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

2007/03/30 by Dmitry Matsnev, Matsnev, Dmitry · 1 citation
Mathematics · #20F65 #20G15 #51K05 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.math/0703923

openalex publication_date 2007/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• K-homology and K-theory of pure Braid groups

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