hyperbolic group

A finitely generated group $G$ is hyperbolic if, for some finite set of generators $A$ of $G$ , the Cayley graph $\Gamma(G,A)$ , considered as a metric space with $d(x,y)$ being the minimum number of edges one must traverse to get from $x$ to $y$ , is a hyperbolic metric space.

Hyperbolicity is a group-theoretic property. That is, if $A$ and $B$ are finite sets of generators of a group $G$ and $\Gamma(G,A)$ is a hyperbolic metric space, then $\Gamma(G,B)$ is a hyperbolic metric space.

Simple examples of hyperbolic groups include finite groups and free groups. If $G$ is a finite group, then for any $x,y \in G$ , we have that $d(x,y) \le |G|$ . (See the entry Cayley graph of $S_3$ for a pictorial example.) If $G$ is a free group, then its Cayley graph is a real tree.