A counterexample to the unit conjecture for group rings

TL;DR

This paper provides a concrete counterexample to Kaplansky's unit conjecture, demonstrating that non-trivial units can exist in group rings of torsion-free groups over certain fields.

Contribution

It presents the first explicit counterexample to the unit conjecture for group rings, involving a virtually abelian group over a field of order two.

Findings

Counterexample disproves the unit conjecture

Non-trivial units exist in the specified group ring

Virtually abelian groups can have non-trivial units

Abstract

The unit conjecture, commonly attributed to Kaplansky, predicts that if $K$ is a field and $G$ is a torsion-free group then the only units of the group ring $K[G]$ are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.