Affine laminations and coaffine representations

We study surface subgroups of $\mathrm{S}\mathrm{L}(4,\mathbb{R})$ acting convex cocompactly on $\mathbb{R}{\text{P}}^3$ with image in the coaffine group. The boundary of the convex core is stratified, and the one dimensional strata form a pair of bending laminations. We show that the bending data on each component consist of a convex $\mathbb{R}{\text{P}}^2$ structure and an affine measured lamination depending on the underlying convex projective structure on $S$ with (Hitchin) holonomy $\rho :{\pi }_{1}S\to \mathrm{S}\mathrm{L}(3,\mathbb{R})$. We study the space ${\mathcal{M}\mathcal{L}}^{\rho }(S)$ of bending data compatible with $\rho$ and prove that its projectivization is a sphere of dimension $6g-7$.