On Foliations in $\text{PSL}(4,\mathbb{R})$-Teichm\"uller Theory

We carry out a detailed study of the structure of domains of discontinuity $\Omega_\rho$ in $\mathbb{RP}^3$ of $\text{PSL}_4(\mathbb{R})$-Hitchin representations $\rho$. We then prove the foliated component $\Omega_\rho^1$ of $\Omega_\rho$ has exactly two group-invariant foliations by properly embedded projective line segments and has a unique foliation by properly embedded convex domains in projective planes. This gives a finiteness counterpart to work of Guichard and Wienhard. We also prove analogues for the non-foliated component $\Omega_\rho^2$ and deduce a rigidity of projective equivalences of properly convex foliated projective structures on unit tangent bundles of surfaces.