The Baum-Connes conjecture and proper group actions on affine buildings

Abstract

We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for linear groups. For any finitely generated linear group over a field of characteristic zero we construct a proper action on a finite-asymptotic-dimensional CAT(0)-space, provided that for such a group its unipotent subgroups have `bounded composition rank'. The CAT(0)-space in our construction is a finite product of symmetric spaces and affine Bruhat-Tits buildings. For the case of finitely generated subgroup of SL(2,C) the result is sharpened to show that the Baum-Connes assembly map is an isomorphism.