Hyperbolic metric space

Description

A very general Metrischer Raum is hyperbolic if all geodesic triangles have a diameter of less then some constant $\delta$.

Definition

Let $X$ be a Metrischer Raum mit Pfadmetrik. It is said to be $\delta$-hyperbolic if for any three Geodesic arcs $\alpha ,\beta ,\gamma$ in $X$ forming a geodesic triangle we have $\alpha$ be contained completely in the $\delta$-neighbourhood of $\beta \cup \gamma$.

There is another definition using the groov product which does not require path metric spaces.

Note, that hyperbolicity does not mean that the group has hyperbolic growth. $\mathbb{Z}$ is a simple example.

Properties

Morse Lemma

See Morse lemma

Quasigeodesity is local

For every $k,ϵ$ there is a universial constant $C(\delta ,k,ϵ)$ such that every map $\varphi :\mathbb{R}\to X$ which restricts on each segment of length $C$ to a $(k,ϵ)$-quasigeodesic is globally $(2k,2ϵ)$-quasigeodesic.

The last theorem state that every curve which is a local geodesic also is a global geodesic (if you include more tolerance).

Examples

Free groups

Free groups are hyperbolic

Fundamental groups of hyperbolic manifolds

Fundamental group of manifolds with Euler characteristic $\chi <0$ are hyperbolic

Random groups

A group $G=⟨X_m|R⟩$ on a finite generating set $X_m$ with a “random” set of relations $R$ with a suitable probability is hyperbolic with probability $1$.

Non-examples

Those are some counter-intuitive non-examples

• Amenable Group
• $SL(n,\mathbb{Z})$ for $n\ge 3$
• Fundamental groups of cusped hyperbolic manifolds of dimension $\ge 3$