The class of hyperbolic groups was introduced and studied in Gromov's paper [125]. In that paper a not necessarily finitely generated group is called hyperbolic if it is hyperbolic as a metric space, in the sense that the space has a hyperbolic inner product, which we define in 6.2.1 below. The group is called word-hyperbolic if it is finitely generated and hyperbolic with respect to the word metric. Since we are only concerned with finitely generated groups in this book, we shall from now on refer to such groups simply as hyperbolic groups.

The fact that there is a large variety of apparently different conditions on a finitely generated group that turn out to be equivalent to hyperbolicity (we present a list of several such conditions in Section 6.6) is itself a strong indication of the fundamental position that these groups occupy in geometric group theory. We have already encountered two of these conditions: groups having a Dehn presentation (or algorithm) in Section 3.5, and strongly geodesically automatic groups in Section 5.8.

Gromov's paper is generally agreed to be difficult to read, but there are several accessible accounts of the basic properties of hyperbolic groups, including those by Alonso et al. [5], Ghys and de la Harpe [94], Bridson and Haefliger [39, Part II, Section Γ] and Neumann and Shapiro [203].

Hyperbolicity conditions

We begin by comparing various notions of hyperbolicity. The definitions that we consider here apply to an arbitrary geodesic metric space, as defined in Section 1.6, but we are mainly interested in the case when the space is the Cayley graph of a finitely generated group.

Let be a geodesic metric space. A geodesic triangle xyz in Γ consists of three points x, y, z together with geodesic paths [xy], [yz] and [zx]. We can define hyperbolicity of Γ in terms of ‘thinness’ properties of geodesic triangles.