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

TL;DR

This paper studies the structure of domains of discontinuity in projective space associated with PSL(4,R)-Hitchin representations, revealing specific foliations and convex structures, and establishing rigidity results for these geometric structures.

Contribution

It provides a detailed analysis of foliated and non-foliated components of these domains, identifying their invariant foliations and convex structures, and proves rigidity of certain projective structures.

Findings

The foliated component has exactly two invariant foliations by projective line segments.

The foliated component admits a unique foliation by convex domains in projective planes.

Rigidity results for projective equivalences of convex foliated structures on tangent bundles.

Abstract

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.