Geometry and Topology Seminar: "The Baum-Connes assembly map for certain subgroups of Z2 ⋊ GL(2, Z)"
Abstract: The Baum-Connes conjecture for a group G predicts that a specific map from equivariant K-homology to K-theory of natural objects constructed from G is an isomorphism. This specific map is natural in G and called the assembly map. While the conjecture is proved for large classes of groups, it is still open for linear groups in general. In this talk we discuss an alternative and explicit method of proving the Baum-Connes conjecture for some semi-direct products G = Z ⋊ Γ, where Γ is a non-amenable subgroup of GL(2, Z). This is feasible thanks to the presence of a 3-dimensional model for the classifying space for proper actions of G combined with detailed understanding of the K-theory of C*-algebras of wallpaper groups. Besides providing a hands-on proof for the Baum-Connes conjecture in these cases, our method elucidates the assembly map for groups we study.
Host: Michael Landry