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

Abstract

Consider the wreath product Γ = F F n = Fn F F n , with F a finite group and 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 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 EΓ. As a result of our K-theoretic computations, we obtain that K 0 (C * r (Γ)) is the free abelian group of countable rank with a basis consisting of projections in C * r ( Fn F ) and K 1 (C * r (Γ)) is the free abelian group of rank n with a basis consisting of the unitaries coming from the free group.

K-homology and K-theory of pure Braid groups

“…Pooya and Valette [FPV17,PV18,Poo19], in which explicit computations of the Baum-Connes correspondence are given for certain discrete groups. We believe that these explicit computations contribute to a deeper understanding of the Baum-Connes correspondence.…”