Geometric Structures for the G₂'-Hitchin Component

We give an explicit geometric structures interpretation of the G₂'-Hitchin component Hit (S, G₂') (₁S, 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 = Ein^2, 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 is explicitly constructed from the unique associated -equivariant alternating almost-complex curve: S 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 G₂'-Fuchsian case and shown to be unrelated to the (G₂', Ein^2, 3) -structures of Guichard-Wienhard.