The passage from the integral to the rational group ring in algebraic $K$-theory

Abstract

An open question is whether the map $\widetilde{K_0}\mathbb{Z}G\to \widetilde{K_0}\mathbb{Q}G$ in reduced $K$-theory from the integral to the rational group ring is trivial for any group $G$. We will show that this is false, with a counterexample given by the group $Q{D}_{32}{\ast }_{Q_{16}}Q{D}_{32}$. We will also show how to compute the image of the map $\widetilde{K_0}\mathbb{Z}G\to \widetilde{K_0}\mathbb{Q}G$ using representation theoretic means, assuming $G$ satisfies the Farrell-Jones conjecture.

47 pages, work done as half of the authors thesis