On the Farrell-Jones Conjecture and its applications

Abstract

We present the status of the Farrell–Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true, and we study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture, and conjectures generalizing Moody’s Induction Theorem. Thus, we considerably extend the class of groups for which these conjectures are known.