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 E ̲ L 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 the assembly map μ i L : K i L ( E ̲ L ) → 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.