The Baum-Connes conjecture and group actions on affine buildings

by 1977- Matsnev, Dmitry Anatolievich

Guentner, Higson, and Weinberger proved using Hilbert space techniques that for any countable linear group the Baum-Connes assembly map is split-injective; for the case of a countable linear group of matrices of size 2 they showed that the Baum-Connes assembly map is an isomorphism. In this thesis we study the the possibility of applying a finite-dimensionality argument in order to prove part of the Baum-Connes conjecture for finitely generated 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 are “boundedly composed”. 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. iii