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An Artin group with no geometric CAT(0) action

OpenAI · September 23, 2026

Lean

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Abstract

We construct an Artin group on 116 generators that admits no proper, cocompact isometric action on a nonempty proper CAT(0) space. This refutes the CAT(0) conjecture for Artin groups.

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Notes

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Automatically collected notes (unreviewed)

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.

Follow-up works

  • frogscooper (GitHub handle), artin-cat0-29 (GitHub repository, accessed 2026-10-21). Presents an edited version of this paper's argument with 11-strand instead of 40-strand braid blocks, reducing the example from 116 to 29 generators, with an adapted copy of the release's Lean project reported as axiom-clean, plus an 11-generator variant that the author says assumes a special case of van der Lek's theorem; the author says the new argument and Lean port were worked out with help from Claude. [AI-generated per source]

Comments and discussion

  • Giulio Tiozzo, contribution to 100+ reactions to 100+ solutions (Proofs and Prompts, 8 October 2026; updated 19 October 2026). Writes "I took their example #254 (an Artin group with no CAT(0) action): their group has 116 generators" and reports that Astra produced a 12-generator example in about ten minutes, which he uses to argue the papers were not reviewed by a human.

Formalization

Possibly relevant literature (found by the agent, not checked)

News and blogs

  • David Roberts, What even is mathematical output? (theHigherGeometer, 12 October 2026). Discusses Tiozzo's reported reduction of the CAT(0) example from 116 to 12 generators as part of a criticism of the release's exposition and refinement process.

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026,
  author = {{OpenAI}},
  title = {{An Artin group with no geometric CAT(0) action}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026/paper.pdf}{OAI:An-Artin-group-with-no-geometric-CAT-0-action-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: