Ideal boundary (${\partial }_{\infty }{\mathbb{H}}^n\cong S^{n-1}$)

The ideal boundary of hyperbolic space ${\mathbb{H}}^n$ records directions in which geodesics escape to infinity. Its points can be represented by geodesic rays, identifying rays that stay a bounded distance apart. With the usual boundary topology it is a sphere $S^{n-1}$.

In the unit-ball model it is the surrounding unit sphere; in the upper-half-space model it is ${\mathbb{R}}^{n-1}\cup \{\infty \}$. For $n=3$ this is the Riemann sphere. These are ideal endpoints, not points at finite hyperbolic distance. For instance, the vertical metric element $dy/y$ makes the distance from height one to height $\epsilon$ equal to $|\log \epsilon |$, which diverges as $\epsilon \to 0$.

A complete hyperbolic geodesic has two distinct ideal endpoints. Hyperbolic isometries act on the ideal boundary, translating geometric questions about geodesics into questions about their endpoint pairs. The boundary topology and the hyperbolic metric play different roles: the closed-ball compactification adds ideal points without assigning them finite distance.