Finite graphs, free groups and Stallings’ foldings

Abstract

[EN] The main goal of the work is to study some basic properties of free groups, by using group theory and topology. For this, graphs play a main role, since we will prove that the fundamental group of a connected graph is a free group. Therefore, basic notions and properties of free groups and graphs are explained firstly. Then, it is established a relation between operations in the category of graphs (the pullback and the pushout) and group theoretic operations between subgroups of free groups (the intersection and the join). After that, it is developed the theory of coverings of graphs. Finally, it is given an algorithm in order to decide whether a finitely generated subgroup of a free group is of finite index.