On the first Betti number of a hyperbolic manifold with an arithmetic fundamental group
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.
scientific article; zbMATH DE number 495424
scientific article