On the non-Archimedean Hitchin map for $\mathrm{SL}_2(F)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914138957021184 |
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| author | Chen, Jiahuang He, Siqi |
| author_facet | Chen, Jiahuang He, Siqi |
| contents | Let $F$ be a non-Archimedean valued field, $Σ$ a closed Riemann surface of genus at least two, and $Γ$ its fundamental group. Building on the theory of equivariant harmonic maps into $\mathbb{R}$-trees, we study the non-Archimedean Hitchin map from the $\mathrm{SL}_2(F)$-character variety $\mathcal{X}_F(Γ)$, equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $Σ$. We prove that this map is continuous and that its image is contained in the space of Jenkins--Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of $Γ$ on the Bruhat--Tits tree of $\mathrm{SL}_2(F)$ is never small. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_03469 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the non-Archimedean Hitchin map for $\mathrm{SL}_2(F)$ Chen, Jiahuang He, Siqi Differential Geometry Let $F$ be a non-Archimedean valued field, $Σ$ a closed Riemann surface of genus at least two, and $Γ$ its fundamental group. Building on the theory of equivariant harmonic maps into $\mathbb{R}$-trees, we study the non-Archimedean Hitchin map from the $\mathrm{SL}_2(F)$-character variety $\mathcal{X}_F(Γ)$, equipped with the non-Archimedean topology, to the space of holomorphic quadratic differentials on $Σ$. We prove that this map is continuous and that its image is contained in the space of Jenkins--Strebel differentials. Moreover, we establish a dynamical characterization of unbounded representations, showing that the induced action of $Γ$ on the Bruhat--Tits tree of $\mathrm{SL}_2(F)$ is never small. |
| title | On the non-Archimedean Hitchin map for $\mathrm{SL}_2(F)$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2511.03469 |