Joint MathematicsComputer Science colloquium Date: Wednesday, 2nd March, 2022 Time: 10:30 am to 11:45 am Seminar Hall (this talk will be held in the hybrid mode) The technique of Stallings graphs Enric Ventura Universitat Politècnica de Catalunya, Spain. 020322 Abstract This talk will be about free groups $F_n$. Globally, the lattice of subgroups of a free group is very complicated, but Stallings popularized a graphical technique very useful to understand it, and to effectively work on it. I will remind the (combinatorial) construction of the free group $F_n$, will introduce Stallings graphs, and will explain how are they related to the lattice of subgroups of $F_n$. Then, we'll use these techniques to solve several classical algorithmic problems in free groups: the membership problem, the intersection problem, etc. We'll finish the talk by briefly explaining the main ideas behind a recent new result using Stallings graphs: given a finitely generated subgroup $H\leq F_n$ and an element $w\in F_n$ one can decide whether $w$ is the solution of some nontrivial equation with coefficients in $H$ and, in the affirmative case, find them all.
