{
  "schema_version": 1,
  "problem_number": "OWR-4428-012",
  "title": "The Square Root of Two Is an ℓ²-Betti Number Arising from the Lamplighter Group",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Grabowski asked whether √2 is an ℓ²-Betti number arising from the lamplighter group Z/2 ≀ Z, that is, the von Neumann dimension of the kernel of a matrix over its integral group ring (Groups Geom. Dyn. 2016, Problem 4). We show that it is. For an explicit matrix T' ∈ M_2(Q[Z/2 ≀ Z]) with 8T' integral we prove dim_vN ker T' = (1/4) Σ_{N≥0} C(2N,N) 8^(−N) = √2/4, by Pontryagin duality and again by Lück's approximation theorem with an exact formula on the finite quotients Z/2 ≀ Z/3P; four copies of 8T' form an integral 8 × 8 matrix whose kernel has dimension √2. Also k√2/4 (k ≥ 1) arises from Z/2 ≀ Z, and √2 arises from Z/p ≀ Z for every even p. The question and the framework are Grabowski's. The mechanism, balanced words counted by central binomial coefficients, is the one by which Ara and Claramunt obtained (1/4)√(2/7) and (1/8)√(2^t/(2^t − 1)), t ≥ 3, as virtual ℓ²-Betti numbers of Z/2 ≀ Z, that is, as differences of two ℓ²-Betti numbers (J. Algebraic Combin. 2021, Theorem 4.13); in their Remark 4.14 they ask for a genuine witness, a concrete matrix whose kernel has dimension (1/4)√(2/7), which we do not provide. It is also the mechanism of our companion note, which shows that √5/20 arises from Z/2 ≀ Z and whose lemmas we use. Lück's theorem and Pontryagin duality are standard. Apart from that companion note, the sources we searched contain no earlier non-virtual irrational algebraic ℓ²-Betti number of a group Z/p ≀ Z; a search that finds nothing is not a proof of novelty. A description of the set of ℓ²-Betti numbers arising from Z/2 ≀ Z remains open. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.GR",
    "math.OA",
    "math.CO"
  ],
  "keywords": [
    "Atiyah problem",
    "l2-Betti numbers",
    "von Neumann dimension",
    "lamplighter group",
    "wreath product",
    "square root of two",
    "Grabowski Problem 4",
    "central binomial coefficients",
    "counter matrices",
    "Pontryagin duality",
    "Lück approximation",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-4428-012",
    "math.GR",
    "math.OA",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-14",
  "publication_date": "2026-10-14",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-14",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-4428-012/",
  "pdf_url": "https://eulersolve.org/papers/owr-4428-012/paper.pdf?v=6cab6b252525",
  "doi": "10.5281/zenodo.23380620",
  "zenodo_record_url": "https://zenodo.org/records/23380620",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Problem 4 of Ł. Grabowski (Groups Geom. Dyn. 2016) affirmatively: an explicit matrix over Q[Z/2 wr Z] has kernel of von Neumann dimension sqrt(2)/4, and four copies give sqrt(2); record OWR-4428-012 (what is C(Z/2 wr Z)?) stays partially solved. The mechanism is the one by which Ara and Claramunt obtained irrational algebraic virtual values, and that of the companion note (sqrt(5)/20). Novelty is not certified.",
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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."
}