Kaplansky Direct-Finiteness Conjecture

The Kaplansky direct-finiteness conjecture asserts that the group algebra of any group over a field is directly finite, meaning that implies for . Unlike the Kaplansky zero-divisor conjecture, it does not assume that is a torsion-free group (Bunina et al. 2026).

Bunina et al. (2026) report that for every odd prime number there is a finitely presented group with satisfying , , and . These relations imply , since would give . The same counterexample persists over any field of field characteristic . The claimed construction would also provide a group that is not a sofic group.

The argument depends on substantial geometric and probabilistic inputs from earlier OpenAI manuscripts. Those dependencies have not been independently verified here, so the claimed refutation is not treated as established. Bunina et al. (2026) acknowledge extensive ChatGPT assistance with mathematical exploration and checking. They explicitly distinguish these checks from formal verification.