Length and Eigenvalue Equivalence

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 hyperboli c n‐manifold has pairs of eigenvalue equivalent finite covers of arbitrar ily large volume ratio. We also show the analogous result for primitive length equivalence.