On free actions, minimal flows, and a problem by Ellis

TLDR

The paper aims to exhibit Polish topological groups whose every continuous action on a compact space has a fixed point, thereby providing a negative solution to Ellis’s 1969 problem. The authors construct Polish groups with diverse action properties—some acting freely on their universal minimal flow while others act effectively but not freely—show that groups like \(S(X)\) and \(U(\mathcal H)\) cannot act freely on any compact space, identify \(\mathbb S^1\) as the universal minimal flow for \(\mathrm{Homeo}_+(\mathbb S^1)\), and prove that closed subgroups of amenable groups need not be amenable.

Abstract

We exhibit natural classes of Polish topological groups $G$ such that every continuous action of $G$ on a compact space has a fixed point, and observe that every group with this property provides a solution (in the negative) to a 1969 problem by Robert Ellis, as the Ellis semigroup $E(U)$ of the universal minimal $G$-flow $U$, being trivial, is not isomorphic with the greatest $G$-ambit. Further refining our construction, we obtain a Polish topological group $G$ acting freely on the universal minimal flow $U$ yet such that $\mathcal {S} (G)$ and $E(U)$ are not isomorphic. We also display Polish topological groups acting effectively but not freely on their universal minimal flows. In fact, we can produce examples of groups of all three types having any prescribed infinite weight. Our examples lead to dynamical conclusions for some groups of importance in analysis. For instance, both the full group of permutations $S(X)$ of an infinite set, equipped with the pointwise topology, and the unitary group $U(\mathcal {H})$ of an infinite-dimensional Hilbert space with the strong operator topology admit no free action on a compact space, and the circle $\mathbb {S}^{1}$ forms the universal minimal flow for the topological group $\mathrm {Homeo} _{+}(\mathbb {S}^{1})$ of orientation-preserving homeomorphisms. It also follows that a closed subgroup of an amenable topological group need not be amenable.