Essays in Group Theory pp 75-263 | Cite as

# Hyperbolic Groups

## 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 ⊂ ℝ^{n} for every n ≥ 5. such that the fundamental group π_{1}(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 π_{1}(P) = Γ. Then embed P into ℝ^{5} and take a regular neighborhood of P ⊂ ℝ^{5} for V.

## Keywords

Fundamental Group Cayley Graph Closed Geodesic Hyperbolic Group Geodesic Segment## Preview

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