Boundaries of Kleinian groups

TL;DR

This paper reviews how the topology of the boundary influences the splittings of hyperbolic groups and applies this understanding to classify limit sets of Kleinian groups up to homeomorphism.

Contribution

It provides explicit examples of boundary phenomena and connects boundary topology with the structure of Kleinian groups.

Findings

Explicit boundary phenomena examples

Classification of Kleinian group limit sets

Connection between boundary topology and group splittings

Abstract

We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.