Let G be a Polish group. A G-flow is a continuous action of G on a compact Hausdorff space. G is amenable if every G-flow has an invariant probability measure. G has metrizable universal minimal flow if every G-flow contains a metrizable G-flow. We consider the property of G having a metrizable Furstenberg boundary. This is a common weakening of amenability and having metrizable universal minimal flow. We prove a characterization of G having metrizable Furstenberg boundary in the spirit of Kechris-Pestov-Todorcevic, Bartosova, Moore, and Zucker. We show mapping class groups of infinite type surfaces (a class of groups arising in geometric topology) never have a metrizable Furstenberg boundary.
This is joint work with George Domat and Forte Shinko.