Globalizations, fiber bundles, and envelopes of holomorphy

We describe a relationship between globalizations of local holomorphic acti ons on Stein manifolds induced by global actions of certain non-compact Lie groups, and holomorphic fiber bundles with smooth Stein base and fiber and connected structure group. To this end we prove a univalence result for pa rticular Stein Riemann domains with a free and properly discontinuous actio n of a discrete group of biholomorphisms, We then derive some consequences on the existence of Stein globalizations.