A geometric invariant of discrete groups
This paper brings together and generalizes several hitherto disjoint lines of research. The authors associate to every finitely generated group \(G\) a sphere \(S(G)\cong S^{n-1}\), where \(n\) is the \(\mathbb{Z}\)-rank of the abelianization of \(G\) and a subset \(\Sigma\) of \(S(G)\), their geometric invariant. The invariant \(\Sigma\) generalizes the Bieri-Strebel invariant defined for metabelian groups \(G\), it is closely related to the set of all finitely generated kernels of infinite cyclic quotients, studied by \textit{W. Neumann}. In case \(G\) is the fundamental group of a 3-manifold \(Y\), \(\Sigma\) is shown to equal a set studied by \textit{W. P. Thurston} [Mem. Am. Math. Soc. 339, 99-130 (1986; Zbl 0585.57006)] and closely related to fibrations \(Y\to S^ 1\). One important result of the paper under review is the following theorem improving a result of Bieri and Strebel: If \(G\) is finitely presented and has no nonabelian free subgroup then \[ \Sigma\cup(-\Sigma)=S(G), \] where \(-\Sigma\) is the image of \(\Sigma\) under the antipodal map. The necessary condition for finite presentability stated in this theorem has been shown to be sufficient for center-by-metabelian \(G\) [\textit{R. Bieri} and \textit{R. Strebel}, Proc. Lond. Math. Soc., III. Ser. 41, 439-464 (1980; Zbl 0448.20029)] and for solvable \(S\)-arithmetic \(G\) over a number field provided the Schur multiplier is finitely generated, an obviously necessary condition [\textit{H. Abels}, Finite presentability of \(S\)-arithmetic groups, Lond. Math. Soc. Lect. Note Ser. 121, 128-134 (1986; Zbl 0607.20027)].
surface bundles over the circle. bivariant theory
finitely generated kernels of infinite cyclic quotients
scientific article; zbMATH DE number 4046701
scientific article