Journal of the
Korean Mathematical Society JKMS

An isometry of hyperbolic space can be written as a composition of the reflection in the isometric sphere and two Euclidean isometries on the boundary at infinity. The isometric sphere is also used to construct the Ford fundamental domains for the action of discrete groups of isometries. In this paper, we study the isometric spheres of isometries acting on hyperbolic $4$-space. This is a new phenomenon which occurs in hyperbolic $4$-space that the two isometric spheres of a parabolic isometry can intersect transversally. We provide one geometric way to classify isometries of hyperbolic $4$-space using the isometric spheres.

Keywords: hyperbolic isometry, isometric sphere

MSC numbers: Primary 30F40; Secondary 20H10