Convex foliated projective structures and the Hitchin component for PSL4(R)

Abstract

In this article, we give a geometric interpretation of the Hitchin component $T^4(\Sigma )\subset \mathrm{Rep}({\pi }_1(\Sigma ),{\mathrm{PSL}}_4(R))$ of a closed oriented surface of genus $g\ge 2$. We show that representations in $T^4(\Sigma )$ are precisely the holonomy representations of properly convex foliated projective structures on the unit tangent bundle of $\Sigma$. From this, we also deduce a geometric description of the Hitchin component $T(\Sigma ,{\mathrm{Sp}}_4(R))$ of representations into the symplectic group