Higher Kazhdan projections and delocalized ${\mathrm{\ell }}^2$-Betti numbers for an amalgamated product group

We establish explicit expressions for the $K$-theory classes of higher Kazhdan projections for amalgamated product groups ${\mathbb{Z}}_m{\ast }_{{\mathbb{Z}}_d}{\mathbb{Z}}_n$. Our approach follows the methodology developed by Pooya and Wang for free product groups ${\mathbb{Z}}_m\ast {\mathbb{Z}}_n$, and naturally generalizes their results on free products. As an application of the $K$-class expressions, we obtain non-vanishing results for delocalized ${\mathrm{\ell }}^2$-Betti numbers of $\mathrm{S}\mathrm{L}(2,\mathbb{Z})$.