From Hyperbolic Paths to Vacuum Space-Times

Given a piecewise geodesic in the hyperbolic plane, we construct a natural future-complete vacuum plane wave space-time. In particular, this gives a straightforward way to embed hyperbolic billiards into general relativity, and so the problem of classifying vacuum space-times is at least as complicated as classifying hyperbolic billiard trajectories.  We shall also discuss how this fits in with work of Hjorth and Kechris on the Borel equivalence problem for hyperbolic 2-manifolds.