Geometriae Dedicata

, Volume 142, Issue 1, pp 139–149 | Cite as

Characterizing hyperbolic spaces and real trees

Original Paper

Abstract

Let X be a geodesic metric space. Gromov proved that there exists ε0 > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε0 · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε0. We also show that if all the triangles \({\Delta \subseteq X}\) satisfy the Rips condition with constant ε0 · pr(Δ), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.

Keywords

Gromov-hyperbolic Real tree Rips condition Asymptotic cone Detour 

Mathematics Subject Classification (2000)

53C23 20F67 

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Scuola Normale SuperiorePisaItaly

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