K-theory and K-homology of the wreath products of finite with free groups

Consider the wreath product $\Gamma = F\wr ${\mathrm{F}}_{\mathrm{n}}$ = \bigoplus_{${\mathrm{F}}_{\mathrm{n}}$}F\rtimes${\mathrm{F}}_{\mathrm{n}}$$,with$F$afinitegroupand$${\mathrm{F}}_{\mathrm{n}}$$thefreegroupon$n$ generators. We study the Baum-Connes conjecture for this group. Our aim is to explicitly describe the Baum-Connes assembly map for $F\wr {\mathrm{F}}_{\mathrm{n}}$. To this end, we compute the topological and the analytical K-groups and exhibit their generators. Moreover, we present a concrete 2-dimensional model for $\underset{‾}{E}\mathrm{\Gamma }$. As a result of our K-theoretic computations, we obtain that $\mathrm K_0(\mathrm C^_{\mathrm r}(\Gamma))$ is the free abelian group of countable rank with a basis consisting of projections in $\mathrm C^{\mathrm r}(\bigoplus{${\mathrm{F}}_{\mathrm{n}}$}F)$and$\mathrm K_1(\mathrm C^*_{\mathrm r}(\Gamma))$isthefreeabeliangroupofrank$n$ with a basis consisting of the unitaries coming from the free group.