A torsion free group \(G\) satisfies the trivial units property for a field \(K\) if the group algebra \(K[G]\) only has the trivial units (the ones of the form \(kg\), where \(k\) is a nonzero field element and \(g\) is in \(G\)). \(G\) satisfies the unique product property if for each pair of finite nonempty subsets \(A\), \(B\) some product in \(AB\) can be written uniquely. The unique product property implies the trivial units property for each field.

We give an overview over these and related properties, and how to formulate them in first-order logic. We discuss Gardam's 2021 result that \(F_2[G]\) (where \(F_2\) is the two-element field) fails the unit conjecture for the Hantzsche-Wendt group \(G\), and the computational methods used to obtain a counterexample.

We discuss work in progress with Heiko Dietrich and Melissa Lee (Monash) that would yield a group \(G\) for which the trivial units property for \(F_2\) holds but the unique product property fails.