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On invariant random subgroups of block-diagonal limits of symmetric groups

Authors: Artem Dudko and Kostya Medynets
Journal: Proc. Amer. Math. Soc. 147 (2019), 2481-2494
MSC (2010): Primary 37B05, 20F50, 37A15
Published electronically: March 1, 2019
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Abstract: We classify the ergodic invariant random subgroups of block-
diagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite-dimensional state spaces. These block-diagonal limits arise as the transformation groups (full groups) of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. Given a simple full group $ G$ admitting only a finite number of ergodic measures on the path-space $ X$ of the associated Bratteli diagram, we prove that every non-Dirac ergodic invariant random subgroup of $ G$ arises as the stabilizer distribution of the diagonal action on $ X^n$ for some $ n\geq 1$. As a corollary, we establish that every group character $ \chi $ of $ G$ has the form $ \chi (g) = \mathrm {Prob}(g\in K)$, where $ K$ is a conjugation-invariant random subgroup of $ G$.

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Additional Information

Artem Dudko
Affiliation: Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland

Kostya Medynets
Affiliation: U.S. Naval Academy, Annapolis, Maryland 21402

Keywords: Invariant random subgroups, topological dynamics, locally finite groups
Received by editor(s): November 16, 2017
Received by editor(s) in revised form: July 14, 2018
Published electronically: March 1, 2019
Additional Notes: The research of the first author was supported by the Polish National Science Centre grant 2016/23/P/ST1/04088 within the EU’s Horizon 2020 R&I programme under the Marie Skłodowska-Curie grant agreement No. 665778.
The research of the second author was supported by NSA YIG H98230258656.
Communicated by: Nimish Shah
Article copyright: © Copyright 2019 American Mathematical Society