On the exponent of the cokernel of the forget-control map on \(K_0\)-groups (Q2773384)

Let \(\Gamma\) be a discrete group. The Farrell-Jones isomorphism conjecture in \(K\)-theory [\textit{F. T. Farrell} and \textit{L. E. Jones}, J. Am. Math. Soc. 6, No. 2, 249--297 (1993; Zbl 0798.57018)] proposes that the algebraic \(K\)-theory of the integral group ring \(\mathbb{Z}\Gamma\) can be computed out of the corresponding \(K\)-groups for the virtually cyclic subgroups of \(\Gamma\) via an assembly map. When we look at the corresponding constructions based on the family of finite subgroups of \(\Gamma\) we have a corresponding assembly map. The authors define the controlled \(K\)-Theory of \(\mathbb{Z}\Gamma\) as the image of the assembly map based on the family of finite subgroups of \(\Gamma\), they denote this by \(\widetilde K(\Gamma)_{c}\).NEWLINENEWLINEThe authors prove the following theorem: Let \(\Gamma\) be a group that satisfies the Farrell-Jones conjecture in lower \(K\) theory (i.e. the assembly map based on the virtually cyclic subgroups of \(\Gamma\) is an isomorphism onto \(K_{i}(\mathbb{Z}\Gamma)\) for \(i\leq 1\)). Then the cokernel of the forget control map \(\varphi: \widetilde K_{0}(\Gamma)_{c} \to \widetilde K_{0} (\mathbb{Z}\Gamma)\) is generated by the \(NK_{0}\)-groups of the finite subgroups which are bases of the infinite virtually cyclic subgroups of \(\Gamma\). Moreover, the torsion of this cokernel is determined by the torsion of \(\Gamma\).