Free quotients and the first Betti number of some hyperbolic manifolds
Let \(M\) be an oriented \(n\)-dimensional hyperbolic manifold of finite volume, and suppose that \(M\) contains a totally geodesic submanifold \(F\) of codimension one. Then it is shown that its fundamental group \(\Gamma=\pi_1(M)\) has a virtually non-Abelian free quotient group. In particular, \(M\) has finite sheeted covers with arbitrarily large first Betti numbers. This result is known in the cases where \(\Gamma=\pi_1(M)\) is an arithmetic lattice in \(PO(n,1)\) provided \(n\neq 3\) and \(n\neq 7\) (cf. \textit{M. S. Raghunathan} [The first Betti number of compact locally symmetric spaces. In: Current Trends in Mathematics and Physics -- A tribute on Harish-Chandra (ed. \textit{S. D. Adhikari}), New Delhi, 116-137 (1995)] for a survey). The new method here is based on some simple group theoretic observations and a geometric result due to \textit{M. Gromov} and \textit{I. I. Piatetski-Shapiro} [Publ. Math., Inst. Hautes Étud. Sci. 66, 93-103 (1988; Zbl 0649.22007)] giving a group theoretic description of \(\Gamma=\pi_1(M)\) as an amalgamated product or an HNN-construction. In turn, this method applies to the non-arithmetic lattices in \(PO(n,1)\) constructed by Gromov and Piatetski-Shapiro and to groups generated by reflections as well.
scientific article; zbMATH DE number 934933
scientific article