On the first Betti number of a hyperbolic manifold with an arithmetic fundamental group
type: Item
instance of: scholarly article
title: On the first Betti number of a hyperbolic manifold with an arithmetic fundamental group
zbMATH Open document ID: 0798.11019
DOI: 10.1215/S0012-7094-93-07115-3
author: Jian-Shu Li
author: John Millson
published in: Duke Mathematical Journal
publication date: 1994-02-09T00:00:00Z
review text: Let \(\Gamma\) be an arithmetic lattice in \(SO(n,1)\). If the lattice \(\Gamma\) is commensurable with the group of units of a quadratic form over a totally real field which has signature \((n,1)\) at one real place and is anisotropic at the remaining real places it is shown in \textit{J. J. Millson} [Ann. Math., II. Ser. 104, 235-247 (1976; Zbl 0364.53020)] that there exists a subgroup \(\Gamma'\subset\Gamma\) of finite index so that the first Betti number \(b_ 1 (\Gamma ') \neq 0\). By the classification of rational structures on \(SO (n,1)\), this settles the case of non- cocompact arithmetic lattices. In the cocompact case, there is, \(n \neq 3,7\), another family of arithmetic lattices, the ones given (up to commensurability) as the group of units of an appropriate skew-hermitian form over a quaternion algebra. This family exists only when \(n\) is odd. In the case \(n>5\), nonvanishing results for the first Betti number of such arithmetic lattices were obtained in \textit{J. S. Li} [J. Reine Angew. Math. 428, 177-217 (1992; Zbl 0749.11032)]. In this paper a new proof is given, including the case \(n=5\) as well. It is based on the study of the period of a suitable theta series (a closed 1-form on the arithmetic quotient \(M\) in question) over a carefully chosen closed geodesic in \(M\). This analysis turns out to be quite delicate and technically demanding.
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 495424
zbMATH Keywords: arithmetic hyperbolic manifolds
zbMATH Keywords: cohomology of arithmetic groups
zbMATH Keywords: arithmetic lattices
zbMATH Keywords: first Betti number
reviewed by: Joachim Schwermer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 495424