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Cheeger constants of arithmetic hyperbolic 3-manifolds. (English) Zbl 1207.58024

Let \(M\) be an \(n\)-dimensional Riemannian manifold of finite volume. Let \(N\) be a codimension \(1\)-submanifold of \(M\) that disconnects \(M\) into two manifolds \(A\) and \(B\). Let
\[ h(M,N):={\text{vol}(N)\over\min(\text{vol}(A),\text{vol}(B))}\,. \]
The Cheeger constant \(h(M)\) is defined to be \(\inf_Nh(M,N)\). Let \(\lambda_1(M)\) be the first eigenvalue of the Laplacian on \(M\).
J. Cheeger [Probl. Analysis, Sympos. in Honor of Salomon Bochner, Princeton Univ. 1969, 195–199 (1970; Zbl 0212.44903)] showed \(\lambda_1(M)\geq{1\over4}h(M)^2\). The authors examine the Cheeger constant of certain families of arithmetic hyperbolic \(3\)-manifolds deriving computable bounds on the Cheeger constants and thereby bounds on the first eigenvalue of the Laplacian. Methods of R. Brooks, P. Perry and P. V. Petersen are used [Comment. Math. Helv. 68, No. 4, 599–621 (1993; Zbl 0809.53049)]. Probabilistic methods of R. Brooks and A. Zuk [J. Differ. Geom. 62, No. 1, 49–78 (2002; Zbl 1065.05091)] are used to obtain sharper asymptotic bounds. The Cheeger constants are quite small; this implies that the Cheeger inequality is in general not sufficient to prove Selberg’s eigenvalue conjecture.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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References:

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