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Finite Transitive Graph Embeddings into a Hyperbolic Metric Space Must Stretch or Squeeze

  • Itai BenjaminiEmail author
  • Oded Schramm
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)

Abstract

The δ-hyperbolicity constant of a finite vertex transitive graph with more than two vertices is proportional to its diameter. This implies that any map from such a graph into a 1-Gromov hyperbolic metric space has to stretch or squeeze the metric.

Keywords

finite Vertex-transitive Graphs Geodesic Triangle Gromov Hyperbolic Groups Quasi Proof Geodesic Segment 
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.

References

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    O. Angel, I. Benjamini, Phase transition for the metric distortion of percolation on the hypercube. Combinatorica 27, 645–658 (2007)Google Scholar
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    I. Benjamini, Expanders are not hyperbolic. Isarel J. Math. 108, 33–36 (1998)Google Scholar
  3. 3.
    E. Ghys, A. Haefliger, A. Verjovsky (eds.), Groups Theory from a Geometrical Viewpoint (World Scientific, Singapore, 1991)Google Scholar
  4. 4.
    M. Gromov, Hyperbolic Groups, Essays in Group Theory, ed. by S. Gersten. MSRI Publications, vol. 8 (Springer, Berlin, 1987), pp. 75–265Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

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