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On the isospectral orbifold–manifold problem for nonpositively curved locally symmetric spaces

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Abstract

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.

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References

  1. Adem, A., Leida, J., Ruan, Y.: Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics, vol. 171. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  2. Dryden, E.B., Gordon, C.S., Greenwald, S.J., Webb, D.L.: Asymptotic expansion of the heat kernel for orbifolds. Mich. Math. J. 56(1), 205–238 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dryden, E.B., Strohmaier, A.: Huber’s theorem for hyperbolic orbisurfaces. Can. Math. Bull. 52(1), 66–71 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29, 39–79 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duistermaat, J.J., Kolk, J.A.C., Varadarajan, V.S.: Spectra of compact locally symmetric manifolds of negative curvature. Invent. Math. 52(1), 27–93 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  6. Garibaldi, S., Rapinchuk, A.: Weakly commensurable \(S\)-arithmetic subgroups in almost simple algebraic groups of types B and C. Algebra Number Theory 7(5), 1147–1178 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  8. Gordon, C.S., Rossetti, J.P.: Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn’t reveal. Ann. Inst. Fourier (Grenoble) 53(7), 2297–2314 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. 76, 165–246 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. AMS, Providence (2001)

    Book  MATH  Google Scholar 

  11. Margulis, G.A.: Arithmeticity of the irreducible lattices in the semi-simple groups of rank greater than 1. Invent. Math. 76, 93–120 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Prasad, G.: Strong rigidity of \({\mathbb{Q}}\)-rank 1 lattices. Invent. Math. 21, 255–286 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Prasad, G., Rapinchuk, A.: Weakly commensurable arithmetic groups and isospectral locally symmetric spaces. Publ. Math. Inst. Hautes Étud. Sci. 109, 113–184 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rossetti, J.P., Schueth, D., Weilandt, M.: Isospectral orbifolds with different maximal isotropy orders. Ann. Glob. Anal. Geom. 34(4), 351–366 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sandoval, M.R.: The wave trace of the basic Laplacian of a Riemannian foliation near a non-zero period. Ann. Glob. Anal. Geom. 44(2), 217–244 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Satake, I.: On a generalization of the notion of manifold. Proc. Natl. Acad. Sci. USA 42, 359–363 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  17. Stanhope, E., Uribe, A.: The spectral function of a Riemannian orbifold. Ann. Glob. Anal. Geom. 40(1), 47–65 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sutton, C.J.: Equivariant isospectrality and Sunada’s method. Arch. Math. (Basel) 95(1), 75–85 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Thurston, W.P.: The Geometry and Topology of 3-Manifolds. Mimeographed Notes. Princeton University, Princeton (1979)

    Google Scholar 

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Acknowledgements

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.

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Correspondence to Benjamin Linowitz.

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Linowitz, B., Meyer, J.S. On the isospectral orbifold–manifold problem for nonpositively curved locally symmetric spaces. Geom Dedicata 188, 165–169 (2017). https://doi.org/10.1007/s10711-016-0210-0

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  • DOI: https://doi.org/10.1007/s10711-016-0210-0

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