Rigid actions need not be strongly ergodic

TL;DR

This paper constructs examples of ergodic, free, measure-preserving actions that are rigid but not strongly ergodic, illustrating nuanced properties of such actions and their quotients.

Contribution

It provides the first known examples of rigid actions that are ergodic but not strongly ergodic, and shows rigid actions can have non-rigid quotients.

Findings

Existence of rigid, ergodic, free actions that are not strongly ergodic

Rigid actions can have non-rigid quotients

Examples demonstrate nuanced properties of rigidity in measure-preserving actions

Abstract

A probability measure preserving action of \Gamma on (X,\mu) is called rigid if the inclusion of L^\infty(X) into the crossed product L^\infty(X) \rtimes \Gamma has the relative property (T) in the sense of Popa. We give examples of rigid, free, probability measure preserving actions that are ergodic but not strongly ergodic. The same examples show that rigid actions may admit non-rigid quotients.