Finite group extensions and the Baum-Connes conjecture

In this note, we exhibit a method to prove the Baum-Connes conjecture (with coefficients) for extensions with finite quotients of certain groups which already satisfy the Baum-Connes conjecture. Interesting examples to which this method applies are torsion-free finite extensions of the pure braid groups, e.g. the full braid groups. The Baum-Connes conjecture (in this note the term will always mean “BaumConnes conjecture with coefficients”) for a group G (all groups in this note are assumed to be discrete and countable) states that the Baum-Connes map μr : K G ∗ (EG,A) → K∗(C ∗ red(G,A)) (1) is an isomorphism for every C∗-algebra A with an action of G by C∗-algebra homomorphisms. Here, the left hand side is the equivariant K-homology with coefficients in A of the universal space EG for proper G-actions, which is homological in nature. The right hand side, the K-theory of the reduced crossed product of A and G, belongs to the world of C∗-algebras and —to some extend— representations of groups. If A = C with the trivial action, the right hand side becomes the K-theory of the reduced C∗-algebra of G. If, in addition, G is torsion-free, the left hand side is the K-homology of the classifying space of G. The Baum-Connes conjecture has many important connections to other questions and areas of mathematics. The injectivity of μr implies the Novikov conjecture about homotopy invariance of higher signatures. It also implies the stable Gromov-Lawson-Rosenberg conjecture about the existence of metrics with positive scalar curvature on spin-manifolds. The surjectivity, on the other hand, gives information in particular about C∗ redG. If G is torsion-free, it implies e.g. that this C∗-algebra contains no idempotents different from zero and one. Since we are only considering the Baum-Connes conjecture with coefficients, all these properties follow for all subgroups of G, as well. We do not want to repeat the construction of the K-groups and the map in the Baum-Connes conjecture (1), instead, the reader is referred to [1, 6]. Recently, Gromov and others have produced counterexamples to the conjecture e-mail: [email protected] www: http://wwwmath.uni-muenster.de/u/lueck/ Fax: ++49 -251/83 38370