$$L^2$$-Betti numbers arising from the lamplighter group

Abstract We apply a construction developed in previous paper by the authors order to obtain formula which enables us compute $$\ell ^2$$ ℓ 2 -Betti numbers coming from family of group algebras representable as crossed product algebras. As an application, we whole irrational arising lamplighter algebra $${\mathbb Q}[{\mathbb Z}_2 \wr {\mathbb Z}]$$ Q [ Z ≀ ] . This procedure is constructive, sense that one has explicit description elements realizing such numbers. extends work made Grabowski, who first computed Z}_n n , where $$n \ge 2$$ ≥ natural number. also techniques generalized odometer $${\mathcal {O}}({\overline{n}})$$ O ( ¯ ) $${\overline{n}}$$ supernatural its $$*$$ ∗ -regular closure, and this allows fully characterize set