over to a transformation from C2 − {0} to N+ which is homogeneous of degree two with respect to real scalings which agrees with () upon restriction to scales in which The components of () are precisely those obtained from the outer product In summary, the action of the restricted Lorentz group SO+(1,3) agrees with that of the Möbius group PSL(2,C). This motivates the following definition. In dimension n ≥ 2, the Möbius group Möb(n) is the group of all orientation-preserving conformal isometries of the round sphere Sn to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space R1,n+1, there is an isomorphism of Möb(n) with the restricted Lorentz group SO+(1,n+1) of Lorentz transformations with positive determinant, preserving the direction of time. Coxeter began instead with the equivalent quadratic form He identified the Lorentz group with transformations for which {x : Q(x) = -1} is stable. Then he interpreted the x's as homogeneous coordinates and {x : Q(x) = 0}, the null cone, as the Cayley absolute for a hyperbolic space of points {x : Q(x) < 0}. Next, Coxeter introduced the variables so that the Lorentz-invariant quadric corresponds to the sphere Coxeter notes that Felix Klein also wrote of this correspondence, applying stereographic projection from (0, 0, 1) to the complex plane Coxeter used the fact that circles of the inversive plane represent planes of hyperbolic space, and the general homography is the product of inversions in two or four circles, corresponding to the general hyperbolic displacement which is the product of inversions in two or four planes. Hyperbolic space As seen above, the Möbius group PSL(2,C) acts on Minkowski space as the group of those isometries that preserve the origin, the orientation of space and the direction of time. Restricting to the points where Q=1 in the positive light cone, which form a model of hyperbolic 3-space H, we see that the Möbius group acts on H as a group of orientation-preserving isometries. In fact, the Möbius group is equal to the group of orientation-preserving isometries of hyperbolic 3-space. If we use the Poincaré ball model, identifying the unit ball in R3 with H, then we can think of the Riemann sphere as the "conformal boundary" of H. Every orientation-preserving isometry of H gives rise to a Möbius transformation on the Riemann sphere and vice versa; this