We study qualitative aspects of the geometry of Hitchin representations of surface groups in PSL(n,R). The thematic core of this thesis is on Thurston-Klein geometric structures with distinguished foliations that are associated to Hitchin representations in SL(3,R) and PSL(4,R). This direction was initiated by Guichard-Wienhard’s work on PSL(4,R)-Hitchin representations.

We establish the following results on these foliated structures. First, we prove a rigidity theorem for the projective geometry of leaves of Guichard-Wienhard’s codimension-1 foliation associated to a PSL(4,R)-Hitchin representation. We then classify all foliations of domains of discontinuity in RP(3) for PSL(4,R)-Hitchin representations that are geometrically similar to those studied by Guichard and Wienhard. Finally, we give a parallel framework to Guichard-Wienhard’s work in PSL(4,R) for geometric structures modeled on the space of full flags in R^3 for SL(3,R)-Hitchin representations.

We give a number of applications of our results and the surrounding circle of ideas.

First, we resolve a question asked by Benzécri in 1960 on the point-set topology of the space of projective equivalence classes of properly convex domains in RP(n) for n at least 2. Next, we give explicit geometric constructions of flows constructed in the dynamical study of Hitchin representations. Finally, we construct asymmetric metrics on PSL(n,R)-Hitchin components for n at least 4 and give a new formulation of Thurston’s asymmetric metric on Teichmüller space.

Finally, we study degree-n complex structures in the sense of Fock and Thomas. These are extensions of complex structures on surfaces whose deformation space T(n,S) is conjectured to be canonically homeomorphic to the PSL(n,R)-Hitchin component. Our main result in this direction is the construction of a canonical homeomorphism from Fock-Thomas spaces T(n,S) of higher complex structures to a bundle B(n,S) of harmonic tensors over Teichmüller space.