matsnev[AT]math.ist.utl.pt | Departamento de Matem�tica
Instituto Superior T�cnico Avenida Rovisco Pais 1049-001 Lisboa, Portugal |

The Baum-Connes conjecture for countable subgroups of SL(2)

We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of SL(2,C). Using finite-dimensional methods, we show that the Baum-Connes assembly map for such groups is an isomorphism.

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

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.

Asymptotic dimension of one-relator groups

We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov and give an estimate of their asymptotic dimension in terms of the relator length.

Thesis: The Baum-Connes conjecture and group actions on affine buildings

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.

Translation groupoid of Skandalis, Tu, and Yu

A note on the Baum-Connes conjecture

Curriculum vitae, research statement, teaching statement, teaching evaluations, teaching with technology.

Old Penn State teaching pages contain various instructional aids, Teaching with Technology portfolio, and the pages for classes I taught there.