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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