A refinement of strong multiplicity one for spectra of hyperbolic manifolds

Abstract

Let M1 and M2 denote two compact hyperbolic man- ifolds. Assume that the multiplicities of eigenvalues of the Lapla- cian acting on L 2 (M1) and L 2 (M2) (respectively, multiplicities of lengths of closed geodesics in M1 and M2) 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.