can be used to classify isometries according to their dynamical behaviour on the boundary, generalising that for trees and classical hyperbolic spaces. Let be an isometry of , then one of the following cases occur: First case: has a bounded orbit on (in case is proper this implies that has a fixed point in ). Then it is called an elliptic isometry. Second case: has exactly two fixed points on and every positive orbit accumulates only at . Then is called an hyperbolic isometry. Third case: has exactly one fixed point on the boundary and all orbits accumulate at this point. Then it is called a parabolic isometry. More examples Subsets of the theory of hyperbolic groups can be used to give more examples of hyperbolic spaces, for instance the Cayley graph of a small cancellation group. It is also known that the Cayley graphs of certain models of random groups (which is in effect a randomly-generated infinite regular graph) tend to be hyperbolic very often. It can be difficult and interesting to prove that certain spaces are hyperbolic. For example, the following hyperbolicity results have led to new phenomena being discovered for the groups acting on them. The hyperbolicity of the curve complex has led to new results on the mapping class group. Similarly, the hyperbolicity of certain graphs associated to the outer automorphism group Out(Fn) has led to new results on this group. See also Negatively curved group Ideal triangle Notes References . Metric geometry Hyperbolic geometry