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

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