Irrational Algebraic Values of Grabowski's Automaton Series and an Irrational Algebraic ℓ²-Betti Number of the Lamplighter Group

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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-4428-011 (Grabowski, On the Atiyah problem for the lamplighter groups, Oberwolfach Report 40/2010, Questions 4–6; the note also concerns record OWR-4428-012, Question 5).