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

TL;DR

This paper derives explicit $K$-theory formulas for higher Kazhdan projections in amalgamated groups and applies these to show non-zero delocalized $ ext{l}^2$-Betti numbers for $ ext{SL}(2, ext{Z})$, extending previous free product results.

Contribution

It generalizes the computation of higher Kazhdan projections from free groups to amalgamated products and links these to non-vanishing delocalized $ ext{l}^2$-Betti numbers.

Findings

Explicit $K$-theory formulas for amalgamated groups.

Non-vanishing delocalized $ ext{l}^2$-Betti numbers for $ ext{SL}(2, ext{Z})$.

Extension of free product results to amalgamated products.

Abstract

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 $\mathrm{SL}(2,\mathbb{Z})$.