Special Cases of the Baum-Connes Conjecture

The Baum-Connes conjecture is now proved for various classes of groups. An early result of Kasparov (which actually preceded the formulation of the Baum-Connes conjecture) proved the conjecture for closed subgroups of SO(n,1). Julg and Kasparov proved the conjecture for closed subgroups of SU(n,1), and then Higson and Kasparov proved the conjecture for Gromov's so-called a-T-menable groups. This class includes the closed subgroups of SO(n,1) and SU(n,1) as well as all (second countable) amenable groups, Coxeter groups, and others. As Gromov's terminology suggests, the class includes no property T group, but in an important breakthrough, Lafforgue proved the conjecture for a class which includes some infinite property T groups, for example uniform lattices in SL(3,R) or SL(3,C). Using a different method, more closely related to the approach for SO(n,1) and SU(n,1), Julg proved the conjecture for the property T group Sp(n,1) and its closed subgroups.

Selected References

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Chabert, Jerome; Echterhoff, Siegfried; Nest, Ryszard. The Connes-Kasparov conjecture for almost connected groups and for linear p-adic groups. Publ. Math. Inst. Hautes Etudes Sci. No. 97, (2003), 239-278.   MR
Connes, A. An analogue of the Thom isomorphism for crossed products of a C*-algebra by an action of R. Adv. in Math. 39 (1981), no. 1, 31-55.   MR
Higson, Nigel; Kasparov, Gennadi. E-theory and KK-theory for groups which act properly and isometrically on Hilbert space. Invent. Math. 144 (2001), no. 1, 23-74.   MR
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Julg, Pierre; Kasparov, Gennadi. Operator K-theory for the group SU(n,1). J. Reine Angew. Math. 463 (1995), 99-152.   MR
Kasparov, G. G. Lorentz groups: K-theory of unitary representations and crossed products. (Russian) Dokl. Akad. Nauk SSSR 275 (1984), no. 3, 541-545.   MR
Lafforgue, V. Banach KK-theory and the Baum-Connes conjecture. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 795-812, Higher Ed. Press, Beijing, 2002.   MR
Lafforgue, Vincent. Banach KK-theory and the Baum-Connes conjecture. European Congress of Mathematics, Vol. II (Barcelona, 2000), 31-46, Progr. Math., 202, Birkhaeuser, Basel, 2001.   MR
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