Every action of a nonamenable group is the factor of a small action

Abstract

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.