Concave Foliated Flag Structures and the $\text{SL}_3(\mathbb{R})$ Hitchin Component
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| Format: | Preprint |
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2024
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| _version_ | 1866914862722973696 |
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| author | Nolte, Alexander Riestenberg, J. Maxwell |
| author_facet | Nolte, Alexander Riestenberg, J. Maxwell |
| contents | We give a geometric characterization of flag geometries associated to Hitchin representations in $\text{SL}_3(\mathbb{R})$. Our characterization is based on distinguished invariant foliations, similar to those studied by Guichard-Wienhard in $\text{PSL}_4(\mathbb{R})$.
We connect to the dynamics of Hitchin representations by constructing refraction flows for all positive roots in general $\mathfrak{sl}_n(\mathbb{R})$ in our setting. For $n = 3$, leaves of our one-dimensional foliations are flow-lines. One consequence is that the highest root flows are $C^{1+α}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_06361 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Concave Foliated Flag Structures and the $\text{SL}_3(\mathbb{R})$ Hitchin Component Nolte, Alexander Riestenberg, J. Maxwell Geometric Topology Differential Geometry 57M50, 20H10, 22E40 We give a geometric characterization of flag geometries associated to Hitchin representations in $\text{SL}_3(\mathbb{R})$. Our characterization is based on distinguished invariant foliations, similar to those studied by Guichard-Wienhard in $\text{PSL}_4(\mathbb{R})$. We connect to the dynamics of Hitchin representations by constructing refraction flows for all positive roots in general $\mathfrak{sl}_n(\mathbb{R})$ in our setting. For $n = 3$, leaves of our one-dimensional foliations are flow-lines. One consequence is that the highest root flows are $C^{1+α}$. |
| title | Concave Foliated Flag Structures and the $\text{SL}_3(\mathbb{R})$ Hitchin Component |
| topic | Geometric Topology Differential Geometry 57M50, 20H10, 22E40 |
| url | https://arxiv.org/abs/2407.06361 |