Inversive Ring Geometry - Transitivity

Transitivity

When the ring is C it is common to construct a mapping that takes a triple of complex numbers to the standard triple 0, 1, ∞.

Suppose p, q, r ∈ A with

t = (r – p)−1 and v = (t + (q – r)−1)−1.

When these inverses t and v exist we say "p, q, and r are separated sufficiently". Up to sufficient separation, the group of projectivities is 3-transitive:

The first two factors put r at U(1, 0) = ∞ where it stays. The third factor moves t, the image of p under the first two factors, to U(0, 1), or zero in the canonical embedding. Finally, the fourth factor has traced q through the first three factors and formation of the rotation with v places U(q, 1) at U(1, 1). Thus the composition displayed places the triple p,q,r at the triple 0,1,∞. Evidently it is the unique such projectivity considering the pivotal use of fixed points of generators to bring the triple to 0,1,∞.

Proposition: If the group of projectivities is sharply 3-transitive, then there is a cross-ratio function which is invariant under the permutation of the projective line by projectivities.

proof: If s and t are two sufficiently separated triples then they correspond to projectivities g and h respectively which map each of s and t to (0,1,∞). Thus the projectivity h−1 g maps s to t .

Denote by (x,p,q,r) the image of x under the projectivity determined by p,q,r as above. This function f(x) is the cross-ratio determined by p,q,r ∈ A. The uniqueness of this function (sharp transitivity) implies that when a single projectivity g ∈ G(A) is used to form another triple g(p), g(q), g(r) from the first one, then the new cross-ratio function h must agree with f g. Hence h g−1 = f so that

As the sharpness does not hold in non-commutative rings like quaternions and biquaternions, there are limits to usage of cross-ratios.