Affine Deformations of Divisible Convex Cones and Affine Spacetimes

Abstract: Given 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 domain in the sense of Benoist, we study two particular convex domains on which the affine action of that subgroup is free and properly discontinuous and whose quotients are naturally endowed with an "affine spacetime" structure that we introduce. In that setting, we show that those quotients are Maximal Globally Hyperbolic affine spacetimes admitting a locally uniformly Convex and Compact Cauchy surface (denoted as a MGHCC affine spacetimes), and that they come with a cosmological time function with Cauchy hypersurfaces foliating the quotient affine spacetime as level sets. Finally, we show that those are the only examples of such MGHCC affine spacetimes. All these results generalise the work of Mess, Barbot and Bonsante on affine deformations of uniform lattices of $\mathrm{SO}_0(d,1)$.