Γ-extensions of the spectrum of an orbifold

Farsi, Carla; Proctor, Emily; Seaton, Christopher

doi:10.1090/S0002-9947-2013-06082-5

$\Gamma$-extensions of the spectrum of an orbifold

Authors: Carla Farsi, Emily Proctor and Christopher Seaton
Journal: Trans. Amer. Math. Soc. 366 (2014), 3881-3905
MSC (2010): Primary 58J53, 57R18; Secondary 53C20.
DOI: https://doi.org/10.1090/S0002-9947-2013-06082-5
Published electronically: December 27, 2013
MathSciNet review: 3192622
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Abstract: We introduce the $\Gamma$-extension of the spectrum of the Laplacian of a Riemannian orbifold, where $\Gamma$ is a finitely generated discrete group. This extension, called the $\Gamma$-spectrum, is the union of the Laplace spectra of the $\Gamma$-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the $\Gamma$-spectra of known examples of isospectral pairs and families of orbifolds and demonstrate that, in many cases, isospectral orbifolds need not be $\Gamma$-isospectral. We additionally prove a version of Sunada’s theorem that allows us to construct pairs of orbifolds that are $\Gamma$-isospectral for any choice of $\Gamma$.

References

Alejandro Adem, Johann Leida, and Yongbin Ruan, Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge University Press, Cambridge, 2007. MR 2359514
Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
Robert S. Cahn and Joseph A. Wolf, Zeta functions and their asymptotic expansions for compact symmetric spaces of rank one, Comment. Math. Helv. 51 (1976), no. 1, 1–21. MR 397801, DOI https://doi.org/10.1007/BF02568140
Yuan-Jen Chiang, Spectral geometry of $V$-manifolds and its application to harmonic maps, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 93–99. MR 1216577, DOI https://doi.org/10.1090/pspum/054.1/1216577
J. H. Conway and N. J. A. Sloane, Four-dimensional lattices with the same theta series, Internat. Math. Res. Notices 4 (1992), 93–96. MR 1159450, DOI https://doi.org/10.1155/S1073792892000102
Dennis M. DeTurck and Carolyn S. Gordon, Isospectral deformations. II. Trace formulas, metrics, and potentials, Comm. Pure Appl. Math. 42 (1989), no. 8, 1067–1095. With an appendix by Kyung Bai Lee. MR 1029118, DOI https://doi.org/10.1002/cpa.3160420803
Harold Donnelly, Spectrum and the fixed point sets of isometries. I, Math. Ann. 224 (1976), no. 2, 161–170. MR 420743, DOI https://doi.org/10.1007/BF01436198

P. G. Doyle and J. P. Rossetti, Laplace-isospectral hyperbolic $2$-orbifolds are representation-equivalent, preprint, arXiv:1103.4372 [math.DG]

Emily B. Dryden, Carolyn S. Gordon, Sarah J. Greenwald, and David L. Webb, Asymptotic expansion of the heat kernel for orbifolds, Michigan Math. J. 56 (2008), no. 1, 205–238. MR 2433665, DOI https://doi.org/10.1307/mmj/1213972406

Emily B. Dryden and Alexander Strohmaier, Huber’s theorem for hyperbolic orbisurfaces, Canad. Math. Bull. 52 (2009), no. 1, 66–71. MR 2494312, DOI https://doi.org/10.4153/CMB-2009-008-0

Whitney Duval, John Schulte, Christopher Seaton, and Bradford Taylor, Classifying closed 2-orbifolds with Euler characteristics, Glasg. Math. J. 52 (2010), no. 3, 555–574. MR 2679914, DOI https://doi.org/10.1017/S001708951000042X

J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 1738431

Carla Farsi, Orbifold spectral theory, Rocky Mountain J. Math. 31 (2001), no. 1, 215–235. MR 1821378, DOI https://doi.org/10.1216/rmjm/1008959678

Carla Farsi and Christopher Seaton, Generalized twisted sectors of orbifolds, Pacific J. Math. 246 (2010), no. 1, 49–74. MR 2645879, DOI https://doi.org/10.2140/pjm.2010.246.49

Carla Farsi and Christopher Seaton, Nonvanishing vector fields on orbifolds, Trans. Amer. Math. Soc. 362 (2010), no. 1, 509–535. MR 2550162, DOI https://doi.org/10.1090/S0002-9947-09-04938-1

Carla Farsi and Christopher Seaton, Generalized orbifold Euler characteristics for general orbifolds and wreath products, Algebr. Geom. Topol. 11 (2011), no. 1, 523–551. MR 2783237, DOI https://doi.org/10.2140/agt.2011.11.523

Carolyn Gordon, Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom. 37 (1993), no. 3, 639–649. MR 1217163

C. S. Gordon and J. P. Rossetti, Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn’t reveal, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7, 2297–2314 (English, with English and French summaries). MR 2044174

Akira Ikeda, On lens spaces which are isospectral but not isometric, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 3, 303–315. MR 597742

Akira Ikeda, On space forms of real Grassmann manifolds which are isospectral but not isometric, Kodai Math. J. 20 (1997), no. 1, 1–7. MR 1443360, DOI https://doi.org/10.2996/kmj/1138043715

R. J. Miatello and J. P. Rossetti, Flat manifolds isospectral on $p$-forms, J. Geom. Anal. 11 (2001), no. 4, 649–667. MR 1861302, DOI https://doi.org/10.1007/BF02930761

R. J. Miatello and J. P. Rossetti, Comparison of twisted $p$-form spectra for flat manifolds with diagonal holonomy, Ann. Global Anal. Geom. 21 (2002), no. 4, 341–376. MR 1910457, DOI https://doi.org/10.1023/A%3A1015651821995

J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR 162204, DOI https://doi.org/10.1073/pnas.51.4.542

Iosif Polterovich, Combinatorics of the heat trace on spheres, Canad. J. Math. 54 (2002), no. 5, 1086–1099. MR 1924714, DOI https://doi.org/10.4153/CJM-2002-040-4

Emily Proctor, Orbifold homeomorphism finiteness based on geometric constraints, Ann. Global Anal. Geom. 41 (2012), no. 1, 47–59. MR 2860396, DOI https://doi.org/10.1007/s10455-011-9270-4

Emily Proctor and Elizabeth Stanhope, An isospectral deformation on an infranil-orbifold, Canad. Math. Bull. 53 (2010), no. 4, 684–689. MR 2761691, DOI https://doi.org/10.4153/CMB-2010-074-8

Dorette Pronk and Laura Scull, Translation groupoids and orbifold cohomology, Canad. J. Math. 62 (2010), no. 3, 614–645. MR 2666392, DOI https://doi.org/10.4153/CJM-2010-024-1

Juan Pablo Rossetti, Dorothee Schueth, and Martin Weilandt, Isospectral orbifolds with different maximal isotropy orders, Ann. Global Anal. Geom. 34 (2008), no. 4, 351–366. MR 2447904, DOI https://doi.org/10.1007/s10455-008-9110-3

Dorothee Schueth, Continuous families of isospectral metrics on simply connected manifolds, Ann. of Math. (2) 149 (1999), no. 1, 287–308. MR 1680563, DOI https://doi.org/10.2307/121026

Naveed Shams Ul Bari, Orbifold lens spaces that are isospectral but not isometric, Osaka J. Math. 48 (2011), no. 1, 1–40. MR 2802590

Naveed Shams, Elizabeth Stanhope, and David L. Webb, One cannot hear orbifold isotropy type, Arch. Math. (Basel) 87 (2006), no. 4, 375–384. MR 2263484, DOI https://doi.org/10.1007/s00013-006-1748-0

E. Stanhope and A. Uribe, The spectral function of a Riemannian orbifold, Ann. Global Anal. Geom. 40 (2011), no. 1, 47–65. MR 2795449, DOI https://doi.org/10.1007/s10455-010-9244-y

Toshikazu Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169–186. MR 782558, DOI https://doi.org/10.2307/1971195

Craig J. Sutton, Equivariant isospectrality and Sunada’s method, Arch. Math. (Basel) 95 (2010), no. 1, 75–85. MR 2671240, DOI https://doi.org/10.1007/s00013-010-0139-8

Hirotaka Tamanoi, Generalized orbifold Euler characteristic of symmetric products and equivariant Morava $K$-theory, Algebr. Geom. Topol. 1 (2001), 115–141. MR 1805937, DOI https://doi.org/10.2140/agt.2001.1.115

Hirotaka Tamanoi, Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces, Algebr. Geom. Topol. 3 (2003), 791–856. MR 1997338, DOI https://doi.org/10.2140/agt.2003.3.791

Marie-France Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2) 112 (1980), no. 1, 21–32 (French). MR 584073, DOI https://doi.org/10.2307/1971319

M. Weilandt, Isospectral orbifolds with different maximal isotropy orders, Diplom. Thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik, 2007.