Furstenberg entropy values for nonsingular actions of groups without property (T)

TL;DR

This paper demonstrates that for any positive real number, there exists a type III_1 ergodic free nonsingular action of a countable infinite group without property (T) with Furstenberg entropy equal to that number.

Contribution

It constructs explicit examples of nonsingular actions with prescribed Furstenberg entropy for groups lacking property (T).

Findings

Existence of actions with arbitrary positive Furstenberg entropy.

Extension of entropy theory to groups without property (T).

Provides a method to realize any positive entropy value.

Abstract

Let $G$ be a discrete countable infinite group that does not have Kazhdan's property ~(T) and let $\kappa$ be a generating probability measure on $G$. Then for each $t>0$, there is a type $III_1$ ergodic free nonsingular $G$-action whose $\kappa$-entropy (or the Furstenberg entropy) is $t$.