Baum-Connes conjecture and extensions (Q2706829)

We study the stability of the Baum-Connes conjecture under extensions. We prove the following result: if \(0\to\Gamma_0 \to\Gamma_1\to \Gamma_2 \to 0\) is an exact sequence of discrete countable groups, we show that \(\Gamma_1\) satisfies the Baum-Connes conjecture if the two following conditions are satisfied:NEWLINENEWLINENEWLINE1. the group \(\Gamma_2\) satisfies the Baum-Connes conjecture;NEWLINENEWLINENEWLINE2. every subgroup of \(\Gamma_1\) containing \(\Gamma_0\) as a subgroup of finite index satisfies the Baum-Connes conjecture.NEWLINENEWLINENEWLINEIn particular, this result has the following consequences:NEWLINENEWLINENEWLINE\(\bullet\) The extension of a group with the Haagerup property by a group satisfying the Baum-Connes conjecture satisfies the Baum-Connes conjecture.NEWLINENEWLINENEWLINE\(\bullet\) A discrete group satisfies the Baum-Connes conjecture if and only if each subgroup which contains the commutator subgroup as a group of finite index satisfies the Baum-Connes conjecture.NEWLINENEWLINENEWLINE\(\bullet\) The direct product of two groups satisfying the Baum-Connes conjecture satisfies the Baum-Connes conjecture.