The transformation group of the universal covering of the real projective line, obtained by lifting ordinary projective transformations, is given explicitly in terms of canonical coordinates. A similar formulation is given of the action of the universal covering of SU(2, 2) upon the universal covering of the Šhilov boundary of its associated bounded Hermitian symmetric domain, structured as R 1 × SU(2). The former group G ̃ , isomorphic to S ̃ U(1, 1) , has a unique and continuous bi-invariant global partial ordering ⩽ (similar to that expressing space-time causality relations) corresponding to its bivariant Lorentzian metric; the partial ordering is the same as that induced by the ordering of the real line which the transformation group preserves. As an application, the compactness of the intervals [g 1, g 2] = {g ϵ G ̃ : g 1 ⩽ g ⩽ g 2} for g 1, g 2 ϵ G ̃ , necessary for global hyperbolicity of the metric, is studied. It is shown that [g 1, g 2] is compact if and only if g ⩽ ζg 1 for all g in a neighborhood of g 2, where ζ is the generator of the center of G ̃ satisfying ζ ⩾ e. In particular, the interior of [ e, ζ] is a maximal open global hyperbolic submanifold; S ̃ U(1, 1) is not globally hyperbolic.