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

Abstract

Let $F$ be a finite group. We consider the lamplighter group $L=F\wr \mathbb{Z}$ over $F$. We prove that $L$ has a classifying space for proper actions $\underline{E}L$ which is a complex of dimension two. We use this to give an explicit proof of the Baum-Connes conjecture (without coefficients), that states that the assembly map ${\mu }_i^L:K_i^L(\underline{E}L)\to K_i(C^{\ast }L)(i=0,1)$ is an isomorphism. Actually, $K_0(C^{\ast }L)$ is free abelian of countable rank, with an explicit basis consisting of projections in $C^{\ast }L$, while $K_1(C^{\ast }L)$ is infinite cyclic, generated by the unitary of $C^{\ast }L$ implementing the shift. Finally we show that, for $F$ abelian, the $C^{\ast }$-algebra $C^{\ast }L$ is completely characterized by $|F|$ up to isomorphism.

32 pages, 3 figures; v2. reference added

References in corpus (2)

• K-theory for the $C^{\ast }$-algebras of the solvable Baumslag-Solitar groups
• $K$-theory for the crossed products of infinite tensor product of $C^{\ast }$-algebras by the shift

Cited by in corpus (4)

• K-theory and K-homology of the wreath products of finite with free groups
• K-homology and K-theory of pure Braid groups
• On the equivariant $K$- and $KO$-homology of some special linear groups
• Equivariant $K$-homology for hyperbolic reflection groups