On 1/13/08, Eugene Salamin <gene_salamin@yahoo.com> wrote:
... I should have been more specific. The Euclidean group in 3-space contains the 3-dimensional abelian normal subgroup of translations. The Mobius group on the 2-sphere (thanks, Dan), being semisimple cannot contain a solvable normal subgroup. The Mobius group has a 2-dimensional abelian subgroup (of translations preserving infinity), but it is not normal, since the fixed point, infinity, can be conjugated to any other point.
In the YouTube video, translation of the sphere normal to the plane induced the magnification Mobius transformation. But magnification by factor m and translation by t do not commute: m(z+t) /= mz + t. So the YouTube map is not an isomorphism. (Is it of any interest to think about mappings between groups that are not homomorphisms?)
Phew --- sorted out properly at last --- many thanks! One way and another, that little bug slipped well under my radar ...
Fred, what is a Lie sphere?
No such thing, I'm afraid --- only a Lie sphere group! There doesn't seem to be a good name for this geometry: Lie seems to have originally been responsible for it; but we can hardly call it the "Lie group". The only modern reference book I know of is T.~E.~Cecil, Lie Sphere Geometry, Springer (1992). The classic on sphere geometry book is J.~L.~Coolidge, A Treatise on the Circle and the Sphere, Clarendon Press (1916); Chelsea (1971) which called it "oriented sphere geometry" or some such. There is a paper in the Coxeter Festschrift: I.~M.~Yaglom, On the Circular Transformations of M\"obius, Laguerre, and Lie; and several available online (which I cannot wholeheartedly recommend) including B.~J.~Zlobec, N.~M.~Kosta, Geometric Constructions on Cycles, Rocky Mt. J. Math.34 (2004) 1565--1585. It's a shame this group is not better known --- I've several times seen claims, often by quite respectable mathematicians, to the effect that the "Moebius group is the largest group preserving spheres" --- which is quite simply untrue! The Lie sphere group is generated by the union of the Moebius and Laguerre groups. In n-space it has dimension (n+3)(n+2)/2, and (modulo various finite quotients about which I don't want to get into arguments here) the spin group is essentially O(2,n). I think of it as a third step in the sequence Euclidean group, Moebius group, Lie sphere group. Apparently physicists have also considered the Lie sphere group --- but just to confuse everybody thoroughly, they refer to it as the "conformal" group ...
While Googling for the answer, I discovered the following Indra's Pearls web site. http://klein.math.okstate.edu/IndrasPearls/
This site has improved a bit since last I inspected it. I've generated some intriguing graphics while investigating the Lie-sphere group, but so far sadly not got around to putting them in any fit state to distribute. A higher priority would be to present the thing in a more transparent and constructive fashion than seems to have been managed up to now ... Fred Lunnon