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For $S$ a closed surface of genus $g\geq2$, we construct a canonicaldiffeomorphism from the degree $3$ Fock-Thomas space $\mathcal{T}^3(S)$ ofhigher complex structures to the $\text{SL}(3,\mathbb{R})$ Hitchin component.Our construction is equivariant with respect to natural actions of the mappingclass group $\text{Mod}(S)$. For all $n \geq 3$, we show that the Fock-Thomasspace $\mathcal{T}^n(S)$ has a canonical vector bundle structure overTeichm\"uller space. We then construct a $\text{Mod}(S)$-equivariant bundleisomorphism from $\mathcal{T}^n(S)$ to a sub-bundle of the restriction of thetangent bundle of the $\text{PSL}(n, \mathbb{R})$ Hitchin component to theFuchsian locus. As consequences, we prove that the higher degree moduli spaceof complex structures is a bundle over the moduli space of Riemann surfaces andthat the action of $\text{Mod}(S)$ on $\mathcal{T}^n(S)$ is a proper action byholomorphic automorphisms with respect to a canonical complex structure. Thecore of our approach is a careful analysis of higher degree diffeomorphismgroups.

Canonical Maps from Spaces of Higher Complex Structures to Hitchin Components