hyperbolic group

A finitely generated group $G$ is hyperbolic if, for some finite set of generators $A$ of $G$, the Cayley graph $\mathrm{\Gamma }(G,A)$, 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 $\mathrm{\Gamma }(G,A)$ is a hyperbolic metric space, then $\mathrm{\Gamma }(G,B)$ is a hyperbolic metric space.

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 $S_3$ (http://planetmath.org/CayleyGraphOfS_3) for a pictorial example.) If $G$ is a free group, then its Cayley graph is a real tree.