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Incomparable actions of free groups
Published online by Cambridge University Press: 12 May 2016
Abstract
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.
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- Research Article
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- © Cambridge University Press, 2016