Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 080, 8 pages      arXiv:1108.3650

The 2-Transitive Transplantable Isospectral Drums

Jeroen Schillewaert a and Koen Thas b
a) Department of Mathematics, Free University of Brussels (ULB), CP 216, Boulevard du Triomphe, B-1050 Brussels, Belgium
b) Department of Mathematics, Ghent University, Krijgslaan 281, S25, B-9000 Ghent, Belgium

Received December 14, 2010, in final form August 08, 2011; Published online August 18, 2011

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 R2 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.

Key words: isospectrality; drums; Riemannian manifold; doubly transitive group; linear group.

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