  We say that a countable group $G$ is McDuff if it admits a free ergodic
probability measure preserving action such that the crossed product is a McDuff
II_1 factor. Similarly, $G$ is said to be stable if it admits such an action
with the orbit equivalence relation being stable. The McDuff property,
stability, inner amenability and property Gamma are subtly related and several
implications and non implications were obtained in
[Ef73,JS85,Va09,Ki12a,Ki12b]. We complete the picture with the remaining
implications and counterexamples.
