On totally hyperbolic non-Fuchsian type-preserving representations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912715553898496 |
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| author | Ryu, Inyoung |
| author_facet | Ryu, Inyoung |
| contents | We identify type-preserving representations $ϕ: π_1(Σ)\to \mathrm{PSL}(2,\mathbb{R})$ of the fundamental group of every punctured surface $Σ= Σ_{g,p}$ that are not Fuchsian yet send all non-peripheral simple closed curves to hyperbolic elements, which give a negative answer to a question of Bowditch. These representations have relative Euler class $e(ϕ) = \pm (χ(Σ) + 1)$, and their $\mathrm{PSL}(2,\mathbb{R})$-conjugacy classes form a full-measure subset of $2p$ connected components of the relative character variety. We further show that, while these representations are not Fuchsian, their restrictions to certain subsurfaces of $Σ$ are Fuchsian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_13989 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On totally hyperbolic non-Fuchsian type-preserving representations Ryu, Inyoung Geometric Topology 57K20, 57M05, 57M50 We identify type-preserving representations $ϕ: π_1(Σ)\to \mathrm{PSL}(2,\mathbb{R})$ of the fundamental group of every punctured surface $Σ= Σ_{g,p}$ that are not Fuchsian yet send all non-peripheral simple closed curves to hyperbolic elements, which give a negative answer to a question of Bowditch. These representations have relative Euler class $e(ϕ) = \pm (χ(Σ) + 1)$, and their $\mathrm{PSL}(2,\mathbb{R})$-conjugacy classes form a full-measure subset of $2p$ connected components of the relative character variety. We further show that, while these representations are not Fuchsian, their restrictions to certain subsurfaces of $Σ$ are Fuchsian. |
| title | On totally hyperbolic non-Fuchsian type-preserving representations |
| topic | Geometric Topology 57K20, 57M05, 57M50 |
| url | https://arxiv.org/abs/2511.13989 |