The Baum-Connes conjecture, noncommutative Poincaré duality, and the boundary of the free group (Q1415096)

Summary: For every hyperbolic group \(\Gamma\) with Gromov boundary \(\partial \Gamma\), one can form the cross product \(C^*\)-algebra \(C(\partial \Gamma) \rtimes \Gamma\). For each such an algebra, we construct a canonical \(K\)-homology class. This class induces a Poincaré duality map \(K_*(C(\partial\gamma) \rtimes \Gamma) \rightarrow K^{*+1}(C(\partial\gamma) \rtimes\Gamma)\). We show that this map is an isomorphism in the case of \(\Gamma =\mathbb{F}_{2}\), the free group on two generators. We point out a direct connection between our constructions and the Baum-Connes conjecture and eventually use the latter to deduce our result.