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Group Actions and Quasi-Isometries

  • Martin R. Bridson
  • André Haefliger
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 319)

Abstract

In this chapter we study group actions on metric and topological spaces. Following some general remarks, we shall describe how to construct a group presentation for an arbitrary group Γ acting by homeomorphisms on a simply connected topological space X. If X is a simply connected length space and Γ is acting properly and cocompactly by isometries, then this construction gives a finite presentation for Γ. In order to obtain a more satisfactory description of the relationship between a length space and any group which acts properly and cocompactly by isometries on it, one should regard the group itself as a metric object; in the second part of this chapter we shall explore this idea. The key notion in this regard is quasi-isometry, an equivalence relation among metric spaces that equates spaces which look the same on the large scale (8.14).

Keywords

Cayley Graph Finite Index Length Space Geodesic Space Finite Presentation 
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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Martin R. Bridson
    • 1
  • André Haefliger
    • 2
  1. 1.Mathematical InstituteUniversity of OxfodOxfordGreat Britain
  2. 2.Section de MathématiquesUniversité de GenèveGenève 24Switzerland

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