Existence of nontrivial units in the rational group ring of a torsion-free group

Determine whether there exists a torsion-free group whose rational group ring contains a nontrivial unit that is not a scalar multiple of a group element.

Background

The paper discusses Kaplansky’s unit conjecture, which asks whether the only units in a group ring of a torsion-free group are scalar multiples of group elements. The Hantzsche–Wendt group provides examples of nontrivial units over finite-characteristic and related coefficient rings, but the corresponding question over the rational group ring remains unresolved. The paper constructs compatible nontrivial unitary units over the rings of integers modulo powers of 2, without resolving the rational-coefficient case.

References

The first is whether there exists a torsion-free group with non-trivial units in $ G$; this is at present open, but there are units in $(/p)$ for all $p$ prime.

— On symmetric and twisted-unitary modular units in group rings  (2609.29273 - Bartholdi et al., 24 Sep 2026) in Section 1, Introduction