The baum-connes conjecture, noncommutative poincaré duality, and the boundary of the free group

Abstract

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 Poincaré duality map K□ (C(∂Γ)□Γ)→K □+1 (C(∂Γ)□Γ). We show that this map is an isomorphism in the case of Γ=𝔽2, 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. Copyright © 2003 Hindawi Publishing Corporation. All rights reserved.