Inner amenability, property Gamma, McDuff II_1 factors and stable equivalence relations

© Cambridge University Press, 2017. We say that a countable group is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff factor. Similarly, 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 Effros [Property and inner amenability. Proc. Amer. Math. Soc. 47 (1975), 483-486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math. 109 (1987), 91-114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math. 208 (2012), 389-394], Kida [Inner amenable groups having no stable action. Geom. Dedicata 173 (2014), 185-192] and Kida [Stability in orbit equivalence for Baumslag-Solitar groups and Vaes groups. Groups Geom. Dyn. 9 (2015), 203-235]. We complete the picture with the remaining implications and counterexamples.

Publication year:2018

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