# An introduction to semistability in geometric group theory

## TL;DR

This paper introduces the concept of semistability at infinity in geometric group theory, discusses techniques for proving it, and demonstrates its application to certain hyperbolic groups using hierarchy and boundary topology methods.

## Contribution

It provides an overview of semistability at infinity and applies a combination theorem to establish semistability for specific relatively hyperbolic groups.

## Key findings

- Semistability at infinity can be proved using boundary topology or group hierarchies.
- The combination theorem applies to hyperbolic groups relative to polycyclic subgroups.
- Semistability is established for a new class of relatively hyperbolic groups.

## Abstract

A finitely presented group is semistable at infinity if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable at infinity. This article provides a brief introduction to the notion of semistability at infinity in geometric group theory. We discuss techniques for proving semistability at infinity that involve either the topology of the boundary or the existence of certain hierarchies of splittings of the group. As an illustration of the second technique, we apply a combination theorem of Mihalik-Tschantz to prove semistability at infinity for groups that are hyperbolic relative to polycyclic subgroups using work of Mihalik-Swenson and Louder-Touikan.