Description
An element is hyperbolic, if there is a quasi-geodesic along which the element acts by translation. Here we understand a quasi-geodesic coarsely, i.e. two quasi-geodesics are considered the same if they are a finite distance apart. This allows the inclusion of non-axial isometries which move the elements up and reflect them, like in a ladder (Gleitspiegelung).
Definition
Let be a group acting on a Hyperbolic metric space by isometries. An isometry is called hyperbolic if it admits an invariant quasi-geodesic as axis. If I understand correctly, the quasigeodesic-axis is seen to be unique up to finite movement.
Definition
Let be a group acting on a Hyperbolic metric space by isometries. An isometry is called hyperbolic if it has two fixed points on the Ideal boundary (Hyperbolic space), one of which being attracting, the other one being repelling.
Definition (Quasi-axis)
The
Properties
Tip
Examples
Axial isometries
An Axial isometry is a special case, since here the axis is even geodesic. The Gleitspiegelung-example shows that there are hyperbolic elements , which are not axial. However a power might be (but doesn’t need to be) axial.