Kaplansky's unit conjecture

Every unit of the group ring \(K[G]\) for torsion-free \(G\) is trivial (a scalar multiple of a group element). (Disproved 2021: Gardam found a counterexample over \(\mathbb{F}_2\).)

Kaplansky's conjectures on group rings — no zero divisors, no nilpotents, no nontrivial idempotents, only trivial units for torsion-free groups — stood for over half a century as the cleanest questions linking groups to their algebras. The zero-divisor conjecture fell first in line only in the sense of speculation; it was the unit conjecture Gardam broke. In 2021 Gardam exhibited a torsion-free group whose group algebra over the field of two elements contains a nontrivial unit — found by a computer search guided by geometric group theory, and small enough to check. The remaining conjectures (zero divisors, idempotents) stay open.