# An implication of the Farrell-Jones conjecture

A ‘well-known’ implication of the Farrell-Jones conjecture (for a given group G) is that the map $\widetilde{K_0(\mathbb{Z}G)} \to \widetilde{K_0(\mathbb{Q}G)}$ in reduced algebraic K-theory is rationally trivial.

What at first might seem as a technical statement about algebraic K-theory turns out to have an interesting geometric consequence. It implies the Bass conjecture, which is equivalent to the following (due to Geoghegan https://doi.org/10.1007/BFb0089704, see also arXiv:0903.4341): A finitely presented group G satisfies the Bass conjecture if and only if every homotopy idempotent self-map on a finite, connected complex with fundamental group G has Nielsen number either zero or one. The latter condition implies that every homotopy idempotent self-map of a closed, smooth, oriented manifold of dimension at least three and with fundamental group G is homotopic to a map with exactly one fixed point.

In a recent preprint (arXiv:2110.01413) Georg Lehner improved the rational vanishing result stated above to the following: If a group G satisfies the Farrell-Jones conjecture, then the image of the map $$\widetilde{K_0(\mathbb{Z}G)} \to \widetilde{K_0(\mathbb{Q}G)}$$ is a 2-torsion subgroup. Additionally, he provided an example of a group where the image of this map is indeed non-trivial.

Now I am wondering: Is there any new (or refined version of the) geometric implication of this strengthening?