The Parabolic Intersection Conjecture for Artin groups
A parabolic subgroup of an Artin group is a conjugate of a standard subgroup generated by a subset of the generators. The Parabolic Intersection Conjecture, formulated precisely in Godelle (2023, Conjecture 1) and Cumplido's survey (Conjecture 4), asserts that the intersection of two parabolic subgroups is parabolic. It was known for right-angled, spherical (Cumplido-Gebhardt-Gonzalez-Meneses-Wiest), large-type, some two-dimensional and some affine and FC types. Is the intersection of any two parabolic subgroups of an arbitrary finite-rank Artin group again parabolic?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Group theory, Artin groups
- Posed by
- E. Godelle, On parabolic subgroups of Artin-Tits groups, J. Algebra 632 (2023), Conjecture 1; also M. Cumplido, survey arXiv:2509.08382, Section 3, Conjecture 4
- Year posed
- 2023
- Years open
- 3y
- 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
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every Coxeter matrix on a finite set, all and , there are with . Corollary 1.2 (via Moller-Paris-Varghese): arbitrary intersections of parabolics are parabolic, already attained by at most members, and every subset has a unique parabolic closure. Corollary 1.4: irreducible Artin groups with an infinite label are acylindrically hyperbolic (already known from Kato-Oguni) with weakly malnormal proper parabolics. The proof uses a relative harmonic height for a framed zigzag braid action and does not use asphericity of the Salvetti complex.
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 with its corollaries, read against the conjecture as cited (Godelle 2023, Cumplido survey). The stabilizer and height arguments were not refereed. Lean-checked on the release's Comparator challenge ArtinParabolicIntersections, found through lean/docs/254.md; this challenge is not in the release's formalization catalogue, but its solution module OAI.Topology.ArtinGroups.ParabolicIntersections.Main exists at the pinned commit. Its statement was read here: theorem unconditional_parabolic_intersections asserts that for every finite Coxeter matrix the intersection of two conjugates of standard subgroups is a conjugate of a standard subgroup; further theorems state the arbitrary-intersection, closure and non-clique dynamics corollaries. That covers the headline. Not rebuilt here.