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

TL;DR

This paper proves the Baum-Connes conjecture for lamplighter groups over finite groups, explicitly describes their K-theory, and characterizes their C*-algebras when the finite group is abelian.

Contribution

It provides an explicit proof of the Baum-Connes conjecture for lamplighter groups over finite groups and characterizes their C*-algebras in the abelian case.

Findings

Classifying space for proper actions is 2-dimensional.

K_0(C^*L) is free abelian with explicit basis.

K_1(C^*L) is infinite cyclic, generated by a specific unitary.

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 $\mu_i^L:K_i^L(\underline{E} L)\rightarrow K_i(C^*L)\;(i=0,1)$ is an isomorphism. Actually, $K_0(C^*L)$ is free abelian of countable rank, with an explicit basis consisting of projections in $C^*L$, while $K_1(C^*L)$ is infinite cyclic, generated by the unitary of $C^*L$ implementing the shift. Finally we show that, for $F$ abelian, the $C^*$-algebra $C^*L$ is completely characterized by $|F|$ up to isomorphism.