On G-sets and isospectrality
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, p. 2307-2329

We study finite G-sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If M is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then M has isospectral non-isometric covers.

Nous étudions les G-ensembles finis et leur produit tensoriel avec des variétés Riemanniennes et obtenons certains résultats sur les quotients et revêtements isospectraux. Nous démontrons en particulier le théorème suivant  : Soit M une variété (ou orbifold) Riemannienne compacte et connexe dont le groupe fondamental possède un quotient fini non cyclique. Alors M admet des revêtements isospectraux non isométriques.

DOI : https://doi.org/10.5802/aif.2831
Classification:  58J53,  58D19
Keywords: isospectrality, laplacian, G-sets, Sunada
@article{AIF_2013__63_6_2307_0,
     author = {Parzanchevski, Ori},
     title = {On $G$-sets and isospectrality},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {6},
     year = {2013},
     pages = {2307-2329},
     doi = {10.5802/aif.2831},
     mrnumber = {3237449},
     zbl = {06325435},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_6_2307_0}
}
Parzanchevski, Ori. On $G$-sets and isospectrality. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2307-2329. doi : 10.5802/aif.2831. http://www.numdam.org/item/AIF_2013__63_6_2307_0/

[1] Band, R.; Parzanchevski, O.; Ben-Shach, G. The isospectral fruits of representation theory: quantum graphs and drums, Journal of Physics A: Mathematical and Theoretical, Tome 42 (2009), pp. 175202 | Article | MR 2539297 | Zbl 1176.58019

[2] Bérard, P. Transplantation et isospectralité I, Mathematische Annalen, Tome 292 (1992) no. 1, pp. 547-559 | Article | MR 1152950 | Zbl 0735.58008

[3] Brooks, R. Some relations between graph theory and Riemann surfaces, Isr. Math. Conf. Proc. 11, Citeseer (1996) | MR 1476704 | Zbl 0890.30027

[4] Buser, P. Isospectral Riemann surfaces, Ann. Inst. Fourier, Tome 36 (1986) no. 2, pp. 167-192 | Article | Numdam | MR 850750 | Zbl 0579.53036

[5] Buser, P.; Conway, J.; Doyle, P.; Semmler, K. D. Some planar isospectral domains, International Mathematics Research Notices, Tome 1994 (1994) no. 9, pp. 391-400 | Article | MR 1301439 | Zbl 0837.58033

[6] Chapman, Sj Drums that sound the same, American Mathematical Monthly, Tome 102 (1995) no. 2, pp. 124-138 | Article | MR 1315592 | Zbl 0849.35084

[7] Deturck, D.M.; Gordon, C.S.; Lee, K.B. Isospectral deformations II: Trace formulas, metrics, and potentials, Communications on Pure and Applied Mathematics, Tome 42 (1989) no. 8, pp. 1067-1095 | Article | MR 1029118 | Zbl 0709.53030

[8] Dipasquale, M. On the Order of a Group Containing Nontrivial Gassmann Equivalent Subgroups, Rose-Hulman Undergraduate Mathematics Journal, Tome 10 (2009) no. 1

[9] Doyle, P.G.; Rossetti, J.P. Laplace-isospectral hyperbolic 2-orbifolds are representation-equivalent, Arxiv preprint arXiv:1103.4372 (2011)

[10] GAP – Groups, Algorithms, and Programming, Version 4.4.12, The GAP Group (2008) http://www.gap-system.org

[11] Gassmann, F. Bemerkungen zur vorstehenden Arbeit von Hurwitz, Math. Z, Tome 25 (1926), pp. 124-143

[12] Gordon, C.; Webb, D.L.; Wolpert, S. One cannot hear the shape of a drum, American Mathematical Society, Tome 27 (1992) no. 1 | MR 1136137 | Zbl 0756.58049

[13] Hall, M. The theory of groups, Chelsea Pub Co (1976) | MR 414669 | Zbl 0354.20001

[14] Hillairet, L. Spectral decomposition of square-tiled surfaces, Mathematische Zeitschrift, Tome 260 (2008) no. 2, pp. 393-408 | Article | MR 2429619 | Zbl 1156.58012

[15] Kac, M. Can one hear the shape of a drum?, The american mathematical monthly, Tome 73 (1966) no. 4, pp. 1-23 | Article | MR 201237 | Zbl 0139.05603

[16] Larsen, M. Determining a semisimple group from its representation degrees, International Mathematics Research Notices, Tome 2004 (2004) no. 38, pp. 1989 | Article | MR 2063567 | Zbl 1073.22009

[17] Lemańczyk, M.; Thouvenot, J.P.; Weiss, B. Relative discrete spectrum and joinings, Monatshefte für Mathematik, Tome 137 (2002) no. 1, pp. 57-75 | Article | MR 1930996 | Zbl 1090.37002

[18] Merling, M.; Perlis, R. Gassmann Equivalent Dessins, Communications in Algebra®, Tome 38 (2010) no. 6, pp. 2129-2137 | Article | MR 2675525 | Zbl 1246.11126

[19] Milnor, J. Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences of the United States of America, Tome 51 (1964) no. 4, pp. 542 | Article | MR 162204 | Zbl 0124.31202

[20] Parzanchevski, O.; Band, R. Linear representations and isospectrality with boundary conditions, Journal of Geometric Analysis, Tome 20 (2010) no. 2, pp. 439-471 | Article | MR 2579517 | Zbl 1187.58032

[21] Serre, J.P. Linear representations of finite groups, Springer Verlag Tome 42 (1977) | MR 450380 | Zbl 0355.20006

[22] Shapira, T.; Smilansky, U. Quantum graphs which sound the same, Non-linear dynamics and fundamental interactions (2006), pp. 17-29 | Article | Zbl 1132.37312

[23] Stark, H.M.; Terras, A.A. Zeta functions of finite graphs and coverings, part II, Advances in Mathematics, Tome 154 (2000) no. 1, pp. 132-195 | Article | MR 1780097 | Zbl 0972.11086

[24] Sunada, T. Riemannian coverings and isospectral manifolds, The Annals of Mathematics, Tome 121 (1985) no. 1, pp. 169-186 | Article | MR 782558 | Zbl 0585.58047