Kaplansky's zero divisor conjecture

Remarks

The conjecture is open. Torsion-freeness is necessary: if $g \in G$ has finite order $n > 1$, then $(1 - g)(1 + g + \cdots + g^{n-1}) = 1 - g^n = 0$ with both factors nonzero. It holds when $G$ has the unique product property: for nonempty finite sets $A, B \subseteq G$, some product $ab$ with $a \in A$, $b \in B$ arises in only one way. For such $G$, take $A$ and $B$ to be the supports of $a$ and $b$. The coefficient of that unique product in $ab$ is a product of two nonzero field elements, so $ab \ne 0$. Left-orderable groups, and in particular torsion-free abelian groups and free groups, have this property. For $G = \Z$, the group ring $K[\Z]$ is the Laurent polynomial ring $K[x, x^{-1}]$, which is a domain. The conjecture is also known for torsion-free elementary amenable groups, whose group rings are Ore domains. In characteristic zero it follows from the strong Atiyah conjecture for $G$.