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The ℓ¹-Bass Conjecture for Discrete Groups
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Abstract
We prove the ℓ1-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ1 group algebra is supported on finitely many conjugacy classes of finite-order elements.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Roman Sauer, contribution to 100+ reactions to 100+ solutions (Proofs and Prompts, posted 2026-10-08). Sauer writes that a trick from his work with Braun and from Alpert's extension of it "now appear in OpenAI's deeply creative solutions of problems 207 and 335"; those works are cited in the companion group-ring paper, not in this one. [family-level]
Formalization
- OpenAI / mathcommunity mirror, Lean scope note 207 (release Lean docs, 2026-10-06). The family's Lean scope note covers the companion paper's complex Bass trace theorem, augmentation-rank identity and characteristic-zero idempotent consequence; it does not cover the ℓ¹(G) statement of Theorem 1.1 here. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Ronghui Ji and Bobby Ramsey, Cyclic Cohomology for Discrete Groups and its Applications (author-hosted survey, undated). Section 3.3 discusses cyclic-homological approaches to Bass trace vanishing and ℓ¹ analogues, background for the claims about support on finite-order conjugacy classes. [family-level]
Digests and expositions
- Orply / Perplexity, The ℓ¹-Bass Conjecture Restricts Idempotent Traces to Finite-Order Classes (Orply, 11 October 2026). This exposition describes the exponential-decay reduction, curvature estimates, and uniform cutoff underlying the manuscript's finite-support claim. [AI-generated per source]
Elsewhere
- Emergent Mind: family 207
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026,
author = {{OpenAI}},
title = {{The $\ell^1$-Bass Conjecture for Discrete Groups}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026/l1-bass-conjecture.pdf}{OAI:The-l1-Bass-Conjecture-for-Discrete-Groups-October-5-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/207.md. The Lean statements have not been checked by people against the paper's claims.
Family 207: The ℓ¹-Bass conjecture for all discrete groups
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $\ell^1(G)$ are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.
Other papers in this family: