A Study of finitely generated Free Groups via the Fundamental Groups

Abstract

Free groups have many applications in Algebraic Topology. In this paper I specifically study the finitely generated free groups by using the covering spaces and fundamental groups. By the Van Kampen's theorem, we have a famous fact that the fundamental group of a wedge sum of circles is a free group. Therefore, to study free groups, we could try to figure out the covering spaces of the wedge sum of circles. And in the appendix B, I prove the Nielsen-Schreier theorem which I will use this to study finitely index subgroups of a finitely generated free group.