This project aims to study equivalence relations in groups. Groups are a class of fundamental objects in Algebra, and more generally, Mathematics. Originally, they were used to define symmetries of patterns --- from how crystals form to the rotations of Rubik’s cube --- but it turns out that many important mathematical structures can be thought of as a group with some extra structure. On the other hand, equivalence relations are relations that describe “sameness”, which is one of most natural questions to ask when given two objects, and whose meaning depend on the exact context. In group theory, there are many natural relations that are equivalence relations. Equivalence relations in group theory have a long history of being one of the most important testing grounds for ideas in computability theory, which has stimulated progress in both areas. This project aims to investigate this interaction, but under the light of a tool, computable reduction, that has seen rapid development in the last two decades. Compared to classical literature, this new perspective allows a finer analysis of the equivalence relations, which allows us to investigate deeper the complexity of these equivalence relations. Furthermore, this new perspective is expected to motivate the development of new tools in both group theory and computability theory, which have the potential to be useful to other problems beyond this project. This project also provides opportunities and support for students to engage i