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The spectral geometry of a tower of coverings. (English) Zbl 0576.58033
Let M be a compact manifold, and \(\{M_ i\}^ a \)family of finite coverings of M. In this paper, we study the behavior of \(\lambda_ 1(M_ i)\) as i tends to \(\infty\). In particular, we show that whether or not \(\lambda_ 1(M_ i)\) tends to 0 as i tends to \(\infty\) depends only on the inclusions \(\pi_ 1(M_ i)\subset \pi_ 1(M)\). Some consequences are: Theorem: Suppose \(\pi_ 1(M)\) has Kazhdan’s property T. Then there is a \(C>0\) such that \(\lambda_ 1(M')>0\) for all finite coverings M’ of M. Theorem: If \(\pi_ 1(M)\) surjects onto Z*Z, then there exists \(C>0\) and infinitely many coverings \(M_ i\) of M such that \(\lambda_ 1(M_ i)>0\) for all i. In particular, there exist \(C>0\) and Riemann surfaces \(S_ i\) of arbitrarily large genus with \(\lambda_ 1(S_ i)>C\).

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
57M10 Covering spaces and low-dimensional topology
57M05 Fundamental group, presentations, free differential calculus
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