The deformations of flat affine structures on the two-torus

TL;DR

This paper studies the complex deformation space of flat affine structures on the two-torus, revealing non-Hausdorff topology and establishing the holonomy map as a local homeomorphism, contrasting with classical Teichmüller theory.

Contribution

It characterizes the deformation space of flat affine structures on the two-torus and analyzes its topological properties, including the non-Hausdorff nature and the holonomy map behavior.

Findings

Deformation space is non-Hausdorff.

Holonomy map is a local homeomorphism.

Contrasts with classical Teichmüller theory.

Abstract

The group action which defines the moduli problem for the deformation space of flat affine structures on the two-torus is the action of the affine group $\Aff(2)$ on $\bbR^2$. Since this action has non-compact stabiliser $\GL(2,\bbR)$, the underlying locally homogeneous geometry is highly non-Riemannian. In this article, we describe the deformation space of all flat affine structures on the two-torus. In this context interesting phenomena arise in the topology of the deformation space, which, for example, is \emph{not} a Hausdorff space. This contrasts with the case of constant curvature metrics, or conformal structures on surfaces, which are encountered in classical Teichm\"uller theory. As our main result on the space of deformations of flat affine structures on the two-torus we prove that the holonomy map from the deformation space to the variety of conjugacy classes of homomorphisms from the fundamental group of the two-torus to the affine group is a local homeomorphism.