×

zbMATH — the first resource for mathematics

On \(G\)-sets and isospectrality. (Sur les \(G\)-ensembles et l’isospectralité.) (English. French summary) Zbl 1364.58017
Summary: 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.

MSC:
58J53 Isospectrality
58D19 Group actions and symmetry properties
Software:
GAP
PDF BibTeX XML Cite
Full Text: DOI Link arXiv
References:
[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, 42, 175202 pp., (2009) · Zbl 1176.58019
[2] Bérard, P., Transplantation et isospectralité I, Mathematische Annalen, 292, 1, 547-559, (1992) · Zbl 0735.58008
[3] Brooks, R., Some relations between graph theory and Riemann surfaces, Isr. Math. Conf. Proc. 11, (1996) · Zbl 0890.30027
[4] Buser, P., Isospectral Riemann surfaces, Ann. Inst. Fourier, 36, 2, 167-192, (1986) · Zbl 0579.53036
[5] Buser, P.; Conway, J.; Doyle, P.; Semmler, K. D., Some planar isospectral domains, International Mathematics Research Notices, 1994, 9, 391-400, (1994) · Zbl 0837.58033
[6] Chapman, SJ, Drums that sound the same, American Mathematical Monthly, 102, 2, 124-138, (1995) · 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, 42, 8, 1067-1095, (1989) · Zbl 0709.53030
[8] DiPasquale, M., On the order of a group containing nontrivial gassmann equivalent subgroups, Rose-Hulman Undergraduate Mathematics Journal, 10, 1, (2009)
[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, (2008)
[11] Gassmann, F., Bemerkungen zur vorstehenden arbeit von Hurwitz, Math. Z, 25, 124-143, (1926) · JFM 52.0156.03
[12] Gordon, C.; Webb, D. L.; Wolpert, S., One cannot hear the shape of a drum, American Mathematical Society, 27, 1, (1992) · Zbl 0756.58049
[13] Hall, M., The theory of groups, (1976), Chelsea Pub Co · Zbl 0354.20001
[14] Hillairet, L., Spectral decomposition of square-tiled surfaces, Mathematische Zeitschrift, 260, 2, 393-408, (2008) · Zbl 1156.58012
[15] Kac, M., Can one hear the shape of a drum?, The american mathematical monthly, 73, 4, 1-23, (1966) · Zbl 0139.05603
[16] Larsen, M., Determining a semisimple group from its representation degrees, International Mathematics Research Notices, 2004, 38, 1989 pp., (2004) · Zbl 1073.22009
[17] Lemańczyk, M.; Thouvenot, J. P.; Weiss, B., Relative discrete spectrum and joinings, Monatshefte für Mathematik, 137, 1, 57-75, (2002) · Zbl 1090.37002
[18] Merling, M.; Perlis, R., Gassmann equivalent dessins, Communications in Algebra®, 38, 6, 2129-2137, (2010) · 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, 51, 4, 542 pp., (1964) · Zbl 0124.31202
[20] Parzanchevski, O.; Band, R., Linear representations and isospectrality with boundary conditions, Journal of Geometric Analysis, 20, 2, 439-471, (2010) · Zbl 1187.58032
[21] Serre, J. P., Linear representations of finite groups, 42, (1977), Springer Verlag · Zbl 0355.20006
[22] Shapira, T.; Smilansky, U., Quantum graphs which sound the same, Non-linear dynamics and fundamental interactions, 17-29, (2006) · Zbl 1132.37312
[23] Stark, H. M.; Terras, A. A., Zeta functions of finite graphs and coverings, part II, Advances in Mathematics, 154, 1, 132-195, (2000) · Zbl 0972.11086
[24] Sunada, T., Riemannian coverings and isospectral manifolds, The Annals of Mathematics, 121, 1, 169-186, (1985) · Zbl 0585.58047
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.