{
  "schema_version": 1,
  "problem_number": "KOU-17.16",
  "title": "On Bardakov's Question on the Linearity of Automorphism Groups of Artin Groups of Finite Type: A Reduction to Six Exceptional Types",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Problem 17.16 of the Kourovka Notebook, posed by V. G. Bardakov, asks whether the automorphism group Aut(A) of an Artin group A of finite type is linear, as A is by the theorems of Bigelow, Krammer, Digne and Cohen–Wales; the Notebook records the affirmative answer of Bardakov and Bryukhanov for braid groups. We give a partial answer: Aut(A) is linear for every Artin group A of finite type without components of types E_6, E_7, E_8, F_4, H_3, H_4 (type D_5 relying on an unrefereed preprint), and the question reduces to linearity of Aut(A/Z(A)) for these six irreducible types. First, Aut(A) is linear over C if and only if Aut(A_i/Z(A_i)) is linear over C for every irreducible component A_i of A. The proof assembles known facts (linearity of A, a lifting argument of Charney and Crisp, parabolic closures after Cumplido, Gebhardt, González-Meneses and Wiest, a lemma of Cumplido and Paris); what we did not find in print is the reduction itself, for reducible groups and in the converse direction. Second, for the types A_n, B_n, D_n and I_2(m), Aut(A) has a subgroup of finite index isomorphic to A/Z(A) or to A/Z(A) × Z. This follows, by the argument of Bardakov and Bryukhanov, from the published computations of Aut(A) by Dyer–Grossman, Charney–Crisp, Gilbert–Howie–Metaftsis–Raptis, Crisp–Paris, Bell–Margalit, Soroko, Castel–Paris and Paris–Soroko (for D_5 a preprint of 2026), and may be regarded as folklore. For the six exceptional types, finiteness of Out(A/Z(A)) would suffice; we found no computation of it in print. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "Artin groups of finite type",
    "spherical Artin groups",
    "automorphism groups",
    "linear groups",
    "faithful linear representations",
    "central quotients",
    "transvections",
    "parabolic closures",
    "braid groups",
    "partial answer",
    "Kourovka Notebook problems",
    "UnsolvedMath",
    "KOU-17.16",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-22",
  "publication_date": "2026-10-22",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-22",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-17-16/",
  "pdf_url": "https://eulersolve.org/papers/kou-17-16/paper.pdf?v=3a11b3f28be1",
  "doi": "10.5281/zenodo.23552948",
  "zenodo_record_url": "https://zenodo.org/records/23552948",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Open for components of types E6, E7, E8, F4, H3, H4; type D5 relies on an unrefereed preprint; the cases A, B, D, I2(m) may be regarded as folklore.",
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    "source.zip": {
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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."
}