Length and eigenvalue equivalence

Christopher J. Leininger; D. B. McReynolds; Walter D. Neumann; Alan W. Reid

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Length and eigenvalue equivalence

Christopher J. Leininger·D. B. McReynolds·Walter D. Neumann·Alan W. Reid

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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.

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BibTeX RIS

Christopher J. Leininger, D. B. McReynolds, Walter D. Neumann, Alan W. Reid. 2006-12-18. Length and eigenvalue equivalence. https://arxiv.org/abs/math/0606343

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