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

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