{
  "schema_version": 1,
  "problem_number": "AMR-109-0231",
  "title": "An Algorithm for the Exponent in Mosher's Theorem on Schottky Subgroups of Mapping Class Groups",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let S be a closed oriented surface of genus at least 2. By a theorem of Mosher (1997), for pairwise independent pseudo-Anosov mapping classes φ1, …, φn of S there is an integer M such that for all integers α1, …, αn ≥ M the classes φi^αi freely generate a subgroup H whose surface group extension Γ_H is word hyperbolic. Problem 3.3 of Mosher's chapter in Farb's problem book (2006) asks whether there is an algorithm which produces M from φ1, …, φn. We answer yes, as a promise problem and modulo published theorems which we list. An explicit search, which reads off finitely many exact distances in the curve graph C(S), halts on every admissible input and outputs an integer M such that for all α1, …, αn ≥ M, chosen independently of each other, the powers φi^αi freely generate a free group of rank n whose orbit map into C(S) is a quasi-isometric embedding; hence H is convex cocompact and Γ_H is word hyperbolic. On inputs that violate the promise the search may run forever, but an output, if there is one, is always correct. Correctness rests on an explicit four-point constant δ = 340 for C(S), derived from the constant 17 of Hensel, Przytycki and Webb, a local-to-global lemma for broken sequences and a lemma on periodic sequences; termination uses the positive translation length of Masur and Minsky, the Morse lemma and Mosher's theorem itself. For n ≥ 3 no M depending only on S exists. The novelty is modest. Every ingredient was in print, and the closest prior work, Kim (2020), gives a semi-algorithm which certifies convex cocompactness of one given subgroup by the same local-to-global method in C(S); what we add is a written, complete proof with a procedure that is uniform in the exponents, and experts may regard the answer as folklore. The constants are crude (δ = 340; M is huge and far from optimal), and the algorithm has not been implemented: no computation in a curve graph was run. The only computations are tests in model spaces (a tree, finite graphs, Cayley graphs of free products of cyclic groups, the hyperbolic plane and hyperbolic 3-space), which are not part of the proof; the test in a tree, where δ = 0, checks only the bookkeeping of the procedure. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "keywords": [
    "mapping class groups",
    "pseudo-Anosov mapping classes",
    "Schottky subgroups",
    "convex cocompact subgroups",
    "surface group extensions",
    "curve graph",
    "local-to-global principle",
    "algorithm",
    "Mosher's Problem 3.3",
    "Problems on Mapping Class Groups and Related Topics",
    "complete answer modulo published theorems",
    "UnsolvedMath",
    "AMR-109-0231",
    "math.GT",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-109-0231/",
  "pdf_url": "https://eulersolve.org/papers/amr-109-0231/paper.pdf?v=19b927e412aa",
  "doi": "10.5281/zenodo.23489383",
  "zenodo_record_url": "https://zenodo.org/records/23489383",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Modulo published theorems listed in the note; the constants are crude (delta = 340), the algorithm is not implemented, and on inputs violating the promise the search may not halt.",
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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."
}