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

This article investigates an explicit description of the Baum–Connes assembly map of the wreath product Γ=F≀Fn=⊕FnF⋊Fn, where F is a finite and Fn is the free group on n generators. In order to do so, we take Davis–Lück’s approach to the topological side which allows computations by means of spectral sequences. Besides describing explicitly the K-groups and their generators, we present a concrete 2-dimensional model for the classifying space E̲Γ. As a result of our computations, we obtain that K0(Cr∗(Γ)) is the free abelian group of countable rank with a basis consisting of projections in Cr∗(⊕FnF), and K1(Cr∗(Γ)) is the free abelian group of rank n with a basis represented by the unitaries coming from the free group.