Baum-Connes conjecture and group actions on trees
type: Item
instance of: scholarly article
title: Baum-Connes conjecture and group actions on trees
zbMATH Open document ID: 1008.19001
DOI: 10.1023/A:1012786413219
author: Hervé Oyono-Oyono
published in: \(K\)-Theory
publication date: 2003-03-31T00:00:00Z
review text: 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.
reviewed by: Peter Haskell
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 58J22
zbMATH DE Number: 1749299
zbMATH Keywords: Baum-Connes conjecture
zbMATH Keywords: tree
zbMATH Keywords: discrete group
zbMATH Keywords: KK-theory
Wikidata QID: Q123028546
MaRDI profile type: Publication
full work available at URL: a:1012786413219
OpenAlex ID: W2021957815
description: scientific article; zbMATH DE number 1749299