On the tree-likeness of hyperbolic spaces

Abstract

AbstractInside any proper hyperbolic geodesic spaceXwe construct a rooted topological${\mathbb R}$-treeTthat reflects the geometry ofXin the following sense. All rays inTare quasi-geodesic inX. Every geodesic ray inXlies eventually close to a ray ofT. The embedding ofTinXextends continuously to their boundaries in a finite-to-one way, the number of boundary points ofTmapping to a given boundary point ofXbeing bounded if the (Assouad) dimension of the boundary ofXis finite.