Geometric group action

In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.
Definition.
In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group "G" on a geometry "X" is geometric if it satisfies the following conditions:
Uniqueness.
If a group "G" acts geometrically upon two geometries "X" and "Y", then "X" and "Y" are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which "G" acts geometrically is quasi-isometric to the Cayley graph of "G".
Examples.
Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.