$(g,k)$-Fermat曲线：模空间的一个嵌入
$(g,k)$-Fermat curves: an embedding of moduli spaces
AI总结：
本文研究闭黎曼曲面的$(g,k)$-Fermat群性质，证明其非超椭圆性与特定素数幂下的唯一性，构造Teichmüller空间嵌入及模空间全纯映射，并给出该模空间映射为单射的充分条件。
摘要：
设$S$为闭黎曼曲面，若其共形自同构群$H \cong {\mathbb Z}_{k}^{2g}$（其中$g,k \geq 2$为整数）作用自由，且商空间$S/H$的亏格为$g$，则称$H$为$(g,k)$-Fermat群。本文研究了这类对象的若干性质：特别地，我们发现$S$是非超椭圆的；并且若$k=p^{r}$（其中$p>84(g-1)$为素数，$r \geq 1$），则$H$是$S$唯一的$(g,k)$-Fermat群。设$\Gamma$为余紧无挠Fuchs群，满足$S/H={\mathbb H}^{2}/\Gamma$。若$\Gamma_{k}$是由$\Gamma$的换位子及其元素的$k$次幂生成的正规子群，则存在$S$与${\mathbb H}^{2}/\Gamma_{k}$之间的双全纯映射，将$H$共轭映射到$\Gamma/\Gamma_{k}$。包含关系$\Gamma_{k} < \Gamma$诱导了对应Teichmüller空间的自然全纯嵌入$\Theta_{k}:{\mathcal T}(\Gamma) \hookrightarrow {\mathcal T}(\Gamma_{k})$。该嵌入在模空间层面诱导了全纯映射$\Phi_{k}:{\mathcal M}(\Gamma) \to {\mathcal M}(\Gamma_{k})$。作为$(g,k)$-Fermat群相关结果的推论，我们给出了$\Phi_{k}$为单射的充分条件。
英文摘要：
A group $H \cong {\mathbb Z}_{k}^{2g}$, where $g,k \geq 2$ are integers, of conformal automorphisms of a closed Riemann surface $S$ is called a $(g,k)$-Fermat group if it acts freely with quotient $S/H$ of genus $g$. We study some properties of these type of objects, in particular, we observe that $S$ is non-hyperelliptic and, if $k=p^{r}$, where $p>84(g-1)$ is a prime integer and $r \geq 1$, then $H$ is the unique $(g,k)$-Fermat group of $S$. Let $Γ$ be a co-compact torsion free Fuchsian group such that $S/H={\mathbb H}^{2}/Γ$. If $Γ_{k}$ is its normal subgroup generated by its commutators and the $k$-powers of its elements, then there is a biholomorphism between $S$ and ${\mathbb H}^{2}/Γ_{k}$ congugating $H$ to $Γ/Γ_{k}$. The inclusion $Γ_{k} < Γ$ induces a natural holomorphic embedding $Θ_{k}:{\mathcal T}(Γ) \hookrightarrow {\mathcal T}(Γ_{k})$ of the corresponding Teichmüller spaces. Such an embedding induces a holomorphic map, at the level of their moduli spaces, $Φ_{k}:{\mathcal M}(Γ) \to {\mathcal M}(Γ_{k})$. As a consequence of the results on $(g,k)$-Fermat groups, we provide sufficient conditions for the injectivity of $Φ_{k}$.
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