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arXiv · 1108.3650

The 2-Transitive Transplantable Isospectral Drums

Jeroen Schillewaert·Koen Thas
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Abstract

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.

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Jeroen Schillewaert, Koen Thas. 2011-08-18. The 2-Transitive Transplantable Isospectral Drums. https://doi.org/10.3842/sigma.2011.080

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