Geometric Structures for the $G_2'$-Hitchin Component

TL;DR

This paper provides an explicit geometric interpretation of the $G_2'$-Hitchin component for surface groups, describing it as a moduli space of geometric structures on a fiber bundle over the surface, and constructs these structures explicitly.

Contribution

It introduces a new geometric structures interpretation of the $G_2'$-Hitchin component as a moduli space of $(G,X)$-structures on a specific fiber bundle, with explicit construction methods.

Findings

Homeomorphism between $Hit(S, G_2')$ and a moduli space of geometric structures.

Explicit construction of geometric structures from $ ho$-equivariant curves.

Analysis of the $G_2'$-Fuchsian case and its relation to known structures.

Abstract

We give an explicit geometric structures interpretation of the $G_2'$-Hitchin component $Hit(S, G_2') \subset \chi(\pi_1S,G_2')$ of a closed oriented surface $S$ of genus $g \geq 2$. In particular, we prove $Hit(S, G_2')$ is naturally homeomorphic to a moduli space $\mathscr{M}$ of $(G,X)$-structures for $G = G_2'$ and $X = Ein^{2,3}$ on a fiber bundle $\mathscr{C}$ over $S$ via the descended holonomy map. Explicitly, $\mathscr{C}$ is the direct sum of fiber bundles $\mathscr{C} = UTS \oplus UTS \oplus \underline{\mathbb{R}_+}$ with fiber $\mathscr{C}_p = UT_p S \times UT_p S \times \mathbb{R}_+$, where $UT S$ denotes the unit tangent bundle. The geometric structure associated to a $G_2'$-Hitchin representation $\rho$ is explicitly constructed from the unique associated $\rho$-equivariant alternating almost-complex curve $\hat{\nu}: \tilde{S} \rightarrow \hat{\mathbb{S}}^{2,4}$; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the $G_2'$-Fuchsian case and shown to be unrelated to the $(G_2', Ein^{2,3})$-structures of Guichard-Wienhard.