Bending Teichmüller spaces and character varieties
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866918508293521408 |
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| author | Baba, Shinpei |
| author_facet | Baba, Shinpei |
| contents | We consider the mapping $b_L\colon\mathcal{T} \to χ$ from the Fricke-Teichmüller space $\mathcal{T}$ into the $\mathrm{PSL}_2\mathbb{C}$-character variety $χ$ of the surface, obtained by bending Fuchsian representations along a fixed measured lamination $L$. We prove that this mapping is an equivariant symplectic real-analytic embedding, and, for almost all measured laminations, proper.
We also show that this ``bending map'' $b_L\colon \mathcal{T} \to χ$ extends continuously almost-everywhere to the canonical inclusion map from the Thurston boundary of $\mathcal{T}$ into the Morgan-Shalen boundary of $χ$.
Moreover, we ``complexify" this bending map in a geometric manner. Namely, we symplectically embed this real-analytic subvariety ${\rm Im} b_L$ into the product variety $χ\times χ$ by the diagonal mapping twisted by complex conjugation. Then we construct a closed $\mathbb{C}$-symplectic complex-analytic subvariety of $χ\times χ$ containing $\mathrm{Im} b_L$ as a half-dimensional real-analytic subvariety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_15394 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bending Teichmüller spaces and character varieties Baba, Shinpei Geometric Topology Complex Variables We consider the mapping $b_L\colon\mathcal{T} \to χ$ from the Fricke-Teichmüller space $\mathcal{T}$ into the $\mathrm{PSL}_2\mathbb{C}$-character variety $χ$ of the surface, obtained by bending Fuchsian representations along a fixed measured lamination $L$. We prove that this mapping is an equivariant symplectic real-analytic embedding, and, for almost all measured laminations, proper. We also show that this ``bending map'' $b_L\colon \mathcal{T} \to χ$ extends continuously almost-everywhere to the canonical inclusion map from the Thurston boundary of $\mathcal{T}$ into the Morgan-Shalen boundary of $χ$. Moreover, we ``complexify" this bending map in a geometric manner. Namely, we symplectically embed this real-analytic subvariety ${\rm Im} b_L$ into the product variety $χ\times χ$ by the diagonal mapping twisted by complex conjugation. Then we construct a closed $\mathbb{C}$-symplectic complex-analytic subvariety of $χ\times χ$ containing $\mathrm{Im} b_L$ as a half-dimensional real-analytic subvariety. |
| title | Bending Teichmüller spaces and character varieties |
| topic | Geometric Topology Complex Variables |
| url | https://arxiv.org/abs/2203.15394 |