A counterexample to the unit conjecture for group rings

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.