Definable subsets and cyclic subgroups of the free group

Chloe Perin, Hebrew University

Abstract: Definable subsets of a group are sets of elements which satisfie a common first-order formula. The simplest example of a definable set is a variety, that is, the set of elements which satisfy a certain equation. In general, though, the first order formula may contain quantifiers of the form $\forall$ and $\exists$. We show that the intersection of a definable subset of a finitely generated free group with a cyclic subgroup C is, up to finitely many elements, a finite union of cosets of subgroups of C. We make extensive use of the "formal solution" techniques developped by Sela to study the first-order theory of free groups. The proofs are surprisingly geometric in nature, and rely on Rips analysis of actions on real trees and Rips and Sela's shortening argument.