Property (T) and the Furstenberg Entropy of Nonsingular Actions

TL;DR

This paper characterizes property (T) of groups through the Furstenberg entropy of nonsingular actions, establishing a new criterion that is both necessary and sufficient, advancing understanding of group actions and entropy.

Contribution

It provides a new characterization of property (T) using Furstenberg entropy, showing the condition is both necessary and sufficient for such groups.

Findings

Furstenberg entropy values are bounded away from zero for property (T) groups.

The boundedness condition is both necessary and sufficient for property (T).

The result links entropy theory with group property (T) in a novel way.

Abstract

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure $\mu$ on a countable group $G$, A. Nevo showed that a necessary condition for $G$ to have property (T) is that the Furstenberg $\mu$-entropy values of the ergodic, properly nonsingular $G$-actions are bounded away from zero. We show that this is also a sufficient condition.