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

TL;DR

This paper demonstrates that the map from reduced algebraic K-theory of integral group rings to rational group rings is non-trivial for certain groups, providing a counterexample and a method for computation under the Farrell-Jones conjecture.

Contribution

It shows the non-triviality of the K-theory map for specific groups and offers a way to compute its image assuming the Farrell-Jones conjecture.

Findings

Counterexample group $QD_{32} *_{Q_{16}} QD_{32}$ where the map is non-trivial

Method to compute the map's image using representation theory

Validation of non-triviality in algebraic K-theory for certain groups

Abstract

An open question is whether the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ 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 $QD_{32} *_{Q_{16}} QD_{32}$. We will also show how to compute the image of the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ using representation theoretic means, assuming $G$ satisfies the Farrell-Jones conjecture.