Incomparable actions of free groups

• Published in: Ergodic theory and dynamical systems 2017-10, Vol.37 (7), p.2084-2098
• Main Authors: CONLEY, CLINTON T., MILLER, BENJAMIN D.
• Format: Article
• Language: English
• Subjects: Borel sets; Ergodic processes
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2016.11

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$ , and $\unicode[STIX]{x1D707}$ is an $E$ -invariant Borel probability measure on $X$ . We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$ , there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$ , generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$ -reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$ , generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.