A characterization of relative Kazhdan Property T for semidirect products with abelian groups

TL;DR

This paper characterizes when a semidirect product of a locally compact abelian group and another group has Kazhdan's Property T, linking it to invariant means on the dual group.

Contribution

It provides a precise criterion for Property T in semidirect products involving abelian groups using invariant means on the dual group.

Findings

Property T holds iff the only H-invariant mean supported near the trivial character is the Dirac measure.

Establishes a connection between Property T and invariant means on the Pontryagin dual.

Characterizes Property T in terms of harmonic analysis on the dual group.

Abstract

Let A be a locally compact abelian group, and H a locally compact group acting on A. Let G=HA be the semidirect product. We prove that the pair (G,A) has Kazhdan's Property T if and only if the only countably approximable H-invariant mean on the Borel subsets of the Pontryagin dual of A, supported at the neighbourhood of the trivial character, is the Dirac measure.