The Baum-Connes conjecture, noncommutative Poincaré duality,and the boundary of the free group

For every hyperbolic group Γ with Gromov boundary ∂Γ, one can form the cross product C∗-algebra C(∂Γ)⋊Γ. For each such algebra, we construct a canonical K-homology class. This class induces a Poincare duality map K∗(C(∂Γ)⋊Γ)→K∗