Affine deformations of quasi-divisible convex cones

TL;DR

This paper studies affine deformations of convex projective structures, classifies associated domains of discontinuity, and relates their moduli space to Teichmüller space, extending known results to a broader class of groups.

Contribution

It introduces a natural condition on translation parts for affine actions, classifies convex domains of discontinuity, and links their moduli space to the Teichmüller space of convex projective structures.

Findings

Convex domains of discontinuity are classified under certain conditions.

Quotients of these domains form affine manifolds with convex surfaces.

The moduli space forms a vector bundle over the Teichmüller space.

Abstract

For any subgroup of $\mathrm{SL}(3,\mathbb{R})\ltimes\mathbb{R}^3$ obtained by adding a translation part to a subgroup of $\mathrm{SL}(3,\mathbb{R})$ which is the fundamental group of a finite-volume convex projective surface, we first show that under a natural condition on the translation parts of parabolic elements, the affine action of the group on $\mathbb{R}^3$ has convex domains of discontinuity that are regular in a certain sense, generalizing a result of Mess for globally hyperbolic flat spacetimes. We then classify all these domains and show that the quotient of each of them is an affine manifold foliated by convex surfaces with constant affine Gaussian curvature. The proof is based on a correspondence between the geometry of an affine space endowed with a convex cone and the geometry of a convex tube domain. As an independent result, we show that the moduli space of such groups is a vector bundle over the moduli space of finite-volume convex projective structures, with rank equaling the dimension of the Teichm\"uller space.