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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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This is an old question but I thought this could be a useful answer to anyone interested in this in the future.

The only class of examples of torsion-free groups with SFC (stably-free cancellation) that I know of in the sense you ask is the surface groups of genus $g>e^{1000000}$. This result is due to Grigori Avramidi in https://arxiv.org/abs/2101.00682, using techniques from hyperbolic geometry. They construct a division algorithm in the integral group ring of surface groups of this genus.

Of course, one might expect there to be far more torsion-free groups with this property. In particular in the case of finite groups and groups with torsion we have lots of examples, for example there is recent progress towards characterising finite examples in John Nicholson’s paper https://arxiv.org/abs/1807.00307. Unfortunately, the techniques from these cases don’t seem to extend nicely to torsion-free groups.

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