The 2-Transitive Transplantable Isospectral Drums

Jeroen Schillewaert ; Koen Thas

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

For Riemannian manifolds there are several examples which are isospectral but not isometric, see e.g. J. Milnor [Proc. Nat. Acad. Sci. USA 51 (1964), 542]; in the present paper, we investigate pairs of domains in $\mathbb R^2$ which are isospectral but not congruent. All known such counter examples to M. Kac's famous question can be constructed by a certain tiling method (“transplantability”) using special linear operator groups which act $2$-transitively on certain associated modules. In this paper we prove that if any operator group acts $2$-transitively on the associated module, no new counter examples can occur. In fact, the main result is a corollary of a result on Schreier coset graphs of $2$-transitive groups.

Détail

Keywords: isospectrality; drums; Riemannian manifold; doubly transitive group; linear group.

Bibliographie

[1] Brooks R., Tse R., “Isospectral surfaces of small genus”, Nagoya Math. J., 107 (1987), 13–24 | MR | Zbl

[2] Buser P., “Isospectral Riemann surfaces”, Ann. Inst. Fourier (Grenoble), 36 (1986), 167–192 | DOI | MR | Zbl

[3] Buser P., Conway J., Doyle P., Semmler K.-D., “Some planar isospectral domains”, Int. Math. Res. Not., 1994:9 (1994), 391–400 ; arXiv: 1005.1839 | DOI | MR | Zbl

[4] Cameron P.J., “Finite permutation groups and finite simple groups”, Bull. London Math. Soc., 13 (1981), 1–22 | DOI | MR | Zbl

[5] Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A., Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, with computational assistance from J.G. Thackray, Oxford University Press, Eynsham, 1985 | MR | Zbl

[6] Dixon J.D., Mortimer B., Permutation groups, Graduate Texts in Mathematics, 163, Springer-Verlag, New York, 1996 | MR | Zbl

[7] Epkenhans M., Gerstengarbe O., “On the Galois number and the minimal degree of doubly transitive groups”, Comm. Algebra, 28 (2000), 4889–4900 | DOI | MR | Zbl

[8] Giraud O., “Finite geometries and diffractive orbits in isospectral billiards”, J. Phys. A: Math. Gen., 38 (2005), L477–L483 ; arXiv: nlin.CD/0503069 | DOI | MR

[9] Giraud O., Thas K., “Hearing shapes of drums – mathematical and physical aspects of isospectrality”, Rev. Modern Phys., 82 (2010), 2213–2255 ; arXiv: 1101.1239 | DOI | MR

[10] Gordon C., Webb D., Wolpert S., “Isospectral plane domains and surfaces via Riemannian orbifolds”, Invent. Math., 110 (1992), 1–22 | DOI | MR | Zbl

[11] Kac M., “Can one hear the shape of the drum?”, Amer. Math. Monthly, 73:4, part II (1966), 1–23 | DOI | MR | Zbl

[12] Milnor J., “Eigenvalues of the Laplace operators on certain manifolds”, Proc. Nat. Acad. Sci. USA, 51 (1964), 542 | DOI | MR | Zbl

[13] Okada Y., Shudo A., “Equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains”, J. Phys. A: Math. Gen., 34 (2001), 5911–5922 ; arXiv: nlin.CD/0105068 | DOI | MR | Zbl

[14] Segre B., Lectures on modern geometry, with an appendix by Lucio Lombardo-Radice, Consiglio Nazionale delle Ricerche Monografie Matematiche, 7, Edizioni Cremonese, Rome, 1961 | MR | Zbl

[15] Sunada T., “Riemannian coverings and isospectral manifolds”, Ann. of Math. (2), 121 (1980), 169–186 | DOI | MR

[16] Thas K., “Kac's question, planar isospectral pairs and involutions in projective space”, J. Phys. A: Math. Gen., 39 (2006), L385–L388 | DOI | MR | Zbl

[17] Thas K., “Kac's question, planar isospectral pairs and involutions in projective space. II. Classification of generalized projective isospectral data”, J. Phys. A: Math. Gen., 39 (2006), 13237–13242 | DOI | MR | Zbl

[18] Thas K., “$\mathrm{PSL}_n(q)$ as operator group of isospectral drums”, J. Phys. A: Math. Gen., 39 (2006), L673–L675 | DOI | MR | Zbl

[19] Thas K., Classification of transplantable isospectral drums in $\mathbb R^2$, in preparation