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

TL;DR

This paper explicitly describes higher Kazhdan projections for certain groups and uses this to compute new delocalised -Betti numbers, including the first non-vanishing results for infinite groups.

Contribution

It provides an explicit description of higher Kazhdan projections and applies this to compute novel delocalised -Betti numbers for specific infinite groups.

Findings

Explicit description of higher Kazhdan projections for free and Cartesian product groups.

New calculations of delocalised -Betti numbers.

First non-vanishing results for infinite groups.

Abstract

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 $\ell^2$-Betti numbers. Notably, we establish the first non-vanishing results for infinite groups.