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JMD

It is well known that if $G$ is a countable amenable group and $G
↷ (Y, \nu)$ factors onto $G ↷ (X, \mu)$, then the entropy of
the first action must be at least the entropy of the
second action. In particular, if $G ↷ (X, \mu)$ has infinite
entropy, then the action $G ↷ (Y, \nu)$ does not admit any finite
generating partition. On the other hand, we prove that if $G$ is a
countable nonamenable group then there exists a finite integer $n$
with the following property: for every probability-measure-preserving
action $G ↷ (X, \mu)$ there is a $G$-invariant probability measure
$\nu$ on $n^G$ such that $G ↷ (n^G, \nu)$ factors onto $G ↷
(X, \mu)$. For many nonamenable groups, $n$ can be chosen to be $4$
or smaller. We also obtain a similar result with respect to continuous
actions on compact spaces and continuous factor maps.

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