Horoball in hyperbolic three-space

An open horoball in hyperbolic three-space is an isometric image of $\{(z,r):r>r_0\}$ , $r_0>0$ , in the upper-half-space model. Its boundary in hyperbolic space is a horosphere; its ideal center is the same as that of the boundary.

Models and convention

For a finite ideal center it is the interior of a Euclidean ball tangent to the boundary plane. A closed horoball also includes its bounding horosphere, but never includes the ideal center as a point of hyperbolic space.

Moving farther into an end

At center $\infty$ , increasing $r_0$ shrinks the horoball. This produces nested neighborhoods of the same cusp end.