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Measures and stabilizers of group Cantor actions

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The first author is supported by DFG grant GR 4899/1-1. The second author is supported by FWF Project P31950-N35

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  • We consider a minimal equicontinuous action of a finitely generated group $ G $ on a Cantor set $ X $ with invariant probability measure $ \mu $, and the stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup $ H $ of $ G $ such that the set of points in $ X $ whose stabilizers are conjugate to $ H $ has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degenerate is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to certain almost one-to-one extensions of equicontinuous actions.

    Mathematics Subject Classification: Primary: 37B05, 37A15, 22F10; Secondary: 22F50, 37E25.


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