{
  "schema_version": 1,
  "problem_number": "OWR-4428-011",
  "title": "Irrational Algebraic Values of Grabowski's Automaton Series and an Irrational Algebraic ℓ²-Betti Number of the Lamplighter Group",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an Oberwolfach report of 2010, Ł. Grabowski asked three questions about the set C(G) of ℓ²-Betti numbers arising from a group G, that is, of von Neumann dimensions of kernels of square matrices over QG. Question 4 asks whether C(G) ⊄ Q holds exactly when G contains a wreath product Z/p ≀ K with K infinite and finitely generated; Question 5 asks what C(Z/2 ≀ Z) is; Question 6 asks which values the series F_A(x) = Σ_i dim ker(M_i) x^{d(M_i)}, summed over the band matrices M_i recognized by a finite automaton A whose letters are integer n × n matrices, takes at rational points, and in particular whether such a value can be irrational and algebraic. We answer the last part of Question 6: an explicit automaton A_2 with n = 2 and entries in {−2, …, 2} recognizes tridiagonal matrices that encode a two-track multiplicative counter, F_{A_2}(x) = x²(1 − 4x⁴)^(−1/2), and F_{A_2}(1/2) = √3/6. Every non-negative rational number is a value, and the values obtained from regular languages with one linear letter-count constraint are algebraic by Furstenberg's theorem on diagonals; the set of values is not determined. Transplanting the counter into the group ring, we construct an explicit T ∈ M_2(Q[Z/2 ≀ Z]) with 4T integral and dim_vN ker T = √5/20, with two proofs: by Pontryagin duality, and by Lück's approximation theorem with an exact formula on the finite quotients Z/2 ≀ Z/K. This answers, for p = 2, Grabowski's question (Invent. Math. 2014) whether C(Z/p ≀ Z) contains irrational algebraic numbers; it does not decide whether √2 ∈ C(Z/2 ≀ Z) (Groups Geom. Dyn. 2016, Problem 4). The questions and the automaton framework are Grabowski's. Ara and Claramunt obtained irrational algebraic virtual ℓ²-Betti numbers of Z/2 ≀ Z, that is, differences of two ℓ²-Betti numbers, from a closely related letter-count language. In the sources we searched we found 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. Question 4, a description of C(Z/2 ≀ Z), and the set of values of the series F_A remain 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",
    "finite automata",
    "regular languages",
    "kernel dimension",
    "central binomial coefficients",
    "central trinomial coefficients",
    "Pontryagin duality",
    "Lück approximation",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-4428-011",
    "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-4428-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-4428-011/paper.pdf?v=5c7ef8c931b7",
  "doi": "10.5281/zenodo.23377729",
  "zenodo_record_url": "https://zenodo.org/records/23377729",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Partial answer to the questions of Ł. Grabowski (Oberwolfach Report 40/2010): an explicit automaton whose series takes the irrational algebraic value sqrt(3)/6 at 1/2 (Question 6), and an explicit matrix over Q[Z/2 wr Z] whose kernel has von Neumann dimension sqrt(5)/20, an irrational algebraic l2-Betti number of the lamplighter group (Question 5, record OWR-4428-012). Question 4, a description of C(Z/2 wr Z) and the value set of the series remain open. Ara and Claramunt obtained irrational algebraic virtual values by a related idea. 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."
}