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On the tree-likeness of hyperbolic spaces

Published online by Cambridge University Press:  10 April 2017

MATTHIAS HAMANN
MATTHIAS HAMANN*
Affiliation:
Department of Mathematics, University of Hamburg, Bundesstraße 55, 20146 Hamburg, Germany. e-mail: matthias.hamann@math.uni-hamburg.de
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Abstract

Inside any proper hyperbolic geodesic space X we construct a rooted topological ${\mathbb R}$-tree T that reflects the geometry of X in the following sense. All rays in T are quasi-geodesic in X. Every geodesic ray in X lies eventually close to a ray of T. The embedding of T in X extends continuously to their boundaries in a finite-to-one way, the number of boundary points of T mapping to a given boundary point of X being bounded if the (Assouad) dimension of the boundary of X is finite.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 2 , March 2018 , pp. 345 - 361
Copyright
Copyright © Cambridge Philosophical Society 2017 

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