Rigid actions need not be strongly ergodic

A probability measure preserving action $\Gamma \curvearrowright (X,\mu )$ is called rigid if the inclusion of $\mathrm {L}^\infty (X)$ into the crossed product $\mathrm {L}^\infty (X) \rtimes \Gamma$ 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.