In 1911, Dehn proposed three decision problems for finitely presented groups: word problem, conjugacy problem, and isomorphism problem. These problems have been central to both group theory and logic, and were each proven to be undecidable in the 50s. There is much current research studying the decidability of these problems in classes of groups.

Although these problems are classically studied as decision problems, each of them is naturally an equivalence relation. In this talk, we study them as equivalence relations and compare them using computable reductions. This leads to a more refined measure of their complexity and brings new results and questions.

Parts of the talk are joint works with Uri Andrews, Matthew Harrison-Trainor, and Luca San Mauro.