Geometric Structures for the $G_2'$-Hitchin Component
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910437140856832 |
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| author | Evans, Parker |
| author_facet | Evans, Parker |
| contents | We give an explicit geometric structures interpretation of the $G_2'$-Hitchin component $Hit(S, G_2') \subset χ(π_1S,G_2')$ of a closed oriented surface $S$ of genus $g \geq 2$. In particular, we prove $Hit(S, G_2')$ is naturally homeomorphic to a moduli space $\mathscr{M}$ of $(G,X)$-structures for $G = G_2'$ and $X = Ein^{2,3}$ on a fiber bundle $\mathscr{C}$ over $S$ via the descended holonomy map. Explicitly, $\mathscr{C}$ is the direct sum of fiber bundles $\mathscr{C} = UTS \oplus UTS \oplus \underline{\mathbb{R}_+}$ with fiber $\mathscr{C}_p = UT_p S \times UT_p S \times \mathbb{R}_+$, where $UT S$ denotes the unit tangent bundle.
The geometric structure associated to a $G_2'$-Hitchin representation $ρ$ is explicitly constructed from the unique associated $ρ$-equivariant alternating almost-complex curve $\hatν: \tilde{S} \rightarrow \hat{\mathbb{S}}^{2,4}$; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the $G_2'$-Fuchsian case and shown to be unrelated to the $(G_2', Ein^{2,3})$-structures of Guichard-Wienhard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric Structures for the $G_2'$-Hitchin Component Evans, Parker Differential Geometry Geometric Topology 57M50, 20H10 (Primary), 53C26, 53C43, 22E40 (Secondary) We give an explicit geometric structures interpretation of the $G_2'$-Hitchin component $Hit(S, G_2') \subset χ(π_1S,G_2')$ of a closed oriented surface $S$ of genus $g \geq 2$. In particular, we prove $Hit(S, G_2')$ is naturally homeomorphic to a moduli space $\mathscr{M}$ of $(G,X)$-structures for $G = G_2'$ and $X = Ein^{2,3}$ on a fiber bundle $\mathscr{C}$ over $S$ via the descended holonomy map. Explicitly, $\mathscr{C}$ is the direct sum of fiber bundles $\mathscr{C} = UTS \oplus UTS \oplus \underline{\mathbb{R}_+}$ with fiber $\mathscr{C}_p = UT_p S \times UT_p S \times \mathbb{R}_+$, where $UT S$ denotes the unit tangent bundle. The geometric structure associated to a $G_2'$-Hitchin representation $ρ$ is explicitly constructed from the unique associated $ρ$-equivariant alternating almost-complex curve $\hatν: \tilde{S} \rightarrow \hat{\mathbb{S}}^{2,4}$; we critically use recent work of Collier-Toulisse on the moduli space of such curves. Our explicit geometric structures are examined in the $G_2'$-Fuchsian case and shown to be unrelated to the $(G_2', Ein^{2,3})$-structures of Guichard-Wienhard. |
| title | Geometric Structures for the $G_2'$-Hitchin Component |
| topic | Differential Geometry Geometric Topology 57M50, 20H10 (Primary), 53C26, 53C43, 22E40 (Secondary) |
| url | https://arxiv.org/abs/2405.04492 |