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

Abstract

Consider the wreath product $\Gamma = F\wr \mathrm{F_n} = \bigoplus_{\mathrm{F_n}}F\rtimes\mathrm{F_n}$, with $F$ a finite group and $\mathrm{F_n}$ the free group on $n$ generators. We study the Baum-Connes conjecture for this group. Our aim is to explicitly describe the Baum-Connes assembly map for $F\wr \mathrm{F_n}$. 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 $\underline{E} \Gamma$. 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 $\mathrm C^*_{\mathrm r}(\bigoplus_{\mathrm{F_n}}F)$ and $\mathrm K_1(\mathrm C^*_{\mathrm r}(\Gamma))$ is the free abelian group of rank $n$ with a basis consisting of the unitaries coming from the free group.