Geometric Structures for the G₂'-Hitchin Component

We give an explicit geometric structures interpretation of the G₂'-Hitchin component Hit(S, G₂') ⊂ χ(π₁S,G₂') of a closed oriented surface S of genus g ≥ 2. In particular, we prove Hit(S, G₂') is naturally homeomorphic to a moduli space M of $(G,X)$-structures for G = G₂' and X = Ein2,3 on a fiber bundle C over S via the descended holonomy map. Explicitly, C is the direct sum of fiber bundles C = UTS ⊕ UTS ⊕ R₊ with fiber Cₚ = UTₚ S × UTₚ S × R₊, where $UT S$ denotes the unit tangent bundle. The geometric structure associated to a G₂'-Hitchin representation ρ is explicitly constructed from the unique associated ρ-equivariant alternating almost-complex curve ν̂: S̃ → Ŝ2,4; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the G₂'-Fuchsian case and shown to be unrelated to the (G₂', Ein2,3)-structures of Guichard-Wienhard.