Nous étudions les -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 une variété (ou orbifold) Riemannienne compacte et connexe dont le groupe fondamental possède un quotient fini non cyclique. Alors admet des revêtements isospectraux non isométriques.
We study finite -sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then has isospectral non-isometric covers.
Keywords: isospectrality, laplacian, G-sets, Sunada
Mot clés : isospectralité, laplacien, G-ensembles, Sunada
@article{AIF_2013__63_6_2307_0, author = {Parzanchevski, Ori}, title = {On $G$-sets and isospectrality}, journal = {Annales de l'Institut Fourier}, pages = {2307--2329}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {63}, number = {6}, year = {2013}, doi = {10.5802/aif.2831}, zbl = {06325435}, mrnumber = {3237449}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2831/} }
TY - JOUR AU - Parzanchevski, Ori TI - On $G$-sets and isospectrality JO - Annales de l'Institut Fourier PY - 2013 SP - 2307 EP - 2329 VL - 63 IS - 6 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2831/ DO - 10.5802/aif.2831 LA - en ID - AIF_2013__63_6_2307_0 ER -
Parzanchevski, Ori. On $G$-sets and isospectrality. Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2307-2329. doi : 10.5802/aif.2831. http://www.numdam.org/articles/10.5802/aif.2831/
[1] The isospectral fruits of representation theory: quantum graphs and drums, Journal of Physics A: Mathematical and Theoretical, Volume 42 (2009), pp. 175202 | DOI | MR | Zbl
[2] Transplantation et isospectralité I, Mathematische Annalen, Volume 292 (1992) no. 1, pp. 547-559 | DOI | MR | Zbl
[3] Some relations between graph theory and Riemann surfaces, Isr. Math. Conf. Proc. 11, Citeseer (1996) | MR | Zbl
[4] Isospectral Riemann surfaces, Ann. Inst. Fourier, Volume 36 (1986) no. 2, pp. 167-192 | DOI | Numdam | MR | Zbl
[5] Some planar isospectral domains, International Mathematics Research Notices, Volume 1994 (1994) no. 9, pp. 391-400 | DOI | MR | Zbl
[6] Drums that sound the same, American Mathematical Monthly, Volume 102 (1995) no. 2, pp. 124-138 | DOI | MR | Zbl
[7] Isospectral deformations II: Trace formulas, metrics, and potentials, Communications on Pure and Applied Mathematics, Volume 42 (1989) no. 8, pp. 1067-1095 | DOI | MR | Zbl
[8] On the Order of a Group Containing Nontrivial Gassmann Equivalent Subgroups, Rose-Hulman Undergraduate Mathematics Journal, Volume 10 (2009) no. 1
[9] 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) http://www.gap-system.org
[11] Bemerkungen zur vorstehenden Arbeit von Hurwitz, Math. Z, Volume 25 (1926), pp. 124-143
[12] One cannot hear the shape of a drum, American Mathematical Society, Volume 27 (1992) no. 1 | MR | Zbl
[13] The theory of groups, Chelsea Pub Co, 1976 | MR | Zbl
[14] Spectral decomposition of square-tiled surfaces, Mathematische Zeitschrift, Volume 260 (2008) no. 2, pp. 393-408 | DOI | MR | Zbl
[15] Can one hear the shape of a drum?, The american mathematical monthly, Volume 73 (1966) no. 4, pp. 1-23 | DOI | MR | Zbl
[16] Determining a semisimple group from its representation degrees, International Mathematics Research Notices, Volume 2004 (2004) no. 38, pp. 1989 | DOI | MR | Zbl
[17] Relative discrete spectrum and joinings, Monatshefte für Mathematik, Volume 137 (2002) no. 1, pp. 57-75 | DOI | MR | Zbl
[18] Gassmann Equivalent Dessins, Communications in Algebra®, Volume 38 (2010) no. 6, pp. 2129-2137 | DOI | MR | Zbl
[19] Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences of the United States of America, Volume 51 (1964) no. 4, pp. 542 | DOI | MR | Zbl
[20] Linear representations and isospectrality with boundary conditions, Journal of Geometric Analysis, Volume 20 (2010) no. 2, pp. 439-471 | DOI | MR | Zbl
[21] Linear representations of finite groups, 42, Springer Verlag, 1977 | MR | Zbl
[22] Quantum graphs which sound the same, Non-linear dynamics and fundamental interactions (2006), pp. 17-29 | DOI | Zbl
[23] Zeta functions of finite graphs and coverings, part II, Advances in Mathematics, Volume 154 (2000) no. 1, pp. 132-195 | DOI | MR | Zbl
[24] Riemannian coverings and isospectral manifolds, The Annals of Mathematics, Volume 121 (1985) no. 1, pp. 169-186 | DOI | MR | Zbl
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