Topología de grafos finitos

Abstract

The goal of this work is to prove some standard theorems on free groups by using graphs and their topology. For this, we have followed Stallings’ paper Topology of Finite Graphs [11]. We will use the fundamental group of a connected graph, which is a free group, and immersions and coverings of graphs to represent subgroups of a free group. In this way, we will see Howson’s Theorem that if A and B are finitely generated subgroups of a free group, then A ∩ B is finitely generated, and M. Hall’s Theorem. This last theorem states that if S is a finitely generated subgroup of a free group F and β1 , ..., β1 ∈ F but β1, ..., β1 ∉ S, then there exists a subgroup S ′ of finite index in F such that S ⊂ S′ , β1 , ..., β1 ∉ S ′ , and there exists a free basis of S ′ having a subset which is a free basis of S. Lastly, we will use core-graphs to proof that if A and B are finitely generated subgroups of a free group and if A ∩ B is of finite index in both A and B, then A ∩ B is of FInite index in A ∨ B, the subgroup generated by A ∪ B.