Inner amenable groups having no stable action

TL;DR

This paper constructs specific inner amenable ICC groups that lack any stable, ergodic, free, probability-measure-preserving actions, addressing a long-standing open problem from 1987.

Contribution

It provides the first examples of such groups, demonstrating the existence of inner amenable ICC groups without stable actions.

Findings

Constructed inner amenable ICC groups with no stable actions

Solved a problem posed by Jones-Schmidt in 1987

Advances understanding of group actions and amenability

Abstract

We construct inner amenable and ICC groups having no ergodic, free, probability-measure-preserving and stable action. This solves a problem posed by Jones-Schmidt in 1987.