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 \(\mathcal{C}(G)\) of \(\ell^2\)-Betti numbers arising from a group \(G\), that is, of von Neumann dimensions of kernels of square matrices over \(\mathbb{Q} G\). Question 4 asks whether \(\mathcal{C}(G)\not\subseteq\mathbb{Q}\) holds exactly when \(G\) contains a wreath product \(\mathbb{Z}/p\mathbin{\wr} K\) with \(K\) infinite and finitely generated; Question 5 asks what \(\mathcal{C}(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z})\) is; Question 6 asks which values the series \(F_A(x)=\sum_i\dim\ker(M_i)\,x^{d(M_i)}\), summed over the band matrices \(M_i\) recognized by a finite automaton \(A\) whose letters are integer \(n\times n\) 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 \(\{-2,\dots,2\}\) recognizes tridiagonal matrices that encode a two-track multiplicative counter, \(F_{A_2}(x)=x^2(1-4x^4)^{-1/2}\), and \(F_{A_2}(\frac12)=\sqrt3/6\). 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 \(T\in M_2(\mathbb{Q}[\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}])\) with \(4T\) integral and \(\dim_{\mathrm{vN}}\ker T=\sqrt5/20\), with two proofs: by Pontryagin duality, and by Lück's approximation theorem with an exact formula on the finite quotients \(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}/K\). This answers, for \(p=2\), Grabowski's question (Invent. Math. 2014) whether \(\mathcal{C}(\mathbb{Z}/p\mathbin{\wr}\mathbb{Z})\) contains irrational algebraic numbers; it does not decide whether \(\sqrt2\in\mathcal{C}(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z})\) (Groups Geom. Dyn. 2016, Problem 4). The questions and the automaton framework are Grabowski's. Ara and Claramunt obtained irrational algebraic virtual \(\ell^2\)-Betti numbers of \(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}\), that is, differences of two \(\ell^2\)-Betti numbers, from a closely related letter-count language. In the sources we searched we found no earlier non-virtual irrational algebraic \(\ell^2\)-Betti number of a group \(\mathbb{Z}/p\mathbin{\wr}\mathbb{Z}\); a search that finds nothing is not a proof of novelty. Question 4, a description of \(\mathcal{C}(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z})\), and the set of values of the series \(F_A\) remain open. This is an unrefereed note.