A finitely generated group is hyperbolic if, for some finite set of generators of , the Cayley graph , considered as a metric space with being the minimum number of edges one must traverse to get from to , is a hyperbolic metric space.
Hyperbolicity is a group-theoretic property. That is, if and are finite sets of generators of a group and is a hyperbolic metric space, then is a hyperbolic metric space.
examples of hyperbolic groups include finite groups and free groups. If is a finite group, then for any , we have that . (See the entry Cayley graph of (http://planetmath.org/CayleyGraphOfS_3) for a pictorial example.) If is a free group, then its Cayley graph is a real tree.
|Date of creation||2013-03-22 17:11:43|
|Last modified on||2013-03-22 17:11:43|
|Last modified by||Wkbj79 (1863)|