The Baum-connes Conjecture for Hyperbolic Groups

The Baum-Connes conjecture states that, for a discrete group G, the K-homology groups of the classifying space for proper G-action is isomorphic to the K-groups of the reduced group C-algebra of G [3, 2]. A positive answer to the Baum-Connes conjecture would provide a complete solution to the problem of computing higher indices of elliptic operators on compact manifolds. The rational injectivity part of the Baum-Connes conjecture implies the Novikov conjecture on homotopy invariance of higher signatures. The Baum-Connes conjecture also implies the Kadison-Kaplansky conjecture that for G torsion free there exists no non-trivial projection in the reduced group C-algebra associated to G. In [7], Higson and Kasparov prove the Baum-Connes conjecture for groups acting properly and isometrically on a Hilbert space. In a recent remarkable work, Vincent Lafforgue proves the Baum-Connes conjecture for strongly bolic groups with property RD [15, 12, 13]. In particular, this implies the Baum-Connes conjecture for the fundamental groups of strictly negatively curved compact manifolds. In [4], Connes and Moscovici prove the rational injectivity part of the Baum-Connes conjecture for hyperbolic groups using cyclic cohomology method. In this paper, we exploit Lafforgue’s work to prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. The main step in the proof is the following theorem.