Une réciproque générique du théorème Sunada

Abstract

Let G a compact group of isometries of a closed riemannian manifold(X,m). Sunada proved that if \(\Gamma _1 {\text{ and }}\Gamma _{\text{2}} \) are twofinite almost-conjugated subgroups of G, then\((\Gamma _1 \backslash X,m)\) and \((\Gamma _2 \backslash X,m)\)are isospectral. We prove that if G is finite, there exists an open dense set in the set of G-invariant metrics for which the converse ofthis resukt is true. If G is infinite, the situations is more complicated and we obtain some partial results.