Topics in Geometric Group Theory. Part I

TL;DR

This survey explores asymptotic topological properties of finitely presented groups, such as QSF and GSC, highlighting recent progress and open questions in geometric group theory from a topological perspective.

Contribution

It provides a comprehensive overview of key topological properties of finitely presented groups and discusses recent advances and open problems in the field.

Findings

Identification of key properties like QSF and GSC in finitely presented groups

Progress in understanding the topological structure at infinity of these groups

Open questions about whether all finitely presented groups satisfy these properties

Abstract

This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of easy groups. As we will explain, these properties are central in the theory of discrete groups seen from a topological viewpoint at infinity. Also, we shall outline the main steps of the achievements obtained in the last years around the very general question whether or not all finitely presented groups satisfy these conditions.