Length and eigenvalue equivalence

Details Length and eigenvalue equivalence Length and eigenvalue equivalence

2006-06-14

arxiv.org/abs/math/0606343v2

International Mathematics Research Notices 2007 (2007), rnm135-24

Mathematics

Geometric Topology

Scientific paper

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.

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