Negative definite functions and a dynamical characterization of property (T) for countable groups

TL;DR

This paper introduces a class of negative definite kernels based on measure spaces to characterize property (T) for countable groups through measure-preserving actions, linking algebraic properties to dynamical behavior.

Contribution

It provides a new dynamical characterization of property (T) using negative definite kernels and measure-preserving actions, expanding understanding of group properties.

Findings

Property (T) characterized via measure-preserving actions

Bounded translation distances imply property (T)

Negative definite kernels linked to group dynamics

Abstract

A class of negative definite kernels is defined in terms of measure spaces. Using this concept, property (T) for a countable group $\G$ is characterized in terms of measure preserving actions of $\G$, as follows. If a set $S$ is translated a finite amount by any fixed element of $\G$, then there is a uniform bound on how far $S$ is translated.