isometric surfaces

Insights into the relationship between spectra and surface geometry

Abstract

This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the following: how many questions do you need to ask a length spectrum to determine it completely? In answering this, a quantitative upper bound is given on the number of isospectral but non-isometric surfaces of a given genus.