  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.
