The 2-Transitive Transplantable Isospectral Drums

TL;DR

This paper investigates isospectral but non-congruent planar domains, showing that 2-transitive group actions prevent the existence of new counterexamples to the isospectral problem.

Contribution

It proves that 2-transitive group actions on associated modules do not produce new isospectral domain pairs, extending understanding of transplantability limitations.

Findings

No new isospectral non-congruent domains can be constructed via 2-transitive groups.

Main result derived from properties of Schreier coset graphs of 2-transitive groups.

Provides a theoretical boundary for constructing isospectral drums.

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.