Essays in Group Theory

Volume 8 of the series Mathematical Sciences Research Institute Publications pp 75-263

Hyperbolic Groups

  • M. GromovAffiliated withIHES

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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.