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CAT(0) actions of Artin groups

Does every finite-rank Artin group act properly and cocompactly by isometries on a nonempty proper CAT(0) metric space? There is no fixed bound on the dimension of the space in this geometric CAT(0) question.

Why it matters and what counts as progress

Why it matters: The question tests whether every finite-rank Artin group admits a global nonpositively curved geometric model.

What counts as progress: Construct a geometric action for a new class, or a finite-rank Artin group admitting no such action in any dimension.

How the problem developed

  1. 2001 · Partial result [1]

    Noel Brady and John Crisp Studied CAT(0) geometry for three-generator Artin groups and showed that some require CAT(0) dimension greater than two while acting geometrically on a nonpositively curved three-complex.

  2. 23 Sept 2026 · Reported counterexample [2]

    OpenAI The manuscript presents an Artin group with no proper cocompact isometric action on any proper CAT(0) space, refuting the geometric CAT(0) action conjecture.

AI contribution

Sources

  1. Brady and Crisp, Two-dimensional Artin groups with CAT(0) dimension three
  2. OpenAI Artin CAT(0) manuscript
  3. An Artin group with no geometric CAT(0) action
  4. Lean proof of the Artin CAT(0) obstruction
  5. OpenAI mathematics release account
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Publication dates and editorial changes.