{
  "schema_version": 1,
  "problem_number": "OWR-12007-010",
  "title": "Elliptic Values of Word Maps on SL(2,R) and a Question of Calegari and Walker on Rotation Numbers",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an Oberwolfach report of 2012, D. Calegari, reporting on joint work with A. Walker, asked whether there is a nontrivial word w in the free group F = ⟨a,b⟩ such that the lifted rotation number rot~(ρ(w)) vanishes for every homomorphism ρ from F to the universal central extension Homeo⁺(S¹)~ of the group of orientation-preserving homeomorphisms of the circle. We show that there is no such word. The main step is elementary: for every nontrivial w in the commutator subgroup of F there are A, B ∈ SL(2,R) with |tr w(A,B)| < 2. The proof combines the classical trace polynomials of Fricke and Vogt with a comparison of signs on the two sides of the surface of reducible characters. Lifting such a pair to the universal cover of PSL(2,R) gives w a rotation number that is not an integer; words outside the commutator subgroup are treated with rigid rotations. All these representations are real-analytic. For w = [a,b] the negative answer is already contained in an example of Calegari and Walker (2011). For the word w0 of length 36 printed in the report we give rational matrices A and B, both hyperbolic, with tr w0(A,B) = 0; with the lifts that have fixed points, rot~(ρ(w0)) = −1/2, and the extremal value R(w0,0,0) equals 1. We also recall, after Calegari and Walker, why rot~(ρ(w0)) = 0 whenever ρ(a) or ρ(b) is elliptic. The second question of the report, on real-analytic representations that realise the extremal rotation number R(w,r,s), is not treated. No printed answer to the first question was found in the sources we searched; this is not a claim of priority. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.GT",
    "math.DS",
    "math.GR"
  ],
  "keywords": [
    "rotation number",
    "free group",
    "circle homeomorphisms",
    "universal cover of PSL(2,R)",
    "trace polynomials",
    "Fricke–Vogt",
    "word maps",
    "elliptic elements",
    "ziggurats",
    "Calegari–Walker",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-12007-010",
    "math.GT",
    "math.DS",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12007-010/",
  "pdf_url": "https://eulersolve.org/papers/owr-12007-010/paper.pdf?v=a23938702356",
  "doi": "10.5281/zenodo.23430860",
  "zenodo_record_url": "https://zenodo.org/records/23430860",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers the first question of Calegari and Walker negatively for every nontrivial word (real-analytic representations); the second question, on real-analytic realisation of R(w,r,s), is not treated.",
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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."
}