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 ≀ Z over F. We prove that L has a classifying space for proper actions EL which is a complex of dimension 2.We use this to give an explicit proof of the Baum–Connes conjecture (without coefficients) that states that theassembly map µLi: KLi(E L) → Ki(C∗L)(i =0, 1) is an isomorphism. Actually, K0(C∗L) is free abelian of countable rank, with an explicit basis consisting of projections in C∗L, while K1(C∗L) is infinite cyclic, generated by the unitary of C∗L implementing t he shift. Finally we show that,for F abelian, the C∗-algebra C∗L is completely characterized by |F | up to isomorphism.