Geometric structures for the G2′-Hitchin component

Abstract

We give an explicit geometric structures interpretation of the $G_2^{\prime }$-Hitchin component $\mathrm{Hit}(S,G_2^{\prime })\subset \chi ({\pi }_{1}S,G_2^{\prime })$ of a closed oriented surface S of genus $g\ge 2$. In particular, we prove $\mathrm{Hit}(S,G_2^{\prime })$ is naturally homeomorphic to a moduli space $M$ of $(G,X)$-structures for $G=G_2^{\prime }$ and $X={\mathrm{Ein}}^{2,3}$ on a fiber bundle $C$ over S via the descended holonomy map. Explicitly, $C$ is the direct sum of fiber bundles

with fiber $C_p=U{T}_{p}S\times U{T}_{p}S\times R_{+}$, where UTS denotes the unit tangent bundle.

The geometric structure associated to a $G_2^{\prime }$-Hitchin representation ρ is explicitly constructed from the unique associated ρ-equivariant alternating almost-complex curve $\hat{\nu }:\tilde{S}\to {\hat{S}}^{2,4}$; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the $G_2^{\prime }$-Fuchsian case and shown to be unrelated to the $(G_2^{\prime },{\mathrm{Ein}}^{2,3})$-structures of Guichard-Wienhard.