Length and eigenvalue equivalence

Research paper by Christopher J. Leininger, D. B. McReynolds, Walter D. Neumann, Alan W. Reid

Indexed on: 18 Dec '06Published on: 18 Dec '06Published in: Mathematics - Geometric Topology

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.