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Cayley graphs and planar isospectral domains. (English) Zbl 0647.53034

Proc. 21st. Int. Taniguchi Symp., Katata/Japan, Conf., Kyoto/Japan 1987, Lect. Notes Math. 1339, 63-77 (1988).
[For the entire collection see Zbl 0638.00022.]
Let G be a finite group which acts freely on a compact Riemannian manifold M. Let \(H_ i\) be subgroups of G and assume \(| [g]\cap H_ 1| =| [g]\cap H_ 2|\) for any conjugacy class [g]\(\in G\). T. Sunada has shown then that the \(M_ i=M/H_ i\) are isospectral. Such manifolds can be constructed from a principal G-bundle over \(M_ 0\) or equivalently from a representation of \(\pi_ 1(M_ 0)\) in G. [Theorem 1 in T. Sunada, Ann. Math., II. Ser. 121, 169-186 (1985; Zbl 0585.58047)].
The author discusses several examples \(({\mathbb{Z}}_ 8\) \(*\cdot {\mathbb{Z}}_ 8\) and SL(3,2)) using the Cayley graph of G. He shows, for example, there exist a pair of flat metrics on the 9-hole sphere which are isospectral for Dirichlet boundary conditions but not isometric; there is a similar result for Neumann boundary conditions.
Reviewer: P.Gilkey

MSC:

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