Affine deformations of a three-holed sphere

Abstract

Associated to every complete affine 3–manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface Σ. We classify these complete affine structures when Σ is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Σ, the deformation space identifies with two opposite octants in ℝ3. 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).