Abstract:
The Myers-Steenrod theorem states that the isometry group of a compact Riemannian manifold is a compact Lie group. In Lorentzian signature, however, there are counterexamples: compact manifolds with non-compact isometry group. In this talk, we will instead consider (non-compact) globally hyperbolic spacetimes satisfying a ``no observer horizons'' condition. Our main result is that the isometry group acts properly on the spacetime. As corollaries, we obtain the existence of an invariant Cauchy time function, and a splitting of the isometry group into two subgroups: a compact one corresponding to spatial isometries, and a trivial, Z, or R factor corresponding to time translations. Time permitting, we will also discuss the conformal groups of these spacetimes. Based on joint work with Abdelghani Zeghib.