From: Robin Chapman Subject: Re: subgroups of SO(n) acting on S^{n-1} Date: Wed, 19 Jan 2000 08:25:32 GMT Newsgroups: sci.math Summary: [missing] In article <863qk2$k6f$1@nnrp1.deja.com>, David Bernier wrote: > Say n=3. I take SO(3) to be the group of 3x3 real-valued > orthogonal matrices with determinant one. Suppose > {g_1, g_2, .... g_K} is a set of K distinct elements of > SO(3), and consider G= = the subgroup of > SO(3) generated by {g_1, g_2, .... g_K}. G acts on the > set S^2 (= {x in R^3, |x|=1} ) in an obvious way. > How large does K have to be so that the point (1, 0, 0) > has a dense orbit in S^2 ? > > I think it's true that any element g of SO(3) other than > the identity has two antipodal fixed points, so the orbit > of any point in S^2 under lies in some plane. Yes, such an element is a "rotation" with an "axis". > Therefore, > dense orbits require K=>2. I seem to remember from > reading about the Banach-Tarski paradox that there are > subgroups of SO(3) having two generators that contain > a free group on two generators. Yes, see Stan Wagon's _The Banach-Tarski Paradox". As an example of a 2-generator free subgroup of SO(3) he gives that generated by rotations of angle arccos(1/3) about the x- and z-axes. -- Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html "`The twenty-first century didn't begin until a minute past midnight January first 2001.'" John Brunner, _Stand on Zanzibar_ (1968) Sent via Deja.com http://www.deja.com/ Before you buy. ============================================================================== From: foltinek@math.utexas.edu (Kevin Foltinek) Subject: Re: subgroups of SO(n) acting on S^{n-1} Date: 19 Jan 2000 11:50:43 -0600 Newsgroups: sci.math In article <863qk2$k6f$1@nnrp1.deja.com> David Bernier writes: > Say n=3. I take SO(3) to be the group of 3x3 real-valued > orthogonal matrices with determinant one. Suppose > {g_1, g_2, .... g_K} is a set of K distinct elements of > SO(3), and consider G= = the subgroup of > SO(3) generated by {g_1, g_2, .... g_K}. G acts on the > set S^2 (= {x in R^3, |x|=1} ) in an obvious way. > How large does K have to be so that the point (1, 0, 0) > has a dense orbit in S^2 ? Given a randomly selected pair {g_1,g_2} (for any reasonable meaning of "randomly selected"), the orbit of any point by the action of the subgroup will be dense in S^2, with probability 1. Heuristic proof: With probability 1, g_1 will be a rotation by an irrational multiple of pi around some axis a_1, and g_2 will be rotation by a non-zero and non-pi angle around an axis a_2 different from a_1. Let p be an arbitrary point on the sphere. If p is on a_1, then rotation by g_2 will move p to a point not on a_1, so we can assume that p is not on a_1. The orbit of p by the action of the subgroup is a set which is dense in some circle of positive radius (because p is not on a_1). Acting on this circle by g_2 gives another circle which is not parallel to the first circle (because a_2 is different from a_1 and the angle is neither zero nor pi). The orbit of this new circle by is dense in a band of positive width around the sphere. Acting again by g_2 and then by increases the width of this band. Eventually (or sooner, if you replace the action by g_2 by an action by (g_2)^m for an appropriate m) you densely cover the sphere. Of course this is not a rigorous proof because you can't "act by " to obtain a dense set, and then do something else; all you can do is write down finite words such as (g1 g2 g1 g1 g2 g1). But you can put repeated strings of g1 into the word which are as long as you want, and that's really all you need to do when you unwind the definition of "dense". A similar construction works for SO(n) acting on S^{n-1} with the same result. Things get more difficult when all of the angles are rational multiples of pi. I seem to recall (but don't quote me on this!) that most of these orbits are also dense. (I think this is because the commutator [g1,g2] will probably be non-zero and small compared to g1 and g2; let g3=[g1,g2], and then probably [g3,g1] will be non-zero and small compared to g3; continuing in this manner, we obtain arbitrarily small elements of . "Small" of course means small angle of rotation.) Which now raises the obvious question: Given g_1 and g_2, elements of SO(3), when is NOT dense in SO(3)? Obvious example is g_1 and g_2 both rotations by pi/2 around orthogonal axes. Any others? Kevin.