{
  "schema_version": 1,
  "problem_number": "AMR-109-0082",
  "title": "Unbounded Multiplicity in the Dilatation Spectrum of Every Term of the Johnson Filtration: An Observation on a Question of Farb",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Farb asked (Question 7.6 of \"Some problems on mapping class groups and moduli space\") whether spec(I_g(k)), the set of the logarithms of the dilatations of the pseudo-Anosov elements of the k-th term of the Johnson filtration of the mapping class group Mod_g, has bounded multiplicity for k ≥ 3; the multiplicity of a value is the number of conjugacy classes of Mod_g which realise it. We observe that the question is answered in the negative: for every g ≥ 2, k ≥ 1 and N there are N pseudo-Anosov classes in I_g(k) with one dilatation which are pairwise non-conjugate in Mod_g, also up to inversion and in the extended mapping class group. This is an observation: the negative answer follows from the mechanism which the source itself gives for the Torelli group and the Johnson kernel (Thurston's representation of the group generated by the twists about two filling curves, equal traces of non-conjugate elements, the two curves chosen separating), applied to a two-generator subgroup of a deep term of the lower central series of that group, which the source itself places in I_g(k), with a reference to Farb, Leininger and Margalit. What the note adds is the comparison of conjugacy in the Fuchsian subgroup with conjugacy in the mapping class group, which the source leaves implicit also for k ≤ 2, written out with proofs, and explicit instances, which are computer-assisted, in exact integer arithmetic: for example, 64 classes in I_2(4) have one dilatation λ, where λ + 1/λ is an integer with 5116 digits. The answer is apparently not recorded in the literature accessible to us, and it may well be known to specialists. Question 7.7 of the source, on the simple length spectrum, is not answered. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "keywords": [
    "mapping class group",
    "Johnson filtration",
    "Torelli group",
    "pseudo-Anosov",
    "dilatation",
    "stretch factor",
    "length spectrum of moduli space",
    "multiplicity",
    "Thurston construction",
    "square-tiled surfaces",
    "trace identities",
    "negative answer (observation)",
    "Farb problem list",
    "UnsolvedMath",
    "AMR-109-0082",
    "math.GT",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-109-0082/",
  "pdf_url": "https://eulersolve.org/papers/amr-109-0082/paper.pdf?v=f04bac1ae0f6",
  "doi": "10.5281/zenodo.23301731",
  "zenodo_record_url": "https://zenodo.org/records/23301731",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "An observation on Question 7.6 of B. Farb's problem list (Proc. Sympos. Pure Math. 74 (2006)): the negative answer follows from the mechanism which the source itself gives for the Torelli group and the Johnson kernel, applied to a two-generator subgroup deep in the lower central series, a construction of Farb, Leininger and Margalit. The note adds the comparison of conjugacy classes, written out with proofs, and explicit instances, which are computer-assisted. The general theorem uses three facts quoted from Leininger (2004); the original papers behind them were not read. The answer may well be known to specialists; novelty is not certified, and it cannot be excluded that the question was meant in another sense. Question 7.7 of the source (simple length spectrum) is not answered.",
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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."
}