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$ \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 $.


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  • [1] Alejandro Adem, Johann Leida, and Yongbin Ruan, Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge University Press, Cambridge, 2007. MR 2359514 (2009a:57044)
  • [2] Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d'une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin, 1971 (French). MR 0282313 (43 #8025)
  • [3] 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 0397801 (53 #1657)
  • [4] 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 (94c:58040)
  • [5] 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 (93e:11047), https://doi.org/10.1155/S1073792892000102
  • [6] 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 (91e:58197), https://doi.org/10.1002/cpa.3160420803
  • [7] Harold Donnelly, Spectrum and the fixed point sets of isometries. I, Math. Ann. 224 (1976), no. 2, 161-170. MR 0420743 (54 #8755)
  • [8] P. G. Doyle and J. P. Rossetti, Laplace-isospectral hyperbolic $ 2$-orbifolds are representation-equivalent, preprint, arXiv:1103.4372 [math.DG]
  • [9] 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 (2009h:58057), https://doi.org/10.1307/mmj/1213972406
  • [10] Emily B. Dryden and Alexander Strohmaier, Huber's theorem for hyperbolic orbisurfaces, Canad. Math. Bull. 52 (2009), no. 1, 66-71. MR 2494312 (2009m:58073), https://doi.org/10.4153/CMB-2009-008-0
  • [11] 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 (2011g:57029), https://doi.org/10.1017/S001708951000042X
  • [12] J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 1738431 (2001j:22008)
  • [13] Carla Farsi, Orbifold spectral theory, Rocky Mountain J. Math. 31 (2001), no. 1, 215-235. MR 1821378 (2001k:58060), https://doi.org/10.1216/rmjm/1008959678
  • [14] Carla Farsi and Christopher Seaton, Generalized twisted sectors of orbifolds, Pacific J. Math. 246 (2010), no. 1, 49-74. MR 2645879 (2011g:57030), https://doi.org/10.2140/pjm.2010.246.49
  • [15] Carla Farsi and Christopher Seaton, Nonvanishing vector fields on orbifolds, Trans. Amer. Math. Soc. 362 (2010), no. 1, 509-535. MR 2550162 (2011a:57051), https://doi.org/10.1090/S0002-9947-09-04938-1
  • [16] 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 (2012h:57048), https://doi.org/10.2140/agt.2011.11.523
  • [17] Carolyn Gordon, Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom. 37 (1993), no. 3, 639-649. MR 1217163 (94b:58098)
  • [18] 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 (2005e:58055)
  • [19] 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 (83a:58091)
  • [20] 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 (98c:58170), https://doi.org/10.2996/kmj/1138043715
  • [21] R. J. Miatello and J. P. Rossetti, Flat manifolds isospectral on $ p$-forms, J. Geom. Anal. 11 (2001), no. 4, 649-667. MR 1861302 (2003d:58053), https://doi.org/10.1007/BF02930761
  • [22] 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 (2003f:58065), https://doi.org/10.1023/A:1015651821995
  • [23] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 542. MR 0162204 (28 #5403)
  • [24] Iosif Polterovich, Combinatorics of the heat trace on spheres, Canad. J. Math. 54 (2002), no. 5, 1086-1099. MR 1924714 (2003g:05015), https://doi.org/10.4153/CJM-2002-040-4
  • [25] Emily Proctor, Orbifold homeomorphism finiteness based on geometric constraints, Ann. Global Anal. Geom. 41 (2012), no. 1, 47-59. MR 2860396, https://doi.org/10.1007/s10455-011-9270-4
  • [26] Emily Proctor and Elizabeth Stanhope, An isospectral deformation on an infranil-orbifold, Canad. Math. Bull. 53 (2010), no. 4, 684-689. MR 2761691 (2012a:58060), https://doi.org/10.4153/CMB-2010-074-8
  • [27] Dorette Pronk and Laura Scull, Translation groupoids and orbifold cohomology, Canad. J. Math. 62 (2010), no. 3, 614-645. MR 2666392 (2011h:55009), https://doi.org/10.4153/CJM-2010-024-1
  • [28] 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 (2009f:58055), https://doi.org/10.1007/s10455-008-9110-3
  • [29] Dorothee Schueth, Continuous families of isospectral metrics on simply connected manifolds, Ann. of Math. (2) 149 (1999), no. 1, 287-308. MR 1680563 (2000c:58063), https://doi.org/10.2307/121026
  • [30] Naveed Shams Ul Bari, Orbifold lens spaces that are isospectral but not isometric, Osaka J. Math. 48 (2011), no. 1, 1-40. MR 2802590 (2012g:58065)
  • [31] 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 (2007g:58039), https://doi.org/10.1007/s00013-006-1748-0
  • [32] E. Stanhope and A. Uribe, The spectral function of a Riemannian orbifold, Ann. Global Anal. Geom. 40 (2011), no. 1, 47-65. MR 2795449 (2012e:58060), https://doi.org/10.1007/s10455-010-9244-y
  • [33] Toshikazu Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2) 121 (1985), no. 1, 169-186. MR 782558 (86h:58141), https://doi.org/10.2307/1971195
  • [34] Craig J. Sutton, Equivariant isospectrality and Sunada's method, Arch. Math. (Basel) 95 (2010), no. 1, 75-85. MR 2671240 (2011f:58060), https://doi.org/10.1007/s00013-010-0139-8
  • [35] Hirotaka Tamanoi, Generalized orbifold Euler characteristic of symmetric products and equivariant Morava $ K$-theory, Algebr. Geom. Topol. 1 (2001), 115-141 (electronic). MR 1805937 (2002e:57052), https://doi.org/10.2140/agt.2001.1.115
  • [36] Hirotaka Tamanoi, Generalized orbifold Euler characteristics of symmetric orbifolds and covering spaces, Algebr. Geom. Topol. 3 (2003), 791-856 (electronic). MR 1997338 (2005j:57025), https://doi.org/10.2140/agt.2003.3.791
  • [37] Marie-France Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math. (2) 112 (1980), no. 1, 21-32 (French). MR 584073 (82b:58102), https://doi.org/10.2307/1971319
  • [38] M. Weilandt, Isospectral orbifolds with different maximal isotropy orders, Diplom. Thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik, 2007.

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Additional Information

Carla Farsi
Affiliation: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395
Email: farsi@euclid.colorado.edu

Emily Proctor
Affiliation: Department of Mathematics, Middlebury College, Middlebury, Vermont 05753
Email: eproctor@middlebury.edu

Christopher Seaton
Affiliation: Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, Tennessee 38112
Email: seatonc@rhodes.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-06082-5
Keywords: Orbifold, isospectral, twisted sector, spectral geometry
Received by editor(s): August 20, 2012
Received by editor(s) in revised form: December 12, 2012
Published electronically: December 27, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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