Orbit equivalence relations and anti-classification results

Classification problems, such as determining when two topological spaces are the same, are central to many fields of mathematics. Invariant descriptive set theory provides a framework with which to measure and compare the difficulty of these problems, which can help determine if certain methods are strong enough to approach them. In this thesis, we study the classification problems which are induced by continuous actions of Polish groups on Polish spaces, which includes most problems arising naturally in other fields of mathematics. 

After an introductory chapter, we prove in Chapters 2 and 3 that there are interesting equivalence relations which are classifiable by CLI Polish groups but not by TSI Polish groups. The main example arises from the group-jump operation of Clemens-Coskey. We also give a characterization of generic ergodicity with respect to equivalence relations classifiable by non-Archimedean TSI Polish groups, as well as characterize the potential Borel complexity spectrum of equivalence relations classifiable by such groups.

In Chapter 4, we show that every countable treeable equivalence relation is classifiable by an abelian Polish group, answering a question of Hjorth in the negative. This provides some progress towards separating classifiability by TSI Polish groups and by abelian Polish groups, and suggests that this is a subtle problem.

In the next two chapters, we study the tame abelian Polish groups. In Chapter 5 show that there is an abelian product group which is D(Π 0/5)-tame but not Π 0/5-tame, answering a question of Ding-Gao. In Chapter 6 we show that there is a tame abelian Polish group which is universal among the tame non-Archimedean abelian Polish groups, answering another question of Ding-Gao.

Finally, in Chapter 7, we give an expository chapter on the Gandy-Harrington topology, a recursion-theoretic tool used in several dichotomy results in the field. Streamlined proofs are given for Hjorth’s dichotomies for turbulence and essential countability, while isolating and abstracting the recursion-theoretic components as much as possible.