On the Betti numbers of a hyperbolic manifold
Let \(\Gamma\) be an arithmetic lattice in \(SO(n,1)\). If \(\Gamma\) 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 then 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 \(b_ 1(\Gamma')\neq 0\). On the other hand, in [\textit{J. J. Millson} and \textit{M. S. Raghunathan}, Geometry and analysis, Pap. dedic. mem. Patodi, 103-123 (1981); also published in Proc. Indian Acad. Sci., Math. Sci. 90, 103-124 (1981; Zbl 0514.22007)] a study of the intersections and intersection numbers of so called special cycles in the attached locally symmetric space was made. In the context of compact arithmetic quotients of, for example, \(SO(p,q)\), they prove by analysing the intersection number of hyperbolic cycles with complementary hyperbolic ones that there are uniform arithmetic subgroups, so that the hyperbolic cycles considered in the corresponding locally symmetric space are non-bounding. Such hyperbolic cycles occur naturally as fixed point components of involutions. In the paper under review this approach to prove nonvanishing results for Betti numbers is taken up to show that for any compact hyperbolic \(n\)- dimensional manifold (\(n\geq 3\)) whose fundamental group \(\Gamma\) is an arithmetic lattice arising from a quadratic form there exists a subgroup \(\Gamma'\subset\Gamma\) of finite index, with non-trivial \(i\)-th Betti number \(b_ i\) (\(i=1,\dots,[{{n+1} \over 2}])\), and \(b_ i\) is bounded from below by a power of its volume with an exponent depending only on \(i\) and \(n\). Some implications of this result for multiplicities of representations occuring in the \(L^ 2\)-spectrum of \(\Gamma\) are discussed. In the recent paper by \textit{J. S. Li} and \textit{J. J. Millson} [``On the first Betti number of a hyperbolic manifold with an arithmetic fundamental group'', Duke Math. J. 71, 365-401 (1993)], a second family of arithmetic lattices in \(SO(n,1)\) is discussed, by a (necessarily) different approach.
scientific article; zbMATH DE number 56876