Baum-Connes conjecture and group actions on trees
The author gives a detailed proof that the Baum-Connes conjecture (with coefficients) is true for a countable discrete group that acts on an oriented tree if and only if it is true for the stabilizer groups of the tree's vertices. \textit{P. Baum}, \textit{A. Connes}, and \textit{N. Higson} [in: \(C^{*}\)-algebras: 1943-1993. A fifty year celebration, Contemp. Math. 167, 241-291 (1994; Zbl 0830.46061)] asserted this theorem, with a reference to the techniques of \textit{M. V. Pimsner} [Invent. Math. 86, 603-634 (1986; Zbl 0638.46049)] as justification. The author has expanded that justification into a proof. The theorem implies that the Baum-Connes conjecture is stable under free products, amalgamated products, and HNN extensions and hence is true for fundamental groups of Haken manifolds, one-relator groups, and pure braid groups. \textit{J.-L. Tu} [K-Theory 17, No.~4, 303-318 (1999; Zbl 0939.19002)] has taken a different approach to get very similar results. Tu identifies a property that permits a proof of the Baum-Connes conjecture via a dual Dirac-Dirac argument. He shows that a discrete group acting on an oriented tree with quotient a finite graph satisfies his property if and only if the stabilizer group of each vertex does.
scientific article; zbMATH DE number 1749299