Metrizable universal minimal flows of Polish groups have a comeagre orbit (Q521604)

Let \(G\) be a topological group. A \textit{\(G\)-flow} is a continuous action of \(G\) on a compact Hausdorff space, and a flow is \textit{minimal} if all of its orbits are dense. For any topological group \(G\) there exists a unique \textit{universal minimal flow} (UMF) \(M(G)\), that is, a minimal \(G\)-flow which maps continuously and equivariantly onto every minimal \(G\)-flow. For locally compact, non-compact groups, the UMF is never metrizable and cannot be described in a meaningful way. However, for many important non-locally compact groups this flow is metrizable, can be computed, and carries interesting information. The systematic study of metrizable UMFs was initiated in 2005 in a paper by \textit{A. S. Kechris} et al. [Geom. Funct. Anal. 15, No. 1, 106--189 (2005; Zbl 1084.54014)]. It turns out that all known concrete computations of UMFs can be carried out using a single technique, first employed by Pestov in 2000: isolating a co-precompact, extremely amenable subgroup \(G^\ast\) of the group of interest \(G\), and then observing that any minimal subflow of the completion \(\widehat{G/G^\ast}\) must be isomorphic to \(M(G)\). In practice, \(\widehat{G/G^\ast}\) is often already minimal and, in fact, the paper under review shows that an appropriate choice of \(G^\ast\) can always be made so that this is the case. Thus computing a metrizable UMF reduces to finding an extremely amenable, co-precompact subgroup, for which a variety of techniques have been developed (for example, in a paper by Pestov from 2005). The main result of the paper under review is the following statement: \textit{if \(G\) is a Polish group whose UMF \(M(G)\) is metrizable, \(M(G)\) has a comeagre orbit}. This, taking into account the work of \textit{J. Melleray} et al. [Int. Math. Res. Not. 2016, No. 5, 1285--1307 (2016; Zbl 1359.37023)], implies that then there exists an extremely amenable, closed, co-precompact subgroup \(G^\ast\) of \(G\) such that \(M(G)=\widehat{G/G^\ast}\). Note that considering Polish groups in the place of arbitrary topological groups is not a real restriction in the setting of the paper under review: if \(M(G)\) is metrizable, then the closure of the image of \(G\) in \(\mathrm{Homeo}(M(G))\) is a Polish group whose minimal flows are exactly the same as those of \(G\), so as long as one is interested only in minimal flows, one can replace one with the other. The above stated main result of the paper under review answers a question by \textit{O. Angel} et al. [J. Eur. Math. Soc. (JEMS) 16, No. 10, 2059--2095 (2014; Zbl 1304.22027)] (``the generic point problem''). The paper is well-written and interesting.