Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface SIGMA. We classify these complete affine structures when SIGMA is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface SIGMA, the deformation space identifies with two opposite octants in R~(3). Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4, Z).
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