Affine Deformations of Divisible Convex Cones and Affine Spacetimes

TL;DR

This paper studies affine deformations of convex cones and their quotients, showing they form maximal globally hyperbolic affine spacetimes with a cosmological time function, generalizing previous Lorentzian spacetime results.

Contribution

It introduces a new class of affine spacetimes arising from convex cone deformations, extending the understanding of their geometric and dynamical properties.

Findings

Quotients are maximal globally hyperbolic affine spacetimes with convex Cauchy surfaces.

Existence of a cosmological time function with Cauchy hypersurfaces as level sets.

Such quotients are characterized as the only examples of these affine spacetimes.

Abstract

Let $G$ be a subgroup of $\mathrm{SL}(\mathbb{R}^{d+1})\ltimes\mathbb{R}^{d+1}$ obtained by adding a translation part to a torsion-free discrete subgroup of $\mathrm{SL}(\mathbb{R}^{d+1})$ dividing a convex cone in the sense of Benoist. We consider the maximal convex domains in $\mathbb{R}^{d+1}$ on which the affine action of $G$ is free and properly discontinuous, and show its quotient by $G$ is naturally endowed with an "affine spacetime" structure, which is a generalisation of the notion of flat Lorentzian spacetime. More precisely, we show that this quotient is a Maximal Globally Hyperbolic affine spacetimes admitting a $C^2$ locally uniformly Convex and Compact Cauchy surface (denoted as a MGHCC affine spacetimes), and that it comes with a cosmological time function with Cauchy hypersurfaces foliating the quotient affine spacetime as level sets. Finally, we show such quotients are the only examples of MGHCC affine spacetimes. All these results generalise the work of Mess, Barbot and Bonsante on affine deformations of uniform lattices of $\mathrm{SO}_0(d,1)$.