Some consequences of Arthur's work on the spectrum and topology of hyperbolic varieties
type: Item
instance of: scholarly article
title: Some consequences of Arthur's work on the spectrum and topology of hyperbolic varieties
zbMATH Open document ID: 1309.22014
author: Nicolas Bergeron
author: Laurent Clozel
published in: Inventiones Mathematicae
publication date: 2013-06-18T00:00:00Z
review text: The paper under review deals with spectral and topological aspects of hyperbolic manifolds. More precisely, let \(G\) be a semisimple \({\mathbb Q}\)-group not coming from a twisted form \(\text{}^3D_4\) or \(\text{}^6D_4\), and such that \(G({\mathbb R})\) is the product of \(\mathrm{SO}(n,1)\) with a compact group. The paper contains several striking results which are interesting on their own, we will mention a few. Let \(M\) be the \(n\)-dimensional manifold \(\Gamma\backslash{\mathbb H}^n\) where \({\mathbb H}^n=\mathrm{SO}(n,1)/\mathrm{SO}(n)\) is the real hyperbolic space and \(\Gamma\subset G\) is a torsion-free congruence discrete subgroup of \(G\). One of the main results in the paper under review can be stated as follows (Theorem 1.2). The \(L^2\)-spectrum of the Laplace operator on functions on \(M\) is contained in the set: \[ \bigcup_{0\leq j<\frac{n-1}{2}}\left\{\left(\tfrac{n-1}{2}\right)^2-\left(\tfrac{n-1}{2}-j\right)^2\right\}\bigcup\;\left[\left(\tfrac{n-1}{2}\right)^2 -\left(\tfrac{1}{2}-\tfrac{1}{N^2+1}\right)^2;+\infty\right[ \] where \(N=n\) if \(n\) is even, and \(N=n+1\) otherwise. This result is connected with a conjecture of \textit{M. Burger} et al. [Bull. Am. Math. Soc., New Ser. 26, No. 2, 253--257 (1992; Zbl 0762.22009)] predicting that the spectrum should be contained in \(\bigcup_{0\leq j<\frac{n-1}{2}}\big\{(\frac{n-1}{2})^2-(\frac{n-1}{2}-j)^2\big\}\bigcup\;[ (\frac{n-1}{2})^2;+\infty[\). As a byproduct, a result on the spectrum of the Laplacian on differential forms is proved. Actually, for \(0\leq k\leq\frac{n}{2}-1\), there exists a positive number \(\varepsilon(n,k)\) such that the first non-zero eigenvalue \(\lambda_1^k(M)\) of the Laplace operator on square integrable \(k\)-forms on \(M\) satisfies the following inequality: \[ \lambda_1^k\geq\varepsilon(n,k). \] Some results relating the (co-)homology of \(M\) with totally geodesic compact submanifolds are obtained. Let \(H\subset G\) be a semisimple group over \({\mathbb Q}\…
reviewed by: Salah Mehdi
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F75
zbMATH DE Number: 6176662
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: congruence subgroup
zbMATH Keywords: Laplacian
zbMATH Keywords: spectrum
zbMATH Keywords: automorphic form
zbMATH Keywords: Lefschetz properties
MaRDI profile type: Publication
full work available at URL: s00222-012-0415-2
OpenAlex ID: W2009172023
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DOI: 10.1007/S00222-012-0415-2
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