An Msci project by Xinya Wang, 2023–2024

Abstract: In Section 1, we give an overview of Borel reducibility of equivalence relations, including foundational definitions, examples, and an introduction to the rich interplay between Borel reducibility and various levels of the Borel hierarchy. We explore the nuances of reducibility, detailing the mechanisms that allow us to compare the complexity of different equivalence relations and their classes. This section sets the stage for a deeper examination of specific types of relations and their behaviors under Borel reducibility.

In Section 2, we introduce conjugacy equivalence relations and give several applications of Borel reducibility to analyze their complexity, including discussions on the structure of the automorphism groups of infinite structures and their conjugacy relations. By providing examples of countable graphs and discussing the automorphism group of the infinite complete graph, the section illustrates the intricate relationship between conjugacy equivalence and the structure of mathematical classifications.

In Section 3, we consider operations on equivalence relations including both a join operation and a jump operation. We show the property of being bireducible with a conjugacy relation is is frequently preserved under these operations.