Euler characteristics, higher Kazhdan projections and delocalised $\ell^2$-Betti numbers

TL;DR

This paper links Euler characteristics with higher Kazhdan projections in non-amenable virtually free groups, enabling explicit K-theory class representations and non-vanishing results for delocalised $ ext{l}^2$-Betti numbers.

Contribution

It establishes a novel connection between combinatorial Euler characteristics and higher Kazhdan projections via the Baum-Connes map, providing explicit K-theory class decompositions.

Findings

Representation of higher Kazhdan projections as sums of averaging projections

Non-vanishing of delocalised $ ext{l}^2$-Betti numbers for certain groups

Explicit formulas relating Euler characteristics and K-theory classes

Abstract

For non-amenable finitely generated virtually free groups, we show that the combinatorial Euler characteristic introduced by Emerson and Meyer is the preimage of the K-theory class of higher Kazhdan projections under the Baum-Connes assembly map. This allows to represent the K-theory class of their higher Kazhdan projection as a finite alternating sum of the K-theory classes of certain averaging projections. The latter is associated to the finite subgroups appearing in the fundamental domain of their Bass-Serre tree. As an immediate application we obtain non-vanishing calculations for delocalised $\ell^2$-Betti numbers for this class of groups.