Ideal boundary (Hyperbolic groups)

Description

Similarly to the Ideal boundary (Hyperbolic space), we can move out into the horizon on the Cayley-Graph and get an ideal boundary that way. This is a generalisation of the Limit set of a Fuchsian group.

Definition (Ideal boundary for groups)

Let $G$ be a finitely generated group with generating set $S$ and Cayley-Graph $C_S(G)$. The ideal boundary $\partial G$ of $G$ is defined as the ideal boundary of $C_S(G)$. The ideal boundary does not depend on the choice of generators.

Properties

Extends to continuous maps

Any quasi-isometric embedding $X\to Y$ between hyperbolic spaces induces a continuous map $\partial X\to \partial Y$

Classification

The action of $g$ in the Cayley graph by left-multiplication induces an action on the ideal boundary. Depending on how an element $g\in G$ acts on the ideal boundary, it is classified into one of three types:

• Elliptic group element: The action on the ideal boundary is finite-order
• Hyperbolic group element: The action fixes two points $p^{\pm }$ on the ideal boundary. On is repelling and another is attracting
• What happened to the parabolic element? According to Calegari it doesn’t exist

Examples

Limit sets of Fuchsian groups

It is known that the Dirichlet region can be identified with the Cayley-Graph. This immediately tells us that the ideal boundary is the Limit set. (Which can be of one of two types)