On Foliations in PSL(4,R)-Teichm\"uller Theory

We carry out a detailed study of the structure of domains of discontinuity Ω_ρ in RP³ of PSL₄(R)-Hitchin representations ρ. We then prove the foliated component Ω_ρ¹ of Ω_ρ 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 Ω_ρ² and deduce a rigidity of projective equivalences of properly convex foliated projective structures on unit tangent bundles of surfaces.