The Baum-Connes conjecture, if it holds for a certain group, provides topological tools to compute the K-theory of its reduced group $\mathrm C^*$-algebra. This conjecture has been confirmed for large classes of groups, such as amenable groups, but also for some Kazhdan's property (T) groups. Property (T) and its strengthening are driving forces in the search for potential counterexamples to the conjecture. Having property (T) for a group is characterised by the existence of a certain projection in the universal group $\mathrm C^*$-algebra of the group, known as the Kazhdan projection. It is this projection and its analogues in other completions of the group ring, which obstruct certain method of proof for the Baum-Connes conjecture.

In this talk, I will introduce a generalisation of Kazhdan projections. Employing these projections we establish a link between surjectivity of the Coarse Baum-Connes assembly map and calculations of ($\ell^2$-)Betti numbers of the group. This is based on joint work with Kang Li and Piotr Nowak.