ABSTRACT
In this chapter we turn back to Polish group actions and orbit equivalence relations. The Borel reducibility hierarchy of analytic equivalence relations has taken shape with the results we have established in the previous chapters. However, it is also noticeable that the results were obtained with a variety of machineries ranging from Baire category methods, measure theoretic methods, group theory, and descriptive set theory. There are still many pairs of equivalence relations we have introduced so far but have not mentioned their reducibility relation, either because it is still an open problem or because its known proof goes beyond the limitation of length or scope of this book. It is thus clear that a sweeping method to prove reducibility or, more challengingly, nonreducibility would be much desirable. The greatest success up to date is Hjorth’s theory of turbulence, which we present in this chapter. It will turn out that orbit equivalence relations from turbulent actions are not reducible to any S∞-orbit equivalence relation. This powerful theorem has many applications in the classification problems of mathematics, and its potential is still being discovered in current research.
