Characterizing hyperbolic spaces and real trees

Roberto Frigerio*, Alessandro Sisto

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

33 Citations (Scopus)


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 Δ ⊆ 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.

Original languageEnglish
Pages (from-to)139-149
Number of pages11
JournalGeometriae Dedicata
Issue number1
Early online date12 Mar 2009
Publication statusPublished - Oct 2009


  • Asymptotic cone
  • Detour
  • Gromov-hyperbolic
  • Real tree
  • Rips condition

ASJC Scopus subject areas

  • Geometry and Topology


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