Let G be the group of projective transformations of R^(n+1) which preserve S^n. Let H be the group of conformal transformations of R^(n+1) which preserve S^n.
Then G and H are isomorphic, and we can choose the isomorphism such that they act on S^n in the same way.
G ~ H is an interesting result --- do you have a reference?
Yes, they're isometries of different models of the same hyperbolic (n+1)-space: G: consider the Beltrami-Klein model. H: consider the Poincare disc model. Lorentz group SO+(n+1,1): consider the hyperboloid model. (You can probably find references for each of these.)
From this, you get isomorphisms such as:
Lorentz group SO+(3,1) ~ Möbius group PSL(2,C) which is intimately entangled with the fact that relativistic boosts cause your celestial sphere to undergo a conformal deformation (stellar aberration).
Your H is just the Moebius(n) group, so it would follow that Moebius(n) is a subgroup of Project(n+1) ; as is Project(n) of course, though acting on a plane rather than a sphere, so again confounding my ambitions for a finite-dimensional representation of the joint conformal-projective group.
Also I ran a simulation generating random 2-space transformations of a fixed point-set which confirms that the dimension exceeds 24 .
I've been skimming through a book found online: "A Taste of Jordan Algebras" Kevin McCrimmon (2004) http://math.nsc.ru/LBRT/a1/files/mccrimmon.pdf which proceeds at a considerably more leisurely pace than Bertram's papers. But all the examples so far seem to be of orthogonal/Hermitian groups ...
Dan wrote << I've been suspecting that in fact the conformal and projective groups on S^n, as I've described them, are identical. Have not proved this yet. But if it's true, it must be well-known.
But (qua topological groups) their dimensions are respectively (n+2)(n+1)/2 versus (n+1)^2 - 1 ?!
Fred Lunnon
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