On G-sets and isospectrality
[Sur les G-ensembles et l’isospectralité]
Parzanchevski, Ori1
1 Hebrew University of Jerusalem
Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2307-2329.

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.

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.

DOI : 10.5802/aif.2831
Classification : 58J53, 58D19
Keywords: isospectrality, laplacian, G-sets, Sunada
Mot clés : isospectralité, laplacien, G-ensembles, Sunada
Parzanchevski, Ori 1

1 Hebrew University of Jerusalem 
@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  - 
%0 Journal Article
%A Parzanchevski, Ori
%T On $G$-sets and isospectrality
%J Annales de l'Institut Fourier
%D 2013
%P 2307-2329
%V 63
%N 6
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2831/
%R 10.5802/aif.2831
%G en
%F AIF_2013__63_6_2307_0
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] Band, R.; Parzanchevski, O.; Ben-Shach, G. 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] Bérard, P. Transplantation et isospectralité I, Mathematische Annalen, Volume 292 (1992) no. 1, pp. 547-559 | DOI | MR | Zbl

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

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

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

[6] Chapman, SJ Drums that sound the same, American Mathematical Monthly, Volume 102 (1995) no. 2, pp. 124-138 | DOI | MR | Zbl

[7] DeTurck, D.M.; Gordon, C.S.; Lee, K.B. 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] DiPasquale, M. On the Order of a Group Containing Nontrivial Gassmann Equivalent Subgroups, Rose-Hulman Undergraduate Mathematics Journal, Volume 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 (2008) http://www.gap-system.org

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

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

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

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

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

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

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

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

[19] Milnor, J. 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] Parzanchevski, O.; Band, R. Linear representations and isospectrality with boundary conditions, Journal of Geometric Analysis, Volume 20 (2010) no. 2, pp. 439-471 | DOI | MR | Zbl

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

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

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

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

Cité par Sources :