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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 $$\lambda_ 1(S_ i)>C$$.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 57M10 Covering spaces and low-dimensional topology 57M05 Fundamental group, presentations, free differential calculus
##### Keywords:
first eigenvalue of the Laplace-Beltrami operator
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