Affine Deformations of Divisible Convex Cones and 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 $\mathrm{SO}_0(d,1)$.