Hyperbolic group

Description

A group is word hyperbolic if it’s Cayley-Graph is a Hyperbolic metric space.

Definition

A group $G$ is word-hyperbolic $\delta$-hyperbolic if there is a finite symmetric generating set $S$ such that the Cayley-Graph $C_S(G)$ is $\delta$-hyperbolic.

Properties

Characterisation by action on ideal boundary

Hyperbolic groups are completely characterised by the dynamics of their action on the Ideal boundary (Hyperbolic groups).
This is a theorem by Bowditch which I won’t write down.

Contains quasi-isometric embeddings of free groups

Let $G$ be hyperbolic and let $\partial G$ contain more than $2$ points. Then a Ping-Pong Lemma-argument tells us that $G$ contains Quasi-isometric embeddings of Free group of arbitrary rank.

Examples

Elemantary hyperbolic groups

A hyperbolic group whose Ideal boundary (Hyperbolic groups) contains at most $2$ points is said to be elementary. A group is elementary i.f.f. it is virtually cyclic.