Explicit Constructions of Universal R-trees and Asymptotic Geometry of Hyperbolic Spaces

We present some explicit constructions of universal R-trees with applications to the asymptotic geometry of hyperbolic spaces. In particular, we show that any asymptotic cone of a complete simply connected manifold of negative curvature is a complete homogeneous R-tree with the valency 20 at every point. It implies that all these asymptotic cones are isometric depending neither on a manifold nor on an ultrafilter. It is also proved that the same tree can be isometrically embedded at infinity into such a manifold or into a non-abelian free group.