The identity component of the isometry group of a compact Lorentz manifold

Abstract

This result may be compared with a Theorem of E. Ghys [Ghy] (see also [Bel]), asserting a similar conclusion, but assuming that M has dimension 3, and that the action is just volume preserving and locally free. The statement there, is that the action of AG may be extended to an action of a finite cover of PSL(2,R), or to an action of the solvable 3-dimensional Lie group SOL. Here we have another motivation. We want to understand the structure of Lie groups acting isometrically on compact Lorentz manifold. The first results in the subject are due to [Zim] and [Gro]. A “final” result is due to [A-S] and [Zeg1], independently. Necessary and sufficient conditions were given in order that a Lie group acts isometrically (and locally faithfully) on a compact Lorentz manifold. Note however, that if a group acts in such a fashion, then its subgroups also act in the same way. For instance, all known examples of isometric actions of AG are obtained by viewing it as a subgroup of SL(2,R). So a natural question is: what are the maximal (connected) Lie groups acting isometrically on a compact Lorentz manifold? Equivalently: