On the Geometry and Topology of Discrete Groups: An Overview

In this paper, we provide a brief introduction to the main notions of geometric group theory and of asymptotic topology of finitely generated groups. We will start by presenting the basis of discrete groups and of the topology at infinity, then we will state some of the main theorems in these fields. Our aim is to give a sample of how the presence of a group action may affect the geometry of the underlying space and how in many cases topological methods may help the determine solutions of algebraic problems which may appear unrelated.