Hopf fibration Free Action The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by G⋅x: The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X. A group action is called free if, for all , implies (i.e., only the identity element fixes any ). In other words, is free if the map sending to is injective , so that implies for all . This means that all stabilizers are trivial. A group with free action is said to act freely. The basic example of a free group action is the action of a group on itself by left multiplication . As long as the group has more than the identity element , there is no element which satisfies for all . An example of a free action which is not transitive is the action of on by , which defines the Hopf map . Nearest neighbor search ( N...