Riemannian coverings and isospectral manifolds.

*(English)*Zbl 0585.58047This paper is a breakthrough in the construction of pairs of compact Riemannian manifolds which are nonisometric but have the same spectrum of the Laplace operator. The great merit is to have found a connetion between Riemannian manifolds and a technique in algebraic number theory which produces nonsolitary number fields. The result is this: Let G be a finite group which has two nonconjugate subgroups \(H_ 1,H_ 2\) which nevertheless are almost conjugate in the sense that for each conjugacy class [g], (g\(\in G)\) there are as many representatives in \(H_ 1\) as in \(H_ 2\). Subgroups of that kind occur e.g. in PSL(2,7) or in \(({\mathbb{Z}}/8{\mathbb{Z}})^{mult}\times ({\mathbb{Z}}/8{\mathbb{Z}})\). Now take any differentiable manifold \(M_ 0\) which admits a finite covering \(M\to M_ 0\) such that the group of covering transformations is G. Then for every Riemannian metric on \(M_ 0\) lifted to M the quotients \(M/H_ 1\), \(M/H_ 2\) are isospectral and, generically, nonisometric. At the end of the paper the author uses some of the technique to estimate eigenvalues of the Laplacian for Riemannian coverings.

Reviewer: P.Buser