Geometriae Dedicata

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

Characterizing hyperbolic spaces and real trees

  • Roberto Frigerio
  • Alessandro Sisto
Original Paper


Let X be a geodesic metric space. Gromov proved that there exists ε 0 > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε 0 · 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 ε 0. We also show that if all the triangles \({\Delta \subseteq X}\) satisfy the Rips condition with constant ε 0 · 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.


Gromov-hyperbolic Real tree Rips condition Asymptotic cone Detour 

Mathematics Subject Classification (2000)

53C23 20F67 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bonk M.: Quasi-geodesic segments and Gromov hyperbolic spaces. Geom. Dedicata 62, 281–298 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Drutu C.: Quasi-isometry invariants and asymptotic cones. Int. J. Alg. Comp. 12, 99–135 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gromov M.: Hyperbolic groups, Essays in group theory (Springer, New York). Math. Sci. Res. Inst. Publ. 8, 75–263 (1987)MathSciNetGoogle Scholar
  5. 5.
    Gromov, M.: Asymptotic invariants of infinite groups. Geometric group theory, vol. 2 (Cambridge Univ. Press, Cambridge). London Math. Soc. Lecture Note Ser. 8, 1–295 (1993).Google Scholar
  6. 6.
    van den Dries L., Wilkie J.: Gromov’s theorem on groups of polynomial growth and elementary logic. J. Algebra 89, 349–374 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Wenger S.: Gromov hyperbolic spaces and the sharp isoperimetric constant. Invent. Math. 171, 227–255 (2008)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

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

Personalised recommendations