Algebraic [formula omitted]-theory over virtually abelian groups

Abstract

Controlled K -theory is used to show that algebraic K -theory of a group mapping to a virtually abelian group is described by an assembly map defined using hyperelementary subgroups (possibly infinite) of the target group. These subgroups are virtually cyclic, so the result is a refinement of the (fibered) Farrell–Jones conjecture that K -theory comes from virtually cyclic groups. A corollary is that for any group, the Farrell–Jones conjecture is equivalent to this refined version.