Every action of a non-amenable group is the factor of a small action

Abstract

It is well known that if $G$ is a countable amenable group and $G\curvearrowright (Y,\nu )$ factors onto $G\curvearrowright (X,\mu )$, then the entropy of the first action must be greater than or equal to the entropy of the second action. In particular, if $G\curvearrowright (X,\mu )$ has infinite entropy, then the action $G\curvearrowright (Y,\nu )$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable non-amenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G\curvearrowright (X,\mu )$ there is a $G$-invariant probability measure $\nu$ on $n^G$ such that $G\curvearrowright (n^G,\nu )$ factors onto $G\curvearrowright (X,\mu )$. For many non-amenable 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.