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There are three conjectures on group rings that bear the name of Kaplansky (see for example this question). The 'unit conjecture' in the title of the present question is the strongest of them, and states that the group ring $\mathbb{C}\Gamma$ of a torsion-free group $\Gamma$ should contain no units besides the obvious ones $\lambda g$ for $\lambda\in\mathbb{C}^\times,\, g\in\Gamma$.

A natural combinatorial property on a group $\Gamma$ under which the conclusion of the unit conjecture is known to hold is the unique products property: one says that $\Gamma$ has unique products if for any two finite (nonempty) subsets $A,B$ in $\Gamma$ there exists $a\in A,b\in B$ such that $ab\not= a'b'$ for all $(a,b)\not= (a',b')\in A\times B$ (informally, $ab$ can be written in only one way as a product). This property has been well-studied, and is known to hold for various classes of groups; it is also known that there are torsion-free groups which have non-unique products (see for example this paper of B. Bowditch for further references).

While the other two conjectures can be approached by a variety of means (see the afore-mentioned MO question for more details), I am not aware of any torsion-free group $\Gamma$ for which the conclusion uf the unit conjecture is known to hold, without the unique products property having been established first.

Hence (at last) my query: is there a known example of a (say finitely generated) torsion-free group $\Gamma$ such that it is known that all units in $\mathbb{C}\Gamma$ are the obvious ones, but for which it is not known that it has unique products (or even better, such that it is known to have non-unique products)?

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    $\begingroup$ Twitter is reporting a counterexample to the unit conjecture over $\mathbb F_2$ with $\Gamma$ the Promislow group $\langle x, y | x^{-1} y^2 x = y^{-2}, y^{-1} x^2 y = x^{-2} \rangle$ due to Giles Gardam. $\endgroup$
    – Pedro
    Feb 22, 2021 at 19:56
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    $\begingroup$ @PedroTamaroff Now on arXiv as arxiv.org/abs/2102.11818 $\endgroup$
    – Yemon Choi
    Mar 1, 2021 at 16:30

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In this preprint of William Carter from 2013, it appears that there are very few known classes of torsion-free groups which are not unique-product groups, and that for these groups the unit conjecture is open.

In this paper of Craven and Pappas from 2013, some preliminary work was done on the fours group. However, I don't know that anyone looked specifically at whether the units conjecture is true for any of these groups when the coefficient field is $\mathbb{C}$.

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  • $\begingroup$ Yes, one of the difficulties in this question is that there are very few examples of groups without the u.p. property, and apart from the fours group they are somehow artificial (apart from Carter's examples all the known ones are generalizations of the original construction of an hyperbolic group without u.p. by Rips--Segev ; see e.g. arxiv.org/abs/1307.0981 by Steenbock). $\endgroup$
    – Jean Raimbault
    Feb 3, 2015 at 15:29
  • $\begingroup$ The Craven--Pappas paper also illustrates the difficulty, since they do a lot of rather involved (though elementary) algebra only to obtain the rather feeble (compared to what we know about zero-divisors) conclusion that a certain kind of unit (with explicit restrictions on the support) cannot exist in this group. Since it is in fact the simplest example of a non u.p. group (it has an index 2 subgroup which does have u.p. this is not encouraging about the general case. $\endgroup$
    – Jean Raimbault
    Feb 3, 2015 at 15:35

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