Is 3D Gay Villa 2 safe?

Are the elements of a 3D rotation group generated by 2 arbitrary elements, with probability 1, dense in the set of all 3D rotations?

• More  specifically, if A and B are two arbitrary 3D rotations such that no concatenation of A's and B's yields the identity, is the group generated by A and B, in general, dense in the group of all 3D rotations? Also assume that A and B don't have the same axis of rotation. Of course there are a handful of cases where the resulting group is finite or small in other ways, I am interested in the general case. Extra credit: if the elements of the group are not dense, describe the geometry of the resulting 3D set of group elements, where a rotation is represented by its rotation axis with length equal to angle of rotation. (That's the unreal part of the log of the quaternion, in case you care.)

• Answer:

  The only proper closed subgroups of SO(3) are either finite or one-dimensional.  So, so long as at least one of A and B is not of finite order, and A and B don’t have the same axes of rotation, <A, B> is dense in SO(3).  Under any reasonable probability measure, both of these degenerate cases are probability zero.* In fact, Googling "closed subgroups of SO(3)" gives an answer to essentially this question: http://math.stackexchange.com/questions/103029/do-two-rotations-densely-generate-so3 * To be specific, the Haar measure is left-invariant under SO(3), so the probability of choosing a rotation with a specific axis is the same as the probability of choosing a  rotation with any other axis; since there are infinitely many axes, this probability must be zero.