Skip to main content
SpringerLink
Log in

Spectra of orbifolds with cyclic fundamental groups

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We give a simple geometric characterization of isospectral orbifolds covered by spheres, complex projective spaces and the quaternion projective line having cyclic fundamental group. The differential operators considered are Laplace–Beltrami operators twisted by characters of the corresponding fundamental group. To prove the characterization, we first give an explicit description of their spectra using generating functions. We also include many isospectral examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bauer, W., Furutani, K.: Zeta regularized determinant of the Laplacian for classes of spherical space forms. J. Geom. Phys. 58(1), 64–88 (2008). doi:10.1016/j.geomphys.2007.09.007

    Article  MathSciNet  MATH  Google Scholar 

  2. Bär, C.: The Dirac operator on space forms of positive curvature. J. Math Soc. Japan 48(1), 69–83 (1996). doi:10.2969/jmsj/04810069

    Article  MathSciNet  MATH  Google Scholar 

  3. Beck, M., Robins, S.: Computing the continuous discretely. Integer-point enumeration in polyhedra. Undergrad. Texts Math., Springer, New York (2007). doi:10.1007/978-0-387-46112-0

  4. Boldt S.: Properties of the Dirac spectrum on three dimensional lens spaces (2015). arXiv:1504.03121

  5. Boldt, S., Lauret, E.A.: An explicit formula for the Dirac multiplicities on lens spaces (2014, accepted). arXiv:1412.2599

  6. Cagliero, L., Tirao, P.: A closed formula for weight multiplicities of representations of \({\rm Sp}_2({\mathbb{C}})\). Manuscr. Math. 115(4), 417–426 (2004). doi:10.1007/s00229-004-0499-0

  7. Carletti, E., Monti Bragadin, G.: On Dirichlet series associated with polynomials. Proc. Am. Math. Soc. 121(1), 33–37 (1994). doi:10.1090/S0002-9939-1994-1179586-5

    Article  MathSciNet  MATH  Google Scholar 

  8. Carletti, E., Monti Bragadin, G.: On Minakshisundaram–Pleijel zeta functions of spheres. Proc. Am. Math. Soc. 122(4), 993–1001 (1994). doi:10.1090/S0002-9939-1994-1249872-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Conway, J.H., Sloane, N.J.A.: Four-dimensional lattices with the same theta series. Int. Math. Res. Not. IMRN 1992(4), 93–96 (1992). doi:10.1155/S1073792892000102

    Article  MathSciNet  MATH  Google Scholar 

  10. DeFord, D., Doyle, P.: Cyclic groups with the same Hodge series (2014). arXiv:1404.2574

  11. Dryden, E., Gordon, C., Greenwald, S., Webb, D.: Asymptotic expansion of the heat kernel for orbifolds. Mich. Math. J. 56(1), 205–238 (2008). doi:10.1307/mmj/1213972406

    Article  MathSciNet  MATH  Google Scholar 

  12. Farsi, C., Proctor, E., Seaton, C.: \(\Gamma \)-extensions of the spectrum of an orbifold. Trans. Am. Math. Soc. 366(7), 3881–3905 (2014). doi:10.1090/S0002-9947-2013-06082-5

    Article  MathSciNet  MATH  Google Scholar 

  13. Fulton, W., Harris, J.: Representation Theory. A first Course. Springer, New York (2004). doi:10.1007/978-1-4612-0979-9

  14. Gilkey, P.: On spherical space forms with meta-cyclic fundamental group which are isospectral but not equivariant cobordant. Compos. Math. 56(2), 171–200 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Gordon, C.: Orbifolds and their spectra. In: Spectral Geometry, pp. 49–71, Proc. Sympos. Pure Math., vol. 84. Amer. Math. Soc., Providence (2012). doi:10.1090/pspum/084

  16. 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). doi:10.5802/aif.2007

    Article  MathSciNet  MATH  Google Scholar 

  17. Gornet, R., McGowan, J.: Lens Spaces, isospectral on forms but not on functions. LMS J. Comput. Math. 9, 270–286 (2006). doi:10.1112/S1461157000001273

    Article  MathSciNet  MATH  Google Scholar 

  18. Hattori, T.: On the residues of spectral zeta functions on spheres. Saitama Math. J. 28, 25–37 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Helgason, S.: Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Mathematical Surveys and Monographs, vol. 83. American Mathematical Society, Providence (2000)

  20. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Graduate Studies inMathematics, vol. 34. American Mathematical Society, Providence (2001)

  21. Ikeda, A.: On lens spaces which are isospectral but not isometric. Ann. Sci. École Norm. Sup. (4) 13(3), 303–315 (1980)

  22. Ikeda, A.: On the spectrum of a riemannian manifold of positive constant curvature. Osaka J. Math. 17, 75–93 (1980)

    MathSciNet  MATH  Google Scholar 

  23. Ikeda, A.: On the spectrum of a riemannian manifold of positive constant curvature II. Osaka J. Math. 17, 691–762 (1980)

    MathSciNet  MATH  Google Scholar 

  24. Ikeda, A.: On spherical space forms which are isospectral but not isometric. J. Math. Soc. Japan 35(3), 437–444 (1983). doi:10.2969/jmsj/03530437

    Article  MathSciNet  MATH  Google Scholar 

  25. Ikeda, A.: Riemannian manifolds \(p\)-isospectral. In: Geometry of Manifolds (Matsumoto, 1988). Perspect. Math., vol. 8, pp. 383–417 (1989)

  26. Ikeda, A.: On space forms of real Grassmann manifolds which are isospectral but not isometric. Kodai Math. J. 20(1), 1–7 (1997). doi:10.2996/kmj/1138043715

    Article  MathSciNet  MATH  Google Scholar 

  27. Ikeda, A., Taniguchi, Y.: Spectra and eigenforms of the Laplacian on \(S^n\) and \(P^n(\mathbb{C})\). Osaka J. Math. 15(3), 515–546 (1978)

    MathSciNet  MATH  Google Scholar 

  28. Ikeda, A., Yamamoto, Y.: On the spectra of 3-dimensional lens spaces. Osaka J. Math. 16(2), 447–469 (1979)

    MathSciNet  MATH  Google Scholar 

  29. Knapp, A.W.: Lie groups beyond an introduction. Progress in Mathematics, vol. 140. Birkhäuser, Boston (2002)

  30. Lauret, E.A., Miatello, R.J., Rossetti, J.P.: Strongly isospectral manifolds with nonisomorphic cohomology rings. Rev. Mat. Iberoam. 29, 611–634 (2013). doi:10.4171/RMI/732

    Article  MathSciNet  MATH  Google Scholar 

  31. Lauret, E.A., Miatello, R.J., Rossetti, J.P.: Spectra of lens spaces from 1-norm spectra of congruence lattices. Int. Math. Res. Not. IMRN (2015, published electronically). doi:10.1093/imrn/rnv159

  32. Lauret, E.A., Miatello, R.J., Rossetti, J.P.: Non-strongly isospectral spherical space forms. In: Proceedings of the First Congress of the Americas, Contemp. Math. (2015). arXiv:1505.02554

  33. Miatello, R.J., Rossetti, J.P.: Comparison of twisted Laplace \(p\)-spectra for flat manifolds with diagonal holonomy. Ann. Global Anal. Geom. 21(4), 341–376 (2002). doi:10.1023/A:1015651821995

    Article  MathSciNet  MATH  Google Scholar 

  34. Milnor, J.: Eigenvalues of the Laplace operator on certain manifolds. Proc. Natl. Acad. Sci. USA 51(4), 542 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mohades, H., Honari, B.: On a relation between spectral theory of lens spaces and Ehrhart theory (2016). arXiv:1601.04256

  36. Pesce, H.: Représentations relativement équivalentes et variétés riemanniennes isospectrales. Comment. Math. Helv. 71, 243–268 (1996). doi:10.1007/BF02566419

    Article  MathSciNet  Google Scholar 

  37. Rossetti, J.P., Schueth, D., Weilandt, M.: Isospectral orbifolds with different maximal isotropy orders. Ann. Global Anal. Geom. 34(4), 351–366 (2008). doi:10.1007/s10455-008-9110-3

    Article  MathSciNet  MATH  Google Scholar 

  38. Stein, W.A., et al.: Sage Mathematics Software (Version 4.3). The Sage Development Team (2009). http://www.sagemath.org

  39. Shams Ul Bari, N.: Orbifold lens spaces that are isospectral but not isometric. Osaka J. Math 48(1), 1–40 (2011)

    MathSciNet  MATH  Google Scholar 

  40. Shams, N., Stanhope, E., Webb, D.: One cannot hear orbifold isotropy type. Arch. Math. (Basel) 87(4), 375–385 (2006). doi:10.1007/s00013-006-1748-0

    Article  MathSciNet  MATH  Google Scholar 

  41. Spreafico, M.: Zeta functions and regularized determinants on projective spaces. Rocky Mt. J. Math. 33(4), 1499–1512 (2003). doi:10.1216/rmjm/1181075478

    Article  MathSciNet  MATH  Google Scholar 

  42. Teo, L.P.: Zeta functions of spheres and real projective spaces (2014). arXiv:1412.0758

  43. Wallach, N.R.: Harmonic analysis on homogeneous spaces. Pure and Applied Mathematics, vol. 19. Marcel Dekker, New York (1973)

  44. Wolf, J.A.: Isospectrality for spherical space forms. Result Math. 40, 321–338 (2001). doi:10.1007/BF03322715

    Article  MathSciNet  MATH  Google Scholar 

  45. Yamamoto, Y.: On the number of lattice points in the square \(x+y\le u\) with a certain congruence condition. Osaka J. Math. 17(1), 9–21 (1980)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author wishes to thank Leandro Cagliero and Jorge Vargas for helpful conversations on representation theory and also Sebastian Boldt and Ramiro Lafuente for helpful comments concerning the algorithms used in the last section. The author also wishes to thank the support of the Oberwolfach Leibniz Fellow programme in May–July 2013 and in August–November 2014, when this project started.

Author information

Authors and Affiliations

  1. CIEM-FaMAF, Universidad Nacional de Córdoba, 5000, Córdoba, Argentina

    Emilio A. Lauret

Authors
  1. Emilio A. Lauret
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio A. Lauret.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lauret, E.A. Spectra of orbifolds with cyclic fundamental groups. Ann Glob Anal Geom 50, 1–28 (2016). https://doi.org/10.1007/s10455-016-9498-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-016-9498-0

Keywords

Mathematics Subject Classification

Search

Navigation