Geometry, Topology and Dynamics Seminar
Chloe Perin
Hebrew Universtiy
Definable subsets and cyclic subgroups of the free group.
Abstract: Definable subsets of a group are sets of elements which satisfy 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 developed 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.
Monday October 26, 2009 at 3:00 PM in SEO 612