Bernoulli actions are weakly contained in any free action

TL;DR

This paper proves that any free probability measure preserving action of a countable group weakly contains all Bernoulli actions, with implications for cost maximization and ergodic properties of such actions.

Contribution

It establishes that all Bernoulli actions are weakly contained in any free action for countable groups, revealing new insights into their structure and cost properties.

Findings

Bernoulli actions are weakly contained in all free actions.

Finitely generated groups have maximal cost on Bernoulli actions.

Ergodic but not strongly ergodic actions are weakly equivalent to their product with a trivial action.

Abstract

We show that for any countable group, any free probability measure preserving action of the group weakly contains all Bernoulli actions of the group. It follows that for a finitely generated groups, the cost is maximal on Bernoulli actions and that all free factors of i.i.d.-s the group have the same cost. We also show that if a probability measure preserving action f is ergodic, but not strongly ergodic, then f is weakly equivalent to f\timesI where I denotes the trivial action on the unit interval. This leads to a relative version of the Glasner-Weiss dichotomy.