Explicit constructions of universal R-trees and asymptotic geometry of hyperbolic spaces

Details Explicit constructions of universal R-trees and asymptotic geometry of hyperbolic spaces Explicit constructions of universal R-trees and asymptotic geometry of hyperbolic spaces

1999-04-23

arxiv.org/abs/math/9904133v2

Mathematics

Differential Geometry

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Scientific paper

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 $2^{\aleph_0}$ 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 R-tree can be isometrically embedded at infinity into such a manifold or into a non-abelian free group.

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