In this talk, we will first review some of the analogies between full groups of measure-preserving equivalence relations and the symmetric group over the integers, which have been used by A. Eisenmann and Y. Glasner to provide interesting examples of invariant random subgroups (IRSs) of the free group.
We will then see how the notion of cost, introduced by G. Levitt, naturally enters this picture. After that, we will explain how a stronger analogy between full groups and the symmetric group over the integers holds in the type III case.
A joint result with A. Kaïchouh which uses this analogy will be presented : full groups of hyperfinite type III equivalence relations have ample generics. This provides a positive answer to a question of A. Kechris and C. Rosendal on the existence of connected Polish group with ample generics.[-]