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.
Categories: math.KT