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The word problem for finite-rank Artin groups

An Artin group is given by generators σs\sigma_s, s∈Ss\in S, and braid relations equating the alternating words of length mstm_{st} in σs,σt\sigma_s,\sigma_t. Garside, Brieskorn-Saito and Deligne solved the word problem for spherical (finite-type) Artin groups, and it is known for further classes such as large type (Holt-Rees) and diagrams without A3A_3 or B3B_3 subdiagrams (Blasco-Garcia et al. 2026), but no algorithm was known in general. Charney's problem list asks (Problem 10) for solvable word and conjugacy problems in infinite type, and Blasco-Garcia, Cumplido, Holt, Morris-Wright and Rees pose the general decidability question explicitly. Is the word problem decidable for every Artin group with finitely many generators?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Combinatorial group theory, Artin groups
Posed by
R. Charney, Problems related to Artin groups, Problem 10 (word and conjugacy problems); posed explicitly in Blasco-Garcia, Cumplido, Holt, Morris-Wright and Rees, J. Algebra 703 (2026)
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Corollary 1.3 of the parabolic-intersections manuscript: there is an algorithm that, given a finite Coxeter matrix MM, a subset XX and words u,vu,v, decides whether v∈uAXu−1v\in uA_Xu^{-1}; with X=∅X=\varnothing this decides the word problem for every finite-rank Artin group. The algorithm uses the framed zigzag-algebra action, shown faithful in Section 8, computed exactly over a cyclotomic field, and tests one transported projective by Gaussian elimination. No complexity bound is asserted, and the conjugacy problem is not treated.

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: Corollary 1.3, its discussion in the introduction and the start of its proof in Section 8, read against the question as the manuscript cites it. The faithfulness of the action, on which everything rests, was not refereed. Not Lean-checked: the release's Comparator challenge for this manuscript (ArtinParabolicIntersections) states the intersection, closure and dynamics results but not the membership algorithm or faithfulness, so this corollary has no formal counterpart. It is a corollary inside a manuscript whose headline is a different conjecture.

Sources

Changelog1 change

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