On the Farrell-Jones Conjecture and its applications

TL;DR

This paper reviews the current status of the Farrell-Jones Conjecture in algebraic K-theory, introduces new groups satisfying the conjecture, and explores its implications for related conjectures and inheritance properties.

Contribution

It extends the class of groups for which the Farrell-Jones Conjecture and related conjectures are known to hold, with new results and applications.

Findings

New groups verified for the Farrell-Jones Conjecture

Enhanced understanding of inheritance properties

Applications to Bass and Kaplansky Conjectures

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 study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture and conjectures generalizing Moody's Induction Theorem. Thus we extend the class of groups for which these conjectures are known considerably.