<< [The Moebius group on n-space] is also isomorphic to Lorentz group in (n+2)-space, i.e. SO(n+1,1). To demo this isomorphism, just fly through space at relativistic velocities. A Lorentz transformation of your spacecraft induces a mapping of the celestial sphere which is a Moebius transformation.

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I suspect the "Moebius group on n-space" is really SO(n+1,1)/{+-I}rather than SO(n+1,1). As I understand it, the original Moebius group is defined as all conformal bijections of the Riemann sphere = S^2 = C u {oo} with itself (conformal meaning orientation-preserving as well as angle-preserving). This is the group of all linear fractional transformations given by z -> (az+b)/(cz+d), a,b,c,d complex. After eliminating repeats, this can be seen to be the group PSL(2,C) = SL(2,C)/{+-I}. And PSL(2,C) can be identified with the orientation-preserving isometries of hyperbolic 3-space. This group is connected, so it must be SO(3,1)/{+-I}. (So in fact it's not natural to say the original Moebius group acts on 2-space, since many transformations take some point to oo and take oo to a (finite) point.) So a natural definition of the "Moebius group on n-space" would be the orientation-preserving conformal transformations of the n-sphere, or alternatively the orientation-preserving isometries of hyperbolic (n+1)-space. One could also make a good case to ignore the orientation-preserving condition, since that may be seen in the original Moebius group as an artifact of the 2-dimensional sphere that it acts on conformally. With orientation-preserving, I think the "n-dimensional Moebius group" of conformal transformation of S^n can be identified with SO(n+1,1)/{+-I}. Without the orientation-preserving condition, I think it would then be O(n+1,1)/{+-I}. (It seems that most authors define the Lorentz group on (n+2)-space as just O(n+1,1).) --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele