Cost is a numerical invariant that serves as a measurable analogue of the minimal number of generators of a probability-measure-preserving equivalence relation. The cost of a group is the infimum of the costs of the orbit equivalence relations arising from its free probability-measure-preserving actions. The free group of rank r has cost r, whereas every infinite amenable group has cost 1. In both cases, all free probability-measure-preserving actions have the same cost. In this talk, I will show that free Burnside groups of sufficiently large odd exponent have cost 1, and consequently that their first \(\ell^2\)-Betti number vanishes. It is not known whether a group of bounded exponent can have cost larger than 1. I will present a normal subgroup theorem showing that, if such a group has finite centralizers of nontrivial elements, then every nontrivial normal subgroup must have finite index, and every ergodic p.m.p. action on a non-atomic probability space is essentially free. This is joint work with Miguel Donoso-Echenique.