Minimal models for free actions (Q2906461)

A `model' for a measure-preserving action of a group \(G\) is a (measurably) isomorphic \(G\) action with some additional desirable structure. There are many ways to find a topological model -- that is, one in which the action is a continuous one on a compact metric space -- but imposing any additional properties presents considerable difficulties. Among these stronger results, the Jewett-Krieger theorem shows that any \(\mathbb{Z}\)-action has a uniquely ergodic topological model [\textit{R. I. Jewett}, J. Math. Mech. 19, 717--729 (1970; Zbl 0192.40601); \textit{W. Krieger}, in: Proc. 6th Berkeley Sympos. Math. Statist. Probab., Univ. Calif. 1970, 2, 327--346 (1972; Zbl 0262.28013)], and similar results have been found for actions of countable amenable groups by \textit{E. Glasner} and \textit{B. Weiss} [in:, Handbook of dynamical systems. Volume 1B. Amsterdam: Elsevier. 597--648 (2006; Zbl 1130.37303)]. Here an extremely general extension of these results is found, showing that any infinite countable group has a `universal minimal action': that is, a minimal topological action with the property that the invariant probability measures represent all free actions of the group on a Lebesgue space up to isomorphism.NEWLINENEWLINEFor the entire collection see [Zbl 1237.37004].