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.
