Length and eigenvalue equivalence

Abstract

Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent when the sets of lengths of their (primitive) closed geodesics are equal. We give a general construction of eigenvalue equivalent and primitive length equivalent Riemannian manifolds. For example, we show that every finite volume hyperbolic n-manifold has pairs of eigenvalue equivalent finite covers of arbitrarily large volume ratio.We also show the analogous result for primitive length equivalence. © The Author 2007.