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Length and eigenvalue equivalence

By C. J. Leininger, D. B. Mcreynolds, W. D. Neumann and A. W. Reid


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.

Year: 2007
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