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

Grabowski asked whether √2 is an ℓ²-Betti number arising from the lamplighter group Z/2 ≀ 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' ∈ M_2(Q[Z/2 ≀ Z]) with 8T' integral we prove dim_vN ker T' = (1/4) Σ_{N≥0} C(2N,N) 8^(−N) = √2/4, by Pontryagin duality and again by Lück's approximation theorem with an exact formula on the finite quotients Z/2 ≀ Z/3P; four copies of 8T' form an integral 8 × 8 matrix whose kernel has dimension √2. Also k√2/4 (k ≥ 1) arises from Z/2 ≀ Z, and √2 arises from Z/p ≀ 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 (1/4)√(2/7) and (1/8)√(2^t/(2^t − 1)), t ≥ 3, as virtual ℓ²-Betti numbers of Z/2 ≀ Z, that is, as differences of two ℓ²-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 (1/4)√(2/7), which we do not provide. It is also the mechanism of our companion note, which shows that √5/20 arises from Z/2 ≀ 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 ℓ²-Betti number of a group Z/p ≀ Z; a search that finds nothing is not a proof of novelty. A description of the set of ℓ²-Betti numbers arising from Z/2 ≀ Z remains 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-4428-012 (Grabowski, On the Atiyah problem for the lamplighter groups, Oberwolfach Report 40/2010, Question 5; the note answers the concrete question of Problem 4 of Grabowski, Groups Geom. Dyn. 10 (2016), and the record remains partially solved).