# Kaplansky's unit conjecture and unique products

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)?

• 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. – Pedro Tamaroff Feb 22 at 19:56
• @PedroTamaroff Now on arXiv as arxiv.org/abs/2102.11818 – Yemon Choi Mar 1 at 16:30

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 $\mathbb{C}$.