{
  "schema_version": 1,
  "problem_number": "OWR-4428-012-B",
  "title": "Genuine Witnesses for Virtual ℓ²-Betti Numbers of Ara and Claramunt and Irrational Algebraic ℓ²-Betti Numbers of Z/p ≀ Z",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Grabowski asked what the set C(Z/2 ≀ Z) of ℓ²-Betti numbers arising from the lamplighter group is, that is, of von Neumann dimensions of kernels of square matrices over its rational group ring, and whether C(Z/p ≀ Z) contains irrational algebraic numbers (Invent. Math. 2014). 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, and asked for a concrete matrix whose kernel has dimension (1/4)√(2/7) (J. Algebraic Combin. 2021, Remark 4.14). We give genuine witnesses for these numbers. An explicit T ∈ M_2(Q[Z/2 ≀ Z]) built from five lamp bits and the shift t^5, with 32T integral, has dim_vN ker T = (1/16) Σ_{N≥0} W_N 32^(−N) = (1/4)√(2/7), where W_N = Σ_k C(N,2k) C(2k,k) 25^k 14^(N−2k); a family T^(n) with n + 2 lamp bits gives (1/8)√(2^n/(2^n − 1)) for every n ≥ 2. So every number of the family listed by Ara and Claramunt lies in C(Z/2 ≀ Z); the other elements of their group of virtual values are not treated. With ρ_j = (1/p)(1 + e_j + ⋯ + e_j^(p−1)) the same template gives, for every integer p ≥ 2, an explicit T_p ∈ M_2(Q[Z/p ≀ Z]) with p²T_p integral and dim_vN ker T_p = ((p−1)/p)² √((p−1)/(p+3)), which is irrational of degree 2 because (p−1)(p+3) = (p+1)² − 4 is never a square; for p = 3 it is 4√3/27. All values are proved twice: by Pontryagin duality with a decomposition into counter matrices, and by Lück's approximation theorem with an exact formula on finite quotients. The questions and the frameworks are Grabowski's and Ara and Claramunt's; the balanced-word counter is the mechanism of Ara and Claramunt and of our two companion notes, which gave √5/20 and √2 for p = 2. Apart from these notes, 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 C(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",
    "virtual l2-Betti numbers",
    "Ara–Claramunt Remark 4.14",
    "Grabowski question",
    "irrational algebraic numbers",
    "balanced words",
    "counter matrices",
    "Pontryagin duality",
    "Lück approximation",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-4428-012",
    "math.GR",
    "math.OA",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-4428-012-b/",
  "pdf_url": "https://eulersolve.org/papers/owr-4428-012-b/paper.pdf?v=99179b55c7fc",
  "doi": "10.5281/zenodo.23391943",
  "zenodo_record_url": "https://zenodo.org/records/23391943",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Second note on record OWR-4428-012 (what is C(Z/2 wr Z)?): explicit matrices realise Ara and Claramunt's virtual values (1/4)sqrt(2/7) and (1/8)sqrt(2^n/(2^n-1)), n at least 2, as genuine l2-Betti numbers of the lamplighter group, answering their Remark 4.14 for the listed family, and give irrational algebraic elements of C(Z/p wr Z) for every p at least 2, answering Grabowski's 2014 question. The record stays partially solved.",
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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."
}