A refinement of strong multiplicity one for spectra of hyperbolic manifolds

TL;DR

This paper refines the strong multiplicity one theorem for hyperbolic manifolds, showing that if the exceptional set of eigenvalues or geodesic lengths is sufficiently sparse, the manifolds are necessarily isospectral.

Contribution

It introduces a density condition on the exceptional set that guarantees isospectrality, improving previous results in spectral geometry of hyperbolic manifolds.

Findings

Density threshold ensures manifolds are iso-spectral

Refined conditions on eigenvalue and geodesic length multiplicities

Applicable to compact hyperbolic manifolds

Abstract

Let $\calM_1$ and $\calM_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\calM_1)$ and $L^2(\calM_2)$ (respectively, multiplicities of lengths of closed geodesics in $\calM_1$ and $\calM_2$) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be iso-spectral.