{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRIC_GROUP_THEORY-0059",
  "title": "Largest Acylindrical Actions for Artin Groups under a Strict Triple Condition",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let A_S be a finitely generated Artin group such that the sum of the reciprocals of the three labels on every triple of generators is strictly less than one, with 1/infinity = 0. This note records a direct consequence of the Martin-Przytycki coned-off Deligne construction and the domination criterion of Abbott-Balasubramanya-Osin. The coned-off action dominates every acylindrical action on a hyperbolic space precisely when every standard generator has a finite-labelled neighbour. In that case it is a largest acylindrical hyperbolic structure. Isolated generators obstruct this particular action, but the known free-product combination theorem still gives a largest action for the group. The argument includes mixed label-2 presentations; the large-type overlap is already covered by existing work. This is an attributed corollary note, not a solution of the general AIM Question 4.5 for arbitrary nonspherical irreducible Artin groups (dataset record AIM-GEOMETRIC_GROUP_THEORY-0059). The general domination criterion, Deligne construction and free-product combination are credited, not claimed as new. This AI-assisted preprint is self-audited and unrefereed. No independent review, formal proof-assistant verification, originality certificate or absolute priority is claimed. Exact finite-presentation regression checks accompany the source; they are not a verification of the infinite-group proof.",
  "result_type": "COMPLETE_SCOPED_COROLLARY",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "Artin groups",
    "acylindrical actions",
    "largest hyperbolic structures",
    "Deligne complex",
    "universal ellipticity",
    "free products"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0059/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0059/paper.pdf?v=022f0014e20c",
  "doi": "10.5281/zenodo.23317442",
  "zenodo_record_url": "https://zenodo.org/records/23317442",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete attributed corollary for finite Artin presentations satisfying the strict reciprocal-sum condition on every triple. The broad AIM Question 4.5 remains open. The unmodified coned-off action is strongly largest exactly without isolated standard generators; a known free-product construction gives a largest action otherwise. ABO, MP and Osin are credited, and the known HMS large-type overlap is disclosed. AI-assisted, self-audited and unrefereed; novelty is not certified and no independent review, formal verification or absolute priority is claimed.",
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      "sha256": "022f0014e20c2c7093b5a5db343b8d497ae3628dcd118c9c6fa1ecae7f77a8d6"
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    "source.zip": {
      "sha256": "b60da34f4850e3cd47cca83274c9cc7a2f70a33e7ddc3e0a19eea06c88e03bb4"
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    "verification_report.md": {
      "sha256": "b1d1668272bdca592873728d3a87e3ee0c037795efe991e5171fd88cf46a641d"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}