Ideal boundary (Hyperbolic space)

Description

If you have a Hyperbolic metric space, it is possible to move out in one direction. We identify a point to ‘“moving to a point at the horizon” which we call the ideal boundary

Definition

Let $X$ be a $\delta$-hyperbolic path metric space (meaning distances can be approximated by paths of similar lengths). There is an ideal boundary $\partial X$ functorially associated to $X$ whose points consist of quasigeodesic rays up to the equivalence relation of being a finite Hausdorff distance apart. There is a natural topology on $\partial X$ for which it is metrizable. If $X$ is proper, $\partial X$ is compact.

We can define the ideal boundary for Hyperbolic group too! See Ideal boundary (Hyperbolic groups)

Properties

Extends to continuous maps

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

Examples

Example