Compactness of the set of isospectral nets and its consequences (Q2564632)

It has been known for a long time that the Laplace spectrum of a closed Riemannian manifold does not in general determine the isometry class of the manifold. By now one even has continuous families of isospectral pairwise nonisometric manifolds [\textit{C. S. Gordon} and \textit{E. N. Wilson}, J. Differ. Geom. 19, 241-256 (1984; Zbl 0532.58032)]. Hence the best uniqueness theorem one can hope for is a compactness result stating that the set of isometry classes of Riemannian manifolds with a fixed spectrum is compact for a suitable topology. Since most known isospectral examples are locally isometric, it is natural to start with this case. Let \(X\) be a Riemannian manifold and let \(G\) be its isometry group. The author introduces a topology on the set \(M_G\) of all discrete subgroups of \(G\) and shows that if \(X\) is analytic, then the set of all \(\Gamma\in M_G\) acting properly discontinuously and cocompactly on \(X\) such that \(\Gamma\backslash X\) has a fixed spectrum is compact. There is a technical assumption on \(G\) which is usually satisfied. Similarly, it is shown that the set of all \(\Gamma\) such that \(\Gamma\backslash X\) has a fixed length spectrum and bounded volume is compact. Finally, some applications to representation theory are given.