On the isospectral orbifold–manifold problem for nonpositively curved locally symmetric spaces
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An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel’s conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups.
Mathematics Subject Classification58J53 22E40
The authors would like to thank Mikhail Belolipetsky for initially bringing this problem to their attention and Carolyn Gordon, Alejandro Uribe and David Webb for useful conversations on the material in this article. The first author was partially supported by an NSF RTG Grant DMS-1045119 and an NSF Mathematical Sciences Postdoctoral Fellowship.
- 7.Gordon, C.: Orbifolds and their spectra. In: Spectral Geometry, Volume 84 of Proceedings of Symposia Pure Mathematics, pp. 49–71. American Mathematical Society, Providence (2012)Google Scholar
- 19.Thurston, W.P.: The Geometry and Topology of 3-Manifolds. Mimeographed Notes. Princeton University, Princeton (1979)Google Scholar