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International Journal of Mathematics and Mathematical Sciences
Volume 2003, Issue 38, Pages 2425-2445
http://dx.doi.org/10.1155/S0161171203209169

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

Heath Emerson1,2

1Indiana University-Purdue University at Indianapolis (IUPUI), 402 North Blackford Street, Indianapolis 46202-3216, IN, USA
2Mathematisches Institut, Westfalische Wilhelms-Universitat, Einsteinstrasse 62, Muenster 48149, Germany

Received 17 September 2002

Copyright © 2003 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.