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Riemannian coverings and isospectral manifolds. (English) Zbl 0585.58047
This 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

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C20 Global Riemannian geometry, including pinching
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