Fractal closures of geodesic planes in Hitchin manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908837039046656 |
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| author | Dey, Subhadip Oh, Hee |
| author_facet | Dey, Subhadip Oh, Hee |
| contents | Ratner's theorem implies topological rigidity of immersed totally geodesic subspaces of noncompact type in finite-volume locally symmetric spaces. In higher rank and infinite volume, however, counter-examples to this rigidity have remained elusive.
We construct the first such examples using \emph{floating geodesic planes}. Specifically, we exhibit a Zariski-dense Hitchin surface group $Γ< \mathrm{SL}_3(\mathbb{R})$ such that the Hitchin manifold $Γ\backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}(3)$ contains immersed floating geodesic planes whose closures are fractal, with non-integer Hausdorff dimensions accumulating at $2$. Moreover, $Γ$ can be chosen inside $\mathrm{SL}_3(\mathbb{Z})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17915 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractal closures of geodesic planes in Hitchin manifolds Dey, Subhadip Oh, Hee Geometric Topology Differential Geometry Dynamical Systems Group Theory Ratner's theorem implies topological rigidity of immersed totally geodesic subspaces of noncompact type in finite-volume locally symmetric spaces. In higher rank and infinite volume, however, counter-examples to this rigidity have remained elusive. We construct the first such examples using \emph{floating geodesic planes}. Specifically, we exhibit a Zariski-dense Hitchin surface group $Γ< \mathrm{SL}_3(\mathbb{R})$ such that the Hitchin manifold $Γ\backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}(3)$ contains immersed floating geodesic planes whose closures are fractal, with non-integer Hausdorff dimensions accumulating at $2$. Moreover, $Γ$ can be chosen inside $\mathrm{SL}_3(\mathbb{Z})$. |
| title | Fractal closures of geodesic planes in Hitchin manifolds |
| topic | Geometric Topology Differential Geometry Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2509.17915 |