Stabilizer rigidity in irreducible group actions

Creators: Hartman, Yair; Tamuz, Omer

Abstract

We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader–Shalom and Nevo–Zimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader–Shalom and Stuck–Zimmer.

Additional Information

© 2016 Hebrew University of Jerusalem. Received: 11 July 2014; Revised: 24 November 2015; First Online: 30 October 2016. We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader–Shalom and Nevo–Zimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader–Shalom and Stuck–Zimmer. We would like to thank Uri Bader, Amos Nevo, Jesse Peterson and Benjamin Weiss for useful discussions and motivating conversations. We would also like to thank Yehuda Shalom and Lewis Bowen for helpful comments on an early draft of this article. Y. Hartman is supported by the European Research Council, grant 239885. O. Tamuz is supported by ISF grant 1300/08, and is a recipient of the Google Europe Fellowship in Social Computing. This research is supported in part by this Google Fellowship.

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