{
  "schema_version": 1,
  "problem_number": "OWR-1274-007",
  "title": "On Bartholdi's Conjecture on the Closures of the Aleshin and Bellaterra Groups",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an Oberwolfach report of 2006, L. Bartholdi conjectured that the closures of the Aleshin automaton group G(A) and of the Bellaterra automaton group G(B) in the automorphism group of the binary rooted tree are Z_2-analytic, and that the closure of G(B) is a pro-2 Sylow subgroup of SL_2(Z_2). We show that the conjecture is false: the closure of the Bellaterra group is 2-adic analytic but is not isomorphic to a pro-2 Sylow subgroup of SL_2(Z_2) (nor of PSL_2(Z_2) or PGL_2(Z_2)), and the closure of the Aleshin group is not 2-adic analytic. The literal Sylow statement fails for an elementary reason: the closure of G(B) contains three distinct involutions, SL_2(Z_2) only one. The substance is in two theorems. The closure of G(B) is isomorphic to {g ∈ GL_2(Z_2) : g ≡ 1 mod 2, det g = ±1}/{±1}, an open subgroup of index 12 of PGL_2(Z_2); the proof uses a Möbius model of the Bellaterra automaton due to Kharlampovich and Vdovina (an unrefereed 2026 preprint whose acknowledgement declares AI assistance; all identities taken from it are verified again here), a lifting lemma for congruence subgroups, and a computation in groups of orders 32 and 256. The closure of G(A) has index 2 in the closure of the group generated by G(B) and the complement map, which in the Möbius coordinate is not differentiable at the fixed point of a prefix contraction; p-adic Lie theory then excludes analyticity. No printed answer to the question was found in the sources searched; two works citing the report could not be read. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "automaton groups",
    "Aleshin automaton",
    "Bellaterra automaton",
    "closures in the automorphism group of the binary tree",
    "profinite groups",
    "2-adic analytic groups",
    "pro-2 Sylow subgroups",
    "PGL2(Z2)",
    "Möbius transformations",
    "congruence subgroups",
    "conjecture of Bartholdi",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-1274-007",
    "math.GR",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1274-007/",
  "pdf_url": "https://eulersolve.org/papers/owr-1274-007/paper.pdf?v=30e9831299d0",
  "doi": "10.5281/zenodo.23585820",
  "zenodo_record_url": "https://zenodo.org/records/23585820",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Unrefereed; Siegenthaler's 2010 thesis could not be read (disclosed in the paper).",
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    "source.zip": {
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    "verification_report.md": {
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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."
}