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Geometric structures for the G2′ -Hitchin component

Advances in Mathematics, ISSN: 0001-8708, Vol: 462, Page: 110091
2025
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Article Description

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 Image 1 with fiber Cp=UTpS×UTpS×R+, where UTS 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 νˆ:S˜→Sˆ2,4 ; 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.

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