The Hyperbolic Plane and Hyperbolic Graphs

  • Itai Benjamini
Part of the Lecture Notes in Mathematics book series (LNM, volume 2100)


The aim of this section is to give a very short introduction to planar hyperbolic geometry. Some good references for parts of this section are [CFKP97] and [ABC+91]. We first discuss the hyperbolic plane. Nets in the hyperbolic plane are concrete examples of the more general hyperbolic graphs. Hyperbolicity is reflected in the behaviour of random walks [Anc88] and percolation as we will see in Chap. 7.


Symmetric Space Hyperbolic Space Cayley Graph Hyperbolic Plane Hyperbolic Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [ABC+91]
    J.M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, H. Short, Notes on word hyperbolic groups, in Group Theory from a Geometrical Viewpoint, Trieste, 1990, ed. by H. Short (World Scientific Publishing, River Edge, 1991), pp. 3–63Google Scholar
  2. [Anc88]
    A. Ancona, Positive harmonic functions and hyperbolicity, in Potential Theory—Surveys and Problems, Prague, 1987. Lecture Notes in Mathematics, vol. 1344 (Springer, Berlin, 1988), pp. 1–23Google Scholar
  3. [BS00]
    M. Bonk, O. Schramm, Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10(2), 266–306 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [CFKP97]
    J.W. Cannon, W.J. Floyd, R. Kenyon, W.R. Parry, Hyperbolic geometry, in Flavors of Geometry. Mathematical Sciences Research Institute Publications, vol. 31 (Cambridge University Press, Cambridge, 1997), pp. 59–115Google Scholar
  5. [GH90]
    É. Ghys, A. Haefliger, Groupes de torsion, in Sur les groupes hyperboliques d’après Mikhael Gromov, Bern, 1988. Progress in Mathematics, vol. 83 (Birkhäuser, Boston, 1990), pp. 215–226Google Scholar
  6. [Shc13]
    V. Shchur, A quantitative version of the Morse lemma and quasi-isometries fixing the ideal boundary. J. Funct. Anal. 264(3), 815–836 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Itai Benjamini
    • 1
  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations