Theories Involving Equations in the Free Group

Abstract

I will introduce equations in the free group (FGEs), outlining their history and properties. I'll give facets of the theory of FGEs that look similar and/or different to that of word equations. Both theories can be studied using techniques from the other; some "adapted results" will be given as proofs-of-concept. Our focus will then settle on definability in theories whose atoms are FGEs. Tools for establishing nondefinability in this setting are due to group theorists and model theorists, yet have the appearance of word-combinatorial lemmas. The tools - combined - yield characterisations of definability, the free monoid analogues of which seem more elusive. I'll end discussing complexity and the possibility of studying FGEs' length abstractions. The work is ongoing, so there'll be plenty of conjectures and unresolved examples.