Genuine Witnesses for Virtual ℓ²-Betti Numbers of Ara and Claramunt and Irrational Algebraic ℓ²-Betti Numbers of Z/p ≀ Z

Grabowski asked what the set \(\mathcal{C}({\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}})\) of \(\ell^2\)-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 \(\mathcal{C}(\mathbb{Z}/p\mathbin{\wr}\mathbb{Z})\) contains irrational algebraic numbers (Invent. Math. 2014). Ara and Claramunt obtained \(\frac14\sqrt{2/7}\) and \(\frac18\sqrt{2^t/(2^t-1)}\), \(t\ge3\), as virtual \(\ell^2\)-Betti numbers of \({\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}}\), that is, as differences of two \(\ell^2\)-Betti numbers, and asked for a concrete matrix whose kernel has dimension \(\frac14\sqrt{2/7}\) (J. Algebraic Combin. 2021, Remark 4.14). We give genuine witnesses for these numbers. An explicit \(T\in M_2(\mathbb{Q}[{\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}}])\) built from five lamp bits and the shift \(t^5\), with \(32T\) integral, has \(\dim_{\mathrm{vN}}\ker T=\frac1{16}\sum_{N\ge0}W_N32^{-N}=\frac14\sqrt{2/7}\), where \(W_N=\sum_k\binom N{2k}\binom{2k}k25^k14^{N-2k}\); a family \(T^{(n)}\) with \(n+2\) lamp bits gives \(\frac18\sqrt{2^n/(2^n-1)}\) for every \(n\ge2\). So every number of the family listed by Ara and Claramunt lies in \(\mathcal{C}({\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}})\); the other elements of their group of virtual values are not treated. With \(\rho_j=\frac1p(1+e_j+\dots+e_j^{p-1})\) the same template gives, for every integer \(p\ge2\), an explicit \(T_p\in M_2(\mathbb{Q}[\mathbb{Z}/p\mathbin{\wr}\mathbb{Z}])\) with \(p^2T_p\) integral and \(\dim_{\mathrm{vN}}\ker T_p=(\frac{p-1}p)^2\sqrt{(p-1)/(p+3)}\), which is irrational of degree \(2\) because \((p-1)(p+3)=(p+1)^2-4\) is never a square; for \(p=3\) it is \(4\sqrt3/27\). 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 \(\sqrt5/20\) and \(\sqrt2\) for \(p=2\). Apart from these notes, the sources we searched contain 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. A description of \(\mathcal{C}({\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}})\) remains open. This is an unrefereed note.