The 2-Transitive Transplantable Isospectral Drums

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.