Affine foliations and covering hyperbolic structures

We describe the relationship between closed affine laminations in a punctur ed surface and some associated hyperbolic structures on certain covers of t he punctured surface, which we call covering hyperbolic structures. Further . in analogy with the theory of William Thurston relating the Teichmuller s pace of a surface to the projective lamination space, we describe a space w ith points representing affine laminations in a given surface and with othe r points representing the associated covering hyperbolic structures.