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Parabolic intersections in Artin groups
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Abstract
We prove that every intersection of parabolic subgroups of any finite-rank Artin group, with arbitrary finite or infinite Coxeter labels, is parabolic. This resolves the Parabolic Intersection Conjecture affirmatively.
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Notes
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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.
Formalization
- OpenAI math release, Lean scope notes for family 254: Classifying spaces and geometric obstructions for Artin groups (lean/docs/254.md, undated). States that the formalization covers arbitrary parabolic intersections, the rank bound on a finite witnessing subfamily, unique parabolic closure, and acylindrical hyperbolicity and weak malnormality for irreducible matrices with an infinite label; the word-problem corollary is not listed. [family-level]
- OpenAI math release, ArtinParabolicIntersections.lean comparator statement (lean/ComparatorChallenges). The comparator statement file for Theorem 1.1 and Corollaries 1.2 and 1.4 (
unconditional_parabolic_intersections,unconditional_arbitrary_intersections,unconditional_parabolic_closure,unconditional_nonclique_dynamics).
Possibly relevant literature (found by the agent, not checked)
- F. Digne, E. Godelle, J. Michel, Retraction to a parabolic subgroup and applications (arXiv:2407.07459, 2024). Uses retractions onto standard parabolics to obtain results on minimal parabolic subgroups and intersections of parabolic subgroups; this is a prior approach to the intersection problem that the paper does not cite in its Section 1.1 survey.
- B. A. Cisneros de la Cruz, M. Cumplido, I. Foniqi, On Artin groups admitting retractions to parabolic subgroups (arXiv:2408.12291, 2024). Extends parabolic retractions to FC type and generalizes the Antolín–Foniqi intersection results; this concerns the FC-type cases the paper discusses before Theorem 1.1.
- William D. Cohen, Acylindrical Visual Splittings and The Tits Alternative For Artin Groups (arXiv:2509.03305, 3 September 2025). Theorem 1.4 classifies acylindrical visual splittings under parabolic-intersection and ribbon-property hypotheses, related to the paper's intersection theorem and its visual-splitting consequences.
- O. Garnier, E. Heng, A. Licata, O. Yacobi, Normalisers of parabolic subgroups of Artin–Tits groups and Tits cone intersections (arXiv:2509.21915, 2025). Studies normalisers of standard parabolic subgroups, by authors of the Heng–Licata categorical framework that the paper's stabilizer argument builds on.
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 254
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Parabolic-intersections-in-Artin-groups-September-23-2026,
author = {{OpenAI}},
title = {{Parabolic Intersections in Artin Groups}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026/paper.pdf}{OAI:Parabolic-intersections-in-Artin-groups-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/254.md. The Lean statements have not been checked by people against the paper's claims.
Family 254: Classifying spaces and geometric obstructions for Artin groups
The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $K(\pi,1)$ conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT$(0)$ space.
Other papers in this family:
- Harmonic heights and the Artin K(pi,1) conjecture · September 23, 2026
- An Artin group with no geometric CAT(0) action · September 23, 2026