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Baum-Connes conjecture and extensions. (English) Zbl 0973.46064
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:
1. the group \(\Gamma_2\) satisfies the Baum-Connes conjecture;
2. every subgroup of \(\Gamma_1\) containing \(\Gamma_0\) as a subgroup of finite index satisfies the Baum-Connes conjecture.
In particular, this result has the following consequences:
\(\bullet\) The extension of a group with the Haagerup property by a group satisfying the Baum-Connes conjecture satisfies the Baum-Connes conjecture.
\(\bullet\) 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.
\(\bullet\) The direct product of two groups satisfying the Baum-Connes conjecture satisfies the Baum-Connes conjecture.

MSC:
46L80 \(K\)-theory and operator algebras (including cyclic theory)
19K35 Kasparov theory (\(KK\)-theory)
20E22 Extensions, wreath products, and other compositions of groups
57M50 General geometric structures on low-dimensional manifolds
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