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 $\Sigma$. 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 $Sp\left(4,\mathbb{Z}\right)$.