The Square Root of Two Is an ℓ²-Betti Number Arising from the Lamplighter Group

Grabowski asked whether \(\sqrt2\) is an \(\ell^2\)-Betti number arising from the lamplighter group \(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}\), that is, the von Neumann dimension of the kernel of a matrix over its integral group ring (Groups Geom. Dyn. 2016, Problem 4). We show that it is. For an explicit matrix \(T'\in M_2(\mathbb{Q}[\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}])\) with \(8T'\) integral we prove \(\dim_{\mathrm{vN}}\ker T'=\frac14\sum_{N\ge0}\binom{2N}N8^{-N}=\frac{\sqrt2}4\), by Pontryagin duality and again by Lück's approximation theorem with an exact formula on the finite quotients \(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}/3P\); four copies of \(8T'\) form an integral \(8\times8\) matrix whose kernel has dimension \(\sqrt2\). Also \(k\sqrt2/4\) \((k\ge1)\) arises from \(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}\), and \(\sqrt2\) arises from \(\mathbb{Z}/p\mathbin{\wr}\mathbb{Z}\) for every even \(p\). The question and the framework are Grabowski's. The mechanism, balanced words counted by central binomial coefficients, is the one by which 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 (J. Algebraic Combin. 2021, Theorem 4.13); in their Remark 4.14 they ask for a genuine witness, a concrete matrix whose kernel has dimension \(\frac14\sqrt{2/7}\), which we do not provide. It is also the mechanism of our companion note, which shows that \(\sqrt5/20\) arises from \(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}\) and whose lemmas we use. Lück's theorem and Pontryagin duality are standard. Apart from that companion note, 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 the set of \(\ell^2\)-Betti numbers arising from \(\mathbb{Z}/2\mathbin{\wr}\mathbb{Z}\) remains open. This is an unrefereed note.