By Wolpert’s theorem almost all compact Riemann surfaces are determined up to isometry by their length spectrum or, equivalently, by their spectrum of the Laplacian. On the other hand, there are surfaces which are only determined up to finitely many. The first such examples were given by Marie France Vignéras [1,2] based on quaternion lattices. Later, in 1984 Sunada  found a general construction of isospectral manifolds, based on finite groups. In Chapter 12 we shall use Sunada’s construction to give numerous examples of isospectral Riemann surfaces. In the present chapter we prove Sunada’s theorem for arbitrary Riemannian manifolds and study its combinatorial aspects in detail. It will turn out that graphs are again a useful tool. Here they occur in the form of Cayley graphs.
KeywordsRiemannian Manifold Finite Group Conjugacy Class Cayley Graph Compact Riemann Surface
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