Highly faithful actions and dense free subgroups in full groups

Abstract

In this paper, we show that every measure-preserving ergodic equivalence relation of cost less than $m$ comes from a “rich” faithful invariant random subgroup of the free group on $m$ generators, strengthening a result of Bowen which had been obtained by a Baire category argument. Our proof is completely explicit: we use our previous construction of topological generators for full groups and observe that these generators induce a totally non free action. We then twist this construction so that the action is moreover amenable onto almost every orbit and highly faithful. In particular, we obtain that the full group of a measure-preserving ergodic equivalence of cost less than m contains a dense free subgroup on m generators.