\(K\)-theory for \(C^*\)-algebras of one-relator groups
The authors compute the K-theory groups of the reduced \(C^*\)-algebra \(C^*_r({\Gamma})\) of a one-relator group \({\Gamma}\) and they prove that every such group is \(K\)-amenable in the sense of Cuntz. For a torsion-free one-relator group \({\Gamma}=\langle X|r\rangle\) such that \(r\) is not a product of commutators a direct proof is given of the fact that the Baum-Connes analytical assembly map \({\mu}_i^{\Gamma}:K_i(B{\Gamma})\to K_i(C^*_r({\Gamma}))\) for \(i=0, 1\) is an isomorphism. From recent results of Oyono and Tu, they deduce that the Baum-Connes conjecture with coefficients holds for any one-relator group, as well as for fundamental groups of Haken 3-manifolds. In particular, if \({\Gamma}\) is a torsion-free group in one of these classes, then \(C^*_r({\Gamma})\) has no nontrivial idempotent.
scientific article; zbMATH DE number 1275467