# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Baum P., Contemp. Math. 167 pp 240– (1994) [2] [B-C-V] B. Bekka, P. A. Cherix, A. Valette, Proper a ne actions of amenable groups, Novikov conjecture, index theorems and rigidity. Vol 2 (Oberwolfach 1993), Soc. Lect. Note Ser. 227 (1995), 1-4. [3] Math. 521 pp 161– (2000) [4] [C-E] J. Chabert, S. Echterho , Twisted equivariant KK-theory and the Baum-Connes conjecture for group extensions, preprint 1999. [5] J. Funct. Anal. 36 pp 1– (1980) [6] Higson N., Electron. Res. Announc. Amer. Math. Soc. 3 pp 131– (1997) [7] [J] P. Jolissaint, Borel cocycles, approximation properties and relative property (T), preprint 1997. · Zbl 0955.22008 [8] Kasparov G. G., Invent. Math 91 pp 147– (1988) [9] Kasparov G. G., Soviet Math. Dokl. 29 pp 256– (1984) [10] [L2] V. La orgue, K-theAorie bivariante pour les algeAbres de Banach et conjecture de Baum-Connes, preprint 1998. [11] [O1] H. Oyono-Oyono, Conjecture de Baum-Connes pour les groupes agissant sur les arbres, TheAse de Doctorat, UniversiteA Claude Bernard-Lyon I, 1997. · Zbl 0918.46062 [12] [O2] H. Oyono-Oyono, Conjecture de Baum-Connes et actions de groupes sur des arbres, C.R.A.S SeArie I Math. 326, n7 (1998), 799-804. · Zbl 0918.46062 [13] [O3] H. Oyono-Oyono, Baum-Connes Conjecture and group actions on trees, preprint 1998. [14] [S] G. Skandalis, Exact sequences for the Kasparov groups of graded algebras, Can. J. Math 37, n2 (1985), 193-216. · Zbl 0603.46064 [15] [V] A. Valette, Introduction to the Baum-Connes conjecture, Course given at the ETH ZuErich, 1999.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.