Splitting With Continuous Control in Algebraic K-theory

Abstract

In this work, the continuously controlled assembly map in algebraic $K$-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups $\Gamma$ that satisfy certain geometric conditions. The group $\Gamma$ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, $K_0(k\Gamma)$ is proved to be isomorphic to the colimit of $K_0(kH)$ over the finite subgroups $H$ of $\Gamma$, when $\Gamma$ is a virtually polycyclic group and $k$ is a field of characteristic zero.