Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known examples of $G$ for which stably free modules are always free, in addition to free groups $F_n$, finitely generated free abelian groups $Z^n$? Any comments are well appreciated.

Actually, if the Farrell-Jones conjecture holds for $G$, all the finitely generated projective modules over $ZG$ will be stably free. It is also known that the trefoil knot group has non-free stably free modules. (the negative side of this question is also discussed in A ring such that all projectives are stably free but not all projectives are free? )


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