Higher Kazhdan projections and delocalised ${\mathrm{\ell }}^2$-Betti numbers

We provide an explicit description of the K-classes of higher Kazhdan projections in degrees greater than 0 for specific free product groups and Cartesian product groups. Employing this description, we obtain new calculations of Lott’s delocalised ${\mathrm{\ell }}^2$-Betti numbers. Notably, we establish the first non-vanishing results for infinite groups.