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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)
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