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

Abstract

Consider the wreath product $\Gamma =F\wr {\mathrm{F}}_{\mathrm{n}}={⨁}_{{\mathrm{F}}_{\mathrm{n}}}F⋊{\mathrm{F}}_{\mathrm{n}}$, with $F$ a finite group and ${\mathrm{F}}_{\mathrm{n}}$ 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 $F\wr {\mathrm{F}}_{\mathrm{n}}$. 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 $\underline{E}\Gamma$. As a result of our K-theoretic computations, we obtain that ${\mathrm{K}}_0({\mathrm{C}}_{\mathrm{r}}^{\ast }(\Gamma ))$ is the free abelian group of countable rank with a basis consisting of projections in ${\mathrm{C}}_{\mathrm{r}}^{\ast }({⨁}_{{\mathrm{F}}_{\mathrm{n}}}F)$ and ${\mathrm{K}}_1({\mathrm{C}}_{\mathrm{r}}^{\ast }(\Gamma ))$ is the free abelian group of rank $n$ with a basis consisting of the unitaries coming from the free group.

18 pages

References in corpus (1)

• K-homology and K-theory for the lamplighter groups of finite groups

Cited by in corpus (3)

• K-homology and K-theory of pure Braid groups
• On the equivariant $K$- and $KO$-homology of some special linear groups
• Bredon homology of wallpaper groups