Affine deformations of a three-holed sphere

TL;DR

This paper classifies complete affine structures on three-holed spheres, describing their deformation spaces, geometric properties, and explicit constructions of proper affine deformations of certain arithmetic groups.

Contribution

It provides a complete classification of affine structures on three-holed spheres and links these to hyperbolic surfaces and arithmetic Fuchsian groups.

Findings

Deformation space is two opposite octants in R^3.

Every manifold admits a fundamental polyhedron bounded by crooked planes.

Constructs proper affine deformations of an arithmetic Fuchsian group.

Abstract

Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such 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).