A local characterization of Kazhdan projections and applications

TL;DR

This paper provides a local characterization of Kazhdan projections for Banach space representations of groups, introduces a generalized topology, and explores applications including stability of rigidity and properties like (T) and (FH).

Contribution

It introduces a local criterion for Kazhdan projections, generalizes the Fell topology to Banach representations, and applies these to stability and property (T) results.

Findings

Kazhdan projections characterized locally for Banach space representations.

Property (T) and (FH) are equivalent for σ-compact groups.

Many Banach strong property (T) forms are open in the space of marked groups.

Abstract

We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group $G$. We also define and study a natural generalization of the Fell topology to arbitrary Banach space representations of a locally compact group. We give several applications in terms of stability of rigidity under perturbations. Among them, we show a Banach-space version of the Delorme--Guichardet theorem stating that property (T) and (FH) are equivalent for $\sigma$-compact locally compact groups. Another kind of applications is that many forms of Banach strong property (T) are open in the space of marked groups, and more generally every group with such a property is a quotient of a compactly presented group with the same property. We also investigate the notions of central and non central Kazhdan projections, and present examples of non central Kazhdan projections coming from hyperbolic groups.