Non-trivial units of complex group rings

One of most important mathematical results from 2021 was Gardam’s refutation of the unit conjecture (blog post): He constructed the first non-trivial unit in a group ring.

The coefficients of the group ring of the first counter-example were \(\mathbb{F}_2\), and for three years the question whether one can generalize his idea to get counter-examples in group ring over fields of characteristic zero remained open. Finally, this was solved today, again by Gardam: link (the coefficients are \(\mathbb{C}\) in this case).