{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRIC_GROUP_THEORY-0044",
  "title": "Spherical Type Is an Isomorphism Invariant of Artin Groups",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We prove that an Artin group abstractly isomorphic to an Artin group of spherical type is itself of spherical type. No condition is imposed on the isomorphism or on the rank of the source presentation. The main observation is that, for an irreducible nonspherical Artin group, the center has zero image in the abelianization. It follows from Digne's finite-support reflection cocycle and the infinitude of every reflection conjugacy class in an irreducible infinite Coxeter group. For a spherical Artin group, the center injects into the abelianization, and Paris's commuting-product theorem implies that every nontrivial abstract direct factor has nontrivial center. These properties exclude all nonspherical source components. We also show that the number of spherical irreducible components of a finite-rank Artin presentation is intrinsic to the group: it is the rank of the image of the center in the abelianization. The argument does not assume the general center or K(pi,1) conjectures.\n\nThis answers AIM Geometry and Topology of Artin Groups, Section 3, Problem 3.1, recorded as AIM-GEOMETRIC_GROUP_THEORY-0044 in UnsolvedMath v1.6.0. The result uses the attributed reflection cocycle and spherical structural theorems; no absolute priority is claimed.\n\nAI-assisted, self-audited and unrefereed preprint. No independent review or formal proof-assistant verification is claimed. Exact finite regression programs accompany the source and do not replace the general written proofs.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "keywords": [
    "Artin groups",
    "Coxeter groups",
    "spherical type",
    "reflection cocycle",
    "abelianization",
    "AIM-GEOMETRIC_GROUP_THEORY-0044"
  ],
  "manuscript_version_date": "2026-10-09",
  "publication_date": "2026-10-09",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-09",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0044/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0044/paper.pdf?v=62f0c5173b06",
  "doi": "10.5281/zenodo.23249590",
  "zenodo_record_url": "https://zenodo.org/records/23249590",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "An Artin group abstractly isomorphic to a spherical Artin group is spherical, with no initial rank restriction on the source. The rank of the center's image in the abelianization counts its spherical irreducible components. The general center and K(pi,1) conjectures are not assumed or proved. Classical reflection-cocycle and spherical structural results retain attribution. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
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      "sha256": "62f0c5173b067a58e6cb1d2aee0f026882c6c53ca3a8760abc60a0409229cdb2"
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    "source.zip": {
      "sha256": "f9d57fd3a5b2ac4b6f4cd161100135cdeead886ff01fa1b623a4f90a4e8833c9"
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    "verification_report.md": {
      "sha256": "b0f75809a85cc0ac7a237850d9432988495ce470113ac3f92592a2603a4f89bc"
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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."
}