On Foliations in $\text{PSL}(4,\mathbb{R})$-Teichmüller Theory
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911940683497472 |
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| author | Nolte, Alexander |
| author_facet | Nolte, Alexander |
| contents | We carry out a detailed study of the structure of domains of discontinuity $Ω_ρ$ in $\mathbb{RP}^3$ of $\text{PSL}_4(\mathbb{R})$-Hitchin representations $ρ$. We then prove the foliated component $Ω_ρ^1$ of $Ω_ρ$ has exactly two group-invariant foliations by properly embedded projective line segments and has a unique foliation by properly embedded convex domains in projective planes. This gives a finiteness counterpart to work of Guichard and Wienhard. We also prove analogues for the non-foliated component $Ω_ρ^2$ and deduce a rigidity of projective equivalences of properly convex foliated projective structures on unit tangent bundles of surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Foliations in $\text{PSL}(4,\mathbb{R})$-Teichmüller Theory Nolte, Alexander Geometric Topology 22E40, 57M50, 20H10 We carry out a detailed study of the structure of domains of discontinuity $Ω_ρ$ in $\mathbb{RP}^3$ of $\text{PSL}_4(\mathbb{R})$-Hitchin representations $ρ$. We then prove the foliated component $Ω_ρ^1$ of $Ω_ρ$ has exactly two group-invariant foliations by properly embedded projective line segments and has a unique foliation by properly embedded convex domains in projective planes. This gives a finiteness counterpart to work of Guichard and Wienhard. We also prove analogues for the non-foliated component $Ω_ρ^2$ and deduce a rigidity of projective equivalences of properly convex foliated projective structures on unit tangent bundles of surfaces. |
| title | On Foliations in $\text{PSL}(4,\mathbb{R})$-Teichmüller Theory |
| topic | Geometric Topology 22E40, 57M50, 20H10 |
| url | https://arxiv.org/abs/2407.02237 |