A brief introduction to Gromov’s notion of hyperbolic groups

Let Γ be a group, and let F be a finite set of elements of Γ. By a word over F we mean a formal product of elements of F and their inverses. Every word over F determines an element of the group Γ, simply using the group operations. The “empty word” is considered a word over F , which corresponds to the identity element of Γ. If z is a word over F , then the length of z is denoted L(z) and is the number of elements of F such that they or their inverses are used in z, counting multiplicities. A word z is said to be irreducible if it does not contain an α ∈ F next to α, i.e., so that all obvious cancellations have been made. If a word z over F corresponds to the identity element of Γ, then z is said to be trivial. A finite subset F of a group Γ is a set of generators of Γ if every element of Γ corresponds to a word over F . A group is said to be finitely-generated if it has a finite set of generators. Let us make the convention that a generating set F of a group Γ should not contain the identity element of Γ. Suppose that Γ is a group and that F is a finite set of generators of Γ. The Cayley graph associated to Γ and F is the graph consisting of the elements of Γ as vertices with the provision that γ1, γ2 in Γ are adjacent if γ2 = γ1 α, where α is an element of F or its inverse. Thus this relation is symmetric in γ1 and γ2. A finite sequence θ0, θ2, . . . , θk of elements of Γ is said to define a path if θj , θj+1 are adjacent in the Cayley graph for each j, 0 ≤ j ≤ k − 1. The