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The unit version of the Kaplansky conjecture is about units in $FG$ where $F$ is a field and $G$ is a torsion free group. In a recent counter example by Giles Gardam, it is given an example of a non trivial unit when $F=F_2$ is the field with 2 elements and G is a finitely presented torsion free group. See this MO post for some discussions on this counter example.

The group G in Giles Gardam counterexample is a result of a group extension $Z^3\to G\to Z_2\times Z_2$. So it would be helpful to study some other extensions of $Z^3$ by this quotient group? Do they generate other examples?

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    $\begingroup$ By Bieberbach's theorem, torsion-free extensions of $\mathbb{Z}^3$ are precisely the fundamental groups of closed, flat 3-manifolds. These are classified, and there are known to be 10 of them (see eg, §4 of Scott's notes: dept.math.lsa.umich.edu/~pscott/8geoms.pdf). Gardam cites a paper of Craven--Pappas (arxiv.org/abs/1010.1144) who explain that any possible counterexample should not be right-orderable. It is not hard to see that, for any closed 3-manifold $M$ with $H_1(M)$ infinite, $\pi_1M$ is locally indicable hence right-orderable. In conclusion... (tbc) $\endgroup$
    – HJRW
    Commented Sep 11, 2023 at 9:20
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    $\begingroup$ (cont'd) one should look for closed flat 3-manifold groups with finite $H_1(M)$. Wikipedia seems to suggest that Gardam's example ("the Hantzsche–Wendt manifold") is the only one -- en.wikipedia.org/wiki/Hantzsche%E2%80%93Wendt_manifold -- ("It is the only closed flat 3-manifold with first Betti number zero.") but it would be nice to find a proper reference. $\endgroup$
    – HJRW
    Commented Sep 11, 2023 at 9:24
  • $\begingroup$ @HJRW Than you very much for these very informative and helpful comments $\endgroup$
    – Ali Taghavi
    Commented Sep 11, 2023 at 10:22
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    $\begingroup$ @HJRW Thanks again for sharing the beautiful papers. They will help me to learn a lot and even better understanding the Giles Gardam ideas $\endgroup$
    – Ali Taghavi
    Commented Sep 11, 2023 at 10:56

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