{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRIC_GROUP_THEORY-0027",
  "title": "Free-by-Cyclic Groups with Unboundedly Many BNS Component Orbits",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "An AIM problem asks for families of free-by-cyclic groups with first Betti number greater than two and many connected components of the Bieri–Neumann–Strebel invariant, even after quotienting by the outer automorphism group. For every m ≥ 2 we construct a linearly growing UPG automorphism of F_{2m+1} whose mapping torus G_m has b₁(G_m) = m+2 and whose BNS invariant is the complement of the m+1 character hyperplanes x+iy = 0 for 0 ≤ i ≤ m. It consequently has exactly 2m+2 connected components. For each component C, we minimize the rank of the kernel of a primitive integral character in C. This minimum is invariant under the full Out(G_m) action. Its m+1 distinct values prove that the components form at least m+1 outer-automorphism orbits. The construction therefore answers the AIM request in the quantitatively unbounded sense. This note is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "keywords": [
    "free-by-cyclic groups",
    "Bieri–Neumann–Strebel invariant",
    "outer automorphism groups",
    "mapping tori",
    "UPG automorphisms",
    "AIM-GEOMETRIC_GROUP_THEORY-0027",
    "open mathematics",
    "mathematical proof",
    "math.GR",
    "math.GT"
  ],
  "manuscript_version_date": "2026-08-29",
  "publication_date": "2026-09-02",
  "publication_date_kind": "first public online release",
  "version": "1.0 (typesetting revision 2026-09-05)",
  "date_modified": "2026-09-05",
  "presentation_revision_only": true,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0027/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0027/paper.pdf?v=7468ce9b6ed6",
  "doi": "10.5281/zenodo.22245655",
  "zenodo_record_url": "https://zenodo.org/records/22245655",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "An anonymous public candidate released 2026-08-31 (DOI 10.5281/zenodo.22201487) contains the same construction and conclusions; the manuscript cites it and makes no novelty or public-priority claim.",
  "files": {
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      "sha256": "7468ce9b6ed62ed648dddc833a0af23c3e166d9e07574a68428450563a8414a5"
    },
    "source.zip": {
      "sha256": "23abfb0ace123c720fc1333e3dd2e37bdd818030e853ebb887ddc3e0580d7def"
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    "verification_report.md": {
      "sha256": "6d47f3cce8684b88cd576eee50e8818272c1a5877b3da4a30ef866ccee0a5380"
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  },
  "ai_use_disclosure": "AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text."
}