# zbMATH — the first resource for mathematics

The Baum-Connes conjecture, noncommutative Poincaré duality, and the boundary of the free group. (English) Zbl 1037.46059
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.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory)
##### Keywords:
Baum-Connes conjecture
Full Text: