# On $G$-sets and isospectrality

Ori Parzanchevski^{[1]}

- [1] Hebrew University of Jerusalem

Annales de l’institut Fourier (2013)

- Volume: 63, Issue: 6, page 2307-2329
- ISSN: 0373-0956

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topParzanchevski, Ori. "On $G$-sets and isospectrality." Annales de l’institut Fourier 63.6 (2013): 2307-2329. <http://eudml.org/doc/275662>.

@article{Parzanchevski2013,

abstract = {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.},

affiliation = {Hebrew University of Jerusalem},

author = {Parzanchevski, Ori},

journal = {Annales de l’institut Fourier},

keywords = {isospectrality; laplacian; G-sets; Sunada; Laplacian; -sets},

language = {eng},

number = {6},

pages = {2307-2329},

publisher = {Association des Annales de l’institut Fourier},

title = {On $G$-sets and isospectrality},

url = {http://eudml.org/doc/275662},

volume = {63},

year = {2013},

}

TY - JOUR

AU - Parzanchevski, Ori

TI - On $G$-sets and isospectrality

JO - Annales de l’institut Fourier

PY - 2013

PB - Association des Annales de l’institut Fourier

VL - 63

IS - 6

SP - 2307

EP - 2329

AB - 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.

LA - eng

KW - isospectrality; laplacian; G-sets; Sunada; Laplacian; -sets

UR - http://eudml.org/doc/275662

ER -

## References

top- R. Band, O. Parzanchevski, G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, Journal of Physics A: Mathematical and Theoretical 42 (2009) Zbl1176.58019MR2539297
- P. Bérard, Transplantation et isospectralité I, Mathematische Annalen 292 (1992), 547-559 Zbl0735.58008MR1152950
- R. Brooks, Some relations between graph theory and Riemann surfaces, Isr. Math. Conf. Proc. 11 (1996) Zbl0890.30027MR1476704
- P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier 36 (1986), 167-192 Zbl0579.53036MR850750
- P. Buser, J. Conway, P. Doyle, K. D. Semmler, Some planar isospectral domains, International Mathematics Research Notices 1994 (1994), 391-400 Zbl0837.58033MR1301439
- SJ Chapman, Drums that sound the same, American Mathematical Monthly 102 (1995), 124-138 Zbl0849.35084MR1315592
- D.M. DeTurck, C.S. Gordon, K.B. Lee, Isospectral deformations II: Trace formulas, metrics, and potentials, Communications on Pure and Applied Mathematics 42 (1989), 1067-1095 Zbl0709.53030MR1029118
- M. DiPasquale, On the Order of a Group Containing Nontrivial Gassmann Equivalent Subgroups, Rose-Hulman Undergraduate Mathematics Journal 10 (2009)
- P.G. Doyle, J.P. Rossetti, Laplace-isospectral hyperbolic 2-orbifolds are representation-equivalent, Arxiv preprint arXiv:1103.4372 (2011)
- GAP – Groups, Algorithms, and Programming, Version 4.4.12, (2008)
- F. Gassmann, Bemerkungen zur vorstehenden Arbeit von Hurwitz, Math. Z 25 (1926), 124-143
- C. Gordon, D.L. Webb, S. Wolpert, One cannot hear the shape of a drum, American Mathematical Society 27 (1992) Zbl0756.58049MR1136137
- M. Hall, The theory of groups, (1976), Chelsea Pub Co Zbl0354.20001MR414669
- L. Hillairet, Spectral decomposition of square-tiled surfaces, Mathematische Zeitschrift 260 (2008), 393-408 Zbl1156.58012MR2429619
- M. Kac, Can one hear the shape of a drum?, The american mathematical monthly 73 (1966), 1-23 Zbl0139.05603MR201237
- M. Larsen, Determining a semisimple group from its representation degrees, International Mathematics Research Notices 2004 (2004) Zbl1073.22009MR2063567
- M. Lemańczyk, J.P. Thouvenot, B. Weiss, Relative discrete spectrum and joinings, Monatshefte für Mathematik 137 (2002), 57-75 Zbl1090.37002MR1930996
- M. Merling, R. Perlis, Gassmann Equivalent Dessins, Communications in Algebra® 38 (2010), 2129-2137 Zbl1246.11126MR2675525
- J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proceedings of the National Academy of Sciences of the United States of America 51 (1964) Zbl0124.31202MR162204
- O. Parzanchevski, R. Band, Linear representations and isospectrality with boundary conditions, Journal of Geometric Analysis 20 (2010), 439-471 Zbl1187.58032MR2579517
- J.P. Serre, Linear representations of finite groups, 42 (1977), Springer Verlag Zbl0355.20006MR450380
- T. Shapira, U. Smilansky, Quantum graphs which sound the same, Non-linear dynamics and fundamental interactions (2006), 17-29 Zbl1132.37312
- H.M. Stark, A.A. Terras, Zeta functions of finite graphs and coverings, part II, Advances in Mathematics 154 (2000), 132-195 Zbl0972.11086MR1780097
- T. Sunada, Riemannian coverings and isospectral manifolds, The Annals of Mathematics 121 (1985), 169-186 Zbl0585.58047MR782558

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