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

This paper presents explicit constructions of universal ℝ-trees as certain spaces of functions, and also proves that a 2 ℵ 0 -universal ℝ-tree can be isometrically embedded at infinity into a complete simply connected manifold of negative curvature, or into a non-abelian free group. In contrast to asymptotic cone constructions, asymptotic spaces are built without using the axiom of choice.