Parametrization of causal actions of universal covering groups and global hyperbolicity

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 ?hilov boundary of its associated bounded Hermitian symmetric domain, structured as R¹ × SU(2). The former group $\text{G}\text{?}$, isomorphic to $\text{S}\text{?}\text{U(1, 1)}$, 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 $\text{g}{}_1\text{, g}{}_2\text{] = {g ?}\text{G}\text{?}\text{: g}{}_1\text{? g ? g}{}_2\text{}}\text{for}\text{g}{}_1\text{, g}{}_2\text{?}\text{G}\text{?}$, necessary for global hyperbolicity of the metric, is studied. It is shown that g₁, g₂] is compact if and only if g ? ζg₁ for all g in a neighborhood of g₂, where ζ is the generator of the center of $\text{G}\text{?}$ satisfying ζ ? e. In particular, the interior of e, ζ] is a maximal open global hyperbolic submanifold; $\text{S}\text{?}\text{U(1, 1)}$ is not globally hyperbolic.