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Topological dynamics of automorphism groups, ultrafilter combinatorics, and the Generic Point Problem

Author: Andy Zucker
Journal: Trans. Amer. Math. Soc. 368 (2016), 6715-6740
MSC (2010): Primary 37B05; Secondary 03C15, 03E15, 05D10, 22F50, 54D35, 54D80, 54H20
Published electronically: November 16, 2015
MathSciNet review: 3461049
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Abstract: For $ G$ a closed subgroup of $ S_{\infty }$, we provide a precise combinatorial characterization of when the universal minimal flow $ M(G)$ is metrizable. In particular, each such instance fits into the framework of metrizable flows developed by Kechris, Pestov, and Todorčević and by Nguyen Van Thé; as a consequence, each $ G$ with metrizable universal minimal flow has the generic point property, i.e. every minimal $ G$-flow has a point whose orbit is comeager. This solves the Generic Point Problem raised by Angel, Kechris, and Lyons for closed subgroups of $ S_\infty $.

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Additional Information

Andy Zucker
Affiliation: Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Keywords: Fra\"{\i}ss\'e theory, Ramsey theory, universal minimal flow, greatest ambit, Generic Point Problem
Received by editor(s): July 4, 2014
Received by editor(s) in revised form: December 7, 2014, and February 8, 2015
Published electronically: November 16, 2015
Additional Notes: The author was partially supported by NSF Grant no. DGE 1252522.
Article copyright: © Copyright 2015 American Mathematical Society

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