Bass trace vanishing for complex group rings
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For every discrete group G, must the Hattori–Stallings trace of each class in K₀(CG) vanish on all infinite-order conjugacy classes? This complex group-ring form of the Bass trace question is distinct from an integral identity-only assertion.
Why it matters and what counts as progress
Why it matters: The unrestricted complex trace-support question reaches beyond linear groups and cannot be replaced by an integral statement without extra input.
What counts as progress: Prove infinite-order vanishing for further group classes or all discrete groups, or construct a trace class violating the support condition.
How the problem developed
- 1976 · Conjecture [1]
Hyman Bass Formulated the integral Bass trace conjecture for Hattori–Stallings traces of projective group-ring modules.
- 2004 · Partial result [2]
Paul Baum, Indira Chatterji and Thomas Schick Established related trace-vanishing results for amenable groups; the arbitrary-group complex group-ring statement remained open.
- 24 Sept 2026 · Reported claim [3]
OpenAI The preprint claims that Hattori–Stallings trace coefficients vanish on all infinite-order conjugacy classes for every discrete group over ℂ.
AI contribution
The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
For every discrete group G, OpenAI claims that the Hattori–Stallings trace of every class in K₀(C G) vanishes on each infinite-order conjugacy class. The claim concerns the complex group ring; an integral identity-only consequence needs a separate known finite-order input.
Sources
- Bass, Euler characteristics and characters of discrete groups
- Berrick, Chatterji and Mislin, From acyclic groups to the Bass conjecture
- OpenAI, The Bass trace conjecture for complex group rings
- The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
- Lean proof of the Bass trace support theorem
- Bass trace comparator and permitted axioms
- OpenAI mathematical research repository
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