{
  "schema_version": 1,
  "problem_number": "AMR-109-0020",
  "title": "Riemann Surfaces of Genus 9 with Nilpotent Automorphism Group of the Largest Order Are Not Unique: On a Question in Farb's Problem 2.19",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Problem 2.19 of Farb's list of problems on mapping class groups asks for properties of automorphism groups that determine a unique point of the moduli space M_g, and asks as an example whether there is a unique Riemann surface of genus g≥2 whose automorphism group is nilpotent and of the largest order possible for a nilpotent group of automorphisms of a surface of genus g. By a theorem of Zomorrodian this largest order is 16(g-1), attained only when g-1 is a power of two. We show that the surface is not unique: there are at least four pairwise non-isomorphic compact Riemann surfaces of genus 9 whose full automorphism group is a 2-group of order 128=16(g-1). The surfaces are quotients of the hyperbolic plane by torsion-free normal subgroups of the triangle group Δ(2,4,8), given by explicit presentations of the four groups of order 128, and the proof that each of them is the full automorphism group uses, besides finite computations, only that the triangle group Δ(2,3,8) is the only Fuchsian group properly containing Δ(2,4,8), which we prove. The presentations are taken from Conder's list of large group actions, and all verifications are finite computations in groups of order 128 and 3072, carried out with programs written for this note. Under the completeness of Conder's list, which two independent enumerations confirm for these genera, the number of such surfaces is 1, 1, 4, 3, 7 and 15 for g=3,5,9,17,33,65, and none for g=2 under our reading of the question. The four groups of genus 9 are also listed, with their generating vectors, in the database LMFDB. The first sentence of Problem 2.19 is open-ended and is not addressed. No statement of the result was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GT",
    "math.GR",
    "math.CV"
  ],
  "keywords": [
    "Riemann surfaces",
    "automorphism groups",
    "nilpotent groups",
    "2-groups",
    "triangle groups",
    "Fuchsian groups",
    "Zomorrodian bound",
    "moduli space",
    "canonical basepoints",
    "Farb problem list",
    "computer-assisted proof",
    "UnsolvedMath",
    "AMR-109-0020",
    "math.GT",
    "math.GR",
    "math.CV",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-109-0020/",
  "pdf_url": "https://eulersolve.org/papers/amr-109-0020/paper.pdf?v=0c0ca9b01eb5",
  "doi": "10.5281/zenodo.23365288",
  "zenodo_record_url": "https://zenodo.org/records/23365288",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the example question of Problem 2.19 of B. Farb's list (Proc. Sympos. Pure Math. 74 (2006)): the Riemann surface of genus g whose automorphism group is nilpotent of the largest possible order 16(g-1) is not unique; four pairwise non-isomorphic surfaces of genus 9 have a full automorphism group of order 128, and, under the completeness of Conder's list (confirmed for these genera by two independent enumerations), the counts are 1, 1, 4, 3, 7, 15 for g = 3, 5, 9, 17, 33, 65. The four groups are listed in the LMFDB. The open-ended first sentence of the problem is not addressed. Novelty is not certified.",
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    "source.zip": {
      "sha256": "6f0eac81b28ada23452f58cf3543a1304740831bc89c420182f5c9f1cb52084c"
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    "verification_report.md": {
      "sha256": "eceea10625f8503bd6baa9e244ff2e79017c26ca774fa17562f03cd0d2279baf"
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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."
}