Conformal actions of semi-simple Lie groups on compact Lorentz manifolds

A result of R. Zimmer going back to the 1980's asserts that up to local isomorphism, SL(2,R) is the only non-compact simple Lie group that can act by isometries on a Lorentz manifold of finite volume. The proof is mainly based on ergodic arguments and it is essential that the group preserves a finite measure.

If we relax the assumption and consider conformal dynamics of Lie groups, we loose the existence of an invariant finite measure, even when the manifold is compact. However, conformal structures are rigid in dimension at least 3, so that it seems possible to describe conformal Lie group actions. In this talk, I will present results investigating the case where the Lie groups are semi-simple, and I will discuss the geometry of the Lorentz manifolds where such dynamics occur.