Outline of “lorentz Spacetimes of Constant Curvature” by Geoff Mess

(d) Domains of dependence and geodesic laminations (§5). This section contains the main theorem in the flat case; essentially that domains of dependence are in one-one correspondence with measured geodesic laminations. First some basic causality; domains of dependence are defined and Proposition 11 describes the structure of the causal horizon. Proposition 12 is one half of the main theorem; given a hyperbolic surface and measured geodesic lamination, there is a corresponding flat spacetime. The proof is long and contains a lot of implicit information on the structure of these examples. Proposition 13 is the other half; that an arbitrary domain of dependence determines a measured geodesic lamination, inverse to the correspondence in Prop. 12. The section concludes with Propositions 14 and 15 which show that the group action on the causal horizon is complicated and uses this to characterize when domains of dependence embed in larger spacetimes.