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The Baum-Connes conjecture with coefficients for word-hyperbolic groups (after Vincent Lafforgue). (English) Zbl 1357.19005

Séminaire Bourbaki. Volume 2012/2013. Exposés 1059–1073. Avec table par noms d’auteurs de 1948/49 à 2012/13. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-785-8/pbk). Astérisque 361, 115-148, Exp. No. 1062 (2014).
Summary: In a recent breakthrough, V. Lafforgue [Beijing: Higher Education Press. 795–812 (2002; Zbl 0997.19003)] verified the Baum-Connes conjecture with coefficients for all word-hyperbolic groups. This provides the first examples of groups with Kazhdan’s Property (T) satisfying the conjecture. His proof (of almost 200 pages) is completely elementary, but of impressive complexity. It makes essential use of group representations of weak exponential growth. These representations are also the topic of Lafforgue’s work [loc.cit.] on strengthened versions of Property (T). His results about these properties for higher rank groups and lattices have interesting applications in graph theory and rigidity theory. They also indicate that it might be very difficult to establish the Baum-Connes conjecture for higher rank lattices with the approaches used so far.
For the entire collection see [Zbl 1294.00022].

MSC:

19K35 Kasparov theory (\(KK\)-theory)
19L47 Equivariant \(K\)-theory
20F67 Hyperbolic groups and nonpositively curved groups
58J22 Exotic index theories on manifolds
46L80 \(K\)-theory and operator algebras (including cyclic theory)

Citations:

Zbl 0997.19003
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