The CAT(0) conjecture for Artin groups
Charney asked which Artin groups are CAT(0) groups (Problem 4 in her Artin-group problem list), and the affirmative expectation that every finite-rank Artin group acts geometrically, that is properly, cocompactly and by isometries, on a proper CAT(0) space is recorded by Haettel (2022) as Conjecture 1.1. It is known for right-angled Artin groups, three-generator large-type groups (Brady-McCammond), XXL type (Haettel) and braid groups on up to seven strands. Does every finite-rank Artin group admit a geometric action on a proper CAT(0) space?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Geometric group theory, CAT(0) groups
- Posed by
- Question: R. Charney, Problems related to Artin groups, Problem 4 (undated); conjecture recorded by T. Haettel, Ann. Inst. Fourier 72 (2022), Conjecture 1.1
- Year posed
- 2022
- Years open
- 4y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 40 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is an explicit Artin matrix on 116 generators, with off-diagonal labels in , whose Artin group admits no proper cocompact isometric action on any nonempty proper CAT(0) space, in any dimension. The group contains the triangle Artin group with three braid relations, extended by three 40-strand braid blocks and two commuting generators making a simultaneous centralizer; stable translation lengths force six Euclidean sectors of angle less than around a cycle, which a winding-number argument rules out. It does not decide which Artin groups are CAT(0).
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against Charney's Problem 4 and Haettel's Conjecture 1.1 as cited. The sector and winding arguments were not refereed. Lean-checked on the release's Comparator challenge ArtinCAT0 (declaration OAI.ArtinCAT0.main, listed in lean/formalization.yaml). Its statement was read here: it defines the explicit 116-letter matrix, checks symmetry, unit diagonal and labels in {2,3,top}, and asserts that for every nonempty proper metric space with an action of the Artin group, the action is not geometric, with CAT(0) defined by the comparison inequality along linear segments and geometric meaning isometric, proper and cocompact. That is the headline claim. Not rebuilt here.