Jashan Bal, University of Waterloo
Projectivity in topological dynamics
A compact space is defined to be projective if it satisfies a certain universal lifting property. Projective objects in the category of compact spaces were characterized as exactly the extremally disconnected compact spaces by Gleason (1958). Analogously, if we fix a topological group G, then one can consider projectivity in the category of G-flows or affine G-flows. We present some new results in this direction, including a characterization of amenability or extreme amenability for closed subgroups of a Polish group via a certain G-flow being projective in the category of affine G-flows or G-flows respectively. Lastly, we introduce a new property, called proximally irreducible, for a G-flow and use it to prove a new dynamical characterization of strong amenability for closed subgroups of a Polish group. In doing so, we answer a question of Zucker by characterizing when the universal minimal proximal flow for a Polish group is metrizable or has a comeager orbit.
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