Hyperbolic Group - Definitions

Definitions

Hyperbolic groups can be defined in several different ways. Many definitions use the Cayley graph of the group and involve a choice of a positive constant δ and first define a δ-hyperbolic group. A group is called hyperbolic if it is δ-hyperbolic for some δ. When translating between different definitions of hyperbolicity, the particular value of δ may change, but the resulting notions of a hyperbolic group turn out to be equivalent.

Let G be a finitely generated group, and T be its Cayley graph with respect to some finite set S of generators. By identifying each edge isometrically with the unit interval in R, the Cayley graph becomes a metric space. The group G acts on T by isometries and this action is simply transitive on the vertices. A path in T of minimal length that connects points x and y is called a geodesic segment and is denoted . A geodesic triangle in T consists of three points x, y, z, its vertices, and three geodesic segments, its sides.

The first approach to hyperbolicity is based on the slim triangles condition and is generally credited to Rips. Let δ > 0 be fixed. A geodesic triangle is δ-slim if each side is contained in a -neighborhood of the other two sides:

The Cayley graph T is δ-hyperbolic if all geodesic triangles are δ-slim, and in this case G is a δ-hyperbolic group. Although a different choice of a finite generating set will lead to a different Cayley graph and hence to a different condition for G to be δ-hyperbolic, it is known that the notion of hyperbolicity, for some value of δ is actually independent of the generating set. In the language of metric geometry, it is invariant under quasi-isometries. Therefore, the property of being a hyperbolic group depends only on the group itself.