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Abstract

Let us start with three equivalent definitions of hyperbolic groups. First observe that for every finitely presented group Γ there exists a smooth bounded (i.e. bounded by a smooth hypersurface) connected domain V ⊂ ℝn for every n ≥ 5. such that the fundamental group π1(V) is isomorphic to Γ. A standard example of such a V is obtained as follows. Fix a finite presentation of Γ and let P be the 2-dimensional cell complex whose 1-cells correspond in the usual way to the generators and the 2-cells to the relations in Γ, such that π1(P) = Γ. Then embed P into ℝ5 and take a regular neighborhood of P ⊂ ℝ5 for V.

Keywords

Fundamental Group Cayley Graph Closed Geodesic Hyperbolic Group 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.

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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • M. Gromov
    • 1
  1. 1.IHESFrance

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