K-theory for C*-algebras of one-relator groups
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We compute the K-theory groups of the reduced C*-algebra C-r*(Gamma) of a one-relator group Gamma. We prove that every such group is K-amenable in the sense of Cuntz. For a torsion-free one-relator group Gamma = [X\r] such that, is not a product of commutators, we give a direct proof of the fact that the Baum-Connes analytical assembly map mu(i)(Gamma): K-i(B Gamma) --> K-i(C-r*(Gamma)) (i = 0,1) is an isomorphism. From recent results of Oyono and Tu, we deduce that the Baum-Connes conjecture with coefficients holds for any one-relator group, as well as for fundamental groups of Haken 3-manifolds (e.g. for all knot groups). In particular, if Gamma is a torsion-free group in one of these classes, then C-r*(Gamma) has no nontrivial idempotent. Mathematics Subject Classifications (1991): 20F05, 20E06, 46L80, 55N15.
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