The lower algebraic \(K\)-theory of virtually infinite cyclic groups
In an earlier paper [Isomorphism conjectures in algebraic \(K\)-theory, J. Am. Math. Soc. 6, No. 2, 249-297 (1993; Zbl 0798.57018)], the authors conjectured that \(K\)-theory for an arbitrary group \(\Gamma\) is ``computable'' from \(K\)-theory of the family of virtually cyclic subgroups of \(\Gamma\). The present paper deals with the ``atoms'', i.e. the virtually cyclic groups. For such a group, it is shown that \(K_n (\mathbb{Z} \Gamma)\) vanishes for \(n \leq -2\) and that \(K_{-1} (\mathbb{Z} \Gamma)\) is generated by the subgroups \(\text{Im} (K_{-1} (\mathbb{Z} F) \to K_{-1} (\mathbb{Z} \Gamma))\) with \(F\) varying over all finite subgroups of \(\Gamma\). The authors first treat the semi-direct product case \(\Gamma = G \rtimes T\) where \(G\) is finite and \(T\) infinite cyclic. Here the methods are algebraic, relying on techniques developed by Bass, Carter and Connolly and da Silva. The extension to the general case uses controlled topology and dynamics arguments developed by the authors in a series of earlier papers.
scientific article; zbMATH DE number 784412
scientific article