{
  "schema_version": 1,
  "problem_number": "AMR-011-0004",
  "title": "Adjoining a Haar-Generic Matrix to a Parabolic-Free Subgroup of SL₂(Qₚ)",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let Γ be a countable subgroup of SL₂(ℚ_p) containing no noncentral element of trace 2 or −2. We prove that, outside a Haar-null set of g ∈ SL₂(ℚ_p), the generated group ⟨Γ,g⟩ has the same property. Thus adjoining one random element preserves the absence of parabolics almost surely, answering Question 4 in Miklós Abért's 2010 list. The key elementary lemma classifies the problematic one-variable generalized words: if all constants lie in a parabolic-free subgroup of SL₂ and the trace of the word is identically 2 or −2, then the word map is identically I or −I. The proof evaluates the word on I+tN over the nilpotent cone. Its two highest coefficients force a central endpoint product and cancellation of the outer exponents, reducing the word by conjugation and induction. This manuscript is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR",
    "math.PR"
  ],
  "keywords": [
    "p-adic groups",
    "SL₂",
    "Haar measure",
    "parabolic elements",
    "word maps",
    "AMR-011-0004",
    "open mathematics",
    "mathematical proof",
    "math.GR",
    "math.PR"
  ],
  "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/amr-011-0004/",
  "pdf_url": "https://eulersolve.org/papers/amr-011-0004/paper.pdf?v=7f7962f17182",
  "doi": "10.5281/zenodo.22245813",
  "zenodo_record_url": "https://zenodo.org/records/22245813",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
      "sha256": "467137e85072fb5b71421cb3448b277c85081e43d38b83bcf0d3bfe9a3397d6e"
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    "verification_report.md": {
      "sha256": "45af30375a0f62906ca42f93f62b85592a727c494a5060378367530e27fc4a24"
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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."
}