Gromov's Centralizer Theorem

We study the properties of rigid geometric structures and their relation with those of finite type. The main result proves that for a noncompact simple Lie group G acting analytically on a manifold M preserving a finite volume and either a connection or a geometric structure of finite type there is a nontrivial space of globally defined Killing vector fields on the universal cover M~M~ that centralize the action of G. Several appplications of this result are provided.