Every Action of a Nonamenable Group Is the Factor of a Small Action

It is well known that if G is a countable amenable group and G y (Y ,ν) factors onto G y (X ,μ), then the entropy of the first action must be at least the entropy of the second action. In particular, if G y (X ,μ) has infinite entropy, then the action G y (Y ,ν) 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 y (X ,μ) there is a G-invariant probability measure ν on nG such that G y (nG ,ν) factors onto G y (X ,μ). 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.