What is the current status of the Kaplansky zero-divisor conjecture for group rings? Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group.
A couple of good resources for this problem that gives some historical overview are:


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*Passman, Donald S. The algebraic structure of group rings. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.

*Passman, Donald S. Group rings, crossed products and Galois theory. CBMS Regional Conference Series in Mathematics, 64. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.
The conjecture has been proven affirmative, when $G$ belongs to special classes of groups. I tried to write down some of the history:


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*Ordered groups (A.I. Malcev 1948 and B.H. Neumann 1949)

*Supersolvable groups (E. Formanek 1973)

*Polycyclic-by-finite groups (K.A. Brown 1976, D.R. Farkas & R.L. Snider 1976)

*Unique product groups (J.M. Cohen, 1974)


Here are my questions:


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*Was Irving Kaplansky the first one to state this conjecture? Can someone provide me with a reference to a paper or book that claims this?

*Since the publications of Passman's expository note (above) in 1986, has there been any major developments on the problem? Are there any new classes of groups that will yield a positive answer to the conjecture? Can someone help me to extend my list above?


The zero-divisor conjecture (let's denote it by "(Z)") is related to the following two conjectures:
(I): If $G$ is torsion-free, then $K[G]$ has no non-trivial idempotents.
(U): If $G$ is torsion-free, then $K[G]$ has no non-trivial units.
Now, if $G$ is torsion-free, then one can show that:
(U) $\Rightarrow$ (Z) $\Rightarrow$ (I).
Has there been any developments, since 1986, to any partial answers on conjecture (U)? Passman claims that "this is not even known for supersolvable groups". Is this still the case?
I want to point out that this post is related to another old MO-post.
 A: It is still the case that (U) is not known for supersoluble groups. In fact, it is not known for the 'easiest' supersoluble group that isn't abelian-by-cyclic (for which (U) is known). The group is an extension of $\mathbb{Z}^3$ by the Klein four group.
Peter Pappas and I, in a paper in the early 2010s, gave some general statements about (U) for supersoluble groups, and a general strategy for how to prove (U) for this particular group, but it hasn't been done, even for this one group, in the intervening decade or so.
A: Apologies for the self-promotion, but there is now a counterexample to the unit conjecture (U) with $K=\mathbb{F}_2$ and virtually abelian $G = \langle a, b \,|\, (a^2)^b=a^{-2}, (b^2)^a=b^{-2} \rangle$ (as mentioned in David Craven's answer) in arXiv:2102.11818
.
A: I suggest you have a look at Chapter 10 of W. Lueck's book, "$L^2$-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002. There he discusses the Atiyah conjecture (conjecture 10.3): if $K$ is a subfield of $\mathbb{C}$, a group $G$ satisfies the Atiyah conjecture with coefficients in $K$ if for any $m\times n$ matrix $A$ with coefficients in $K[G]$, the von Neumann dimension of the kernel of the operator $r_A:\ell^2(G)^m\rightarrow \ell^2(G)^n:x\mapsto Ax$, is an integer. 
Lueck then proves (lemma 10.15) that if $G$ is torsion-free and satisfies the Atiyah conjecture with coefficients in $K$, then it satisfies Kaplansky's conjecture (Z). For $K=\mathbb{C}$ and $G$ amenable, the converse is true (lemma 10.16).
Lueck goes on to prove (Theorem 10.19) a remarkable result by P. A. Linnell (Division rings and group von Neumann algebras. Forum Math., 5(6):561-576,1993): Let $\cal{C}$ be the smallest class of groups containing free groups and closed under directed unions and extensions with elementary amenable quotients; if $G$ is in $\cal{C}$ and has finite subgroups of bounded order, then the Atiyah conjecture with coefficients in $\mathbb{C}$ holds.
For further reading and more recent results, try typing "Atiyah conjecture" on Google or in the ArXiV.
