论文标题
通用可相称性增强Teichmüller空间和模量空间
Universal commensurability augmented Teichmüller space and moduli space
论文作者
论文摘要
众所周知，每个有限的无分支覆盖$α：\ widetilde {s} _ {g（α）} \ riemann surface $ s $的riemann surface $ s $带有$ g \ geq2 $ $ t（\ widetilde {s} _ {g（α）}）$。实际上，已经表明，等距嵌入$γ_α$可以等距地扩展到$ t（s）$的增强teichmüllerspace $ \ widehat {t}（t}（s）$。使用此结果，我们构建了一个限制$ \ wideHat {t} _ {\ infty}（s）$的增强teichmüller空间的$，该索引在所有有限的无分支封面上的$ s $中运行。然后，我们表明，通用可高度性模块化组$ mod _ {\ infty}（s）$的动作可以在$ \ wideHat {t} _ {\ infty}（s）$上等差扩展。此外，对于任何$ x _ {\ infty} \在t _ {\ infty}（s）$中，它是通用性模块化模块组$ mod $ mod _ {\ infty}（s）$在通用通用性可相当性增强teichmülllerspace $ \ wide $ foodhat $ is的行为的范围。最后，我们还通过特征塔构建了一个定向的限制$ \ widehat {m} _ {\ infty}（\ infty}（s）$，通过特征塔来构建$ caut（π_{1}（s）$ mod _ mod _ {\ infty}（\ infty}（s）$ act的$ \ \ \ \ \ \ \ wide s的（π_{1}（s）$）{产生$ \ wideHat {m} _ {\ infty}（s）$作为商。
It is known that every finitely unbranched covering $α:\widetilde{S}_{g(α)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $Γ_α$ from the Teichmüller space $T(S)$ to the Teichüller space $T(\widetilde{S}_{g(α)})$. Actually, it has been showed that the isometric embedding $Γ_α$ can be extended isometrically to the augmented Teichmüller space $\widehat{T}(S)$ of $T(S)$. Using this result, we construct a directed limit $\widehat{T}_{\infty}(S)$ of augmented Teichmüller spaces, where the index runs over all finitely unbranched coverings of $S$. Then, we show that the action of the universal commensurability modular group $Mod_{\infty}(S)$ can extend isometrically on $\widehat{T}_{\infty}(S)$. Furthermore, for any $X_{\infty}\in T_{\infty}(S)$, its orbit of the action of the universal commensurability modular group $Mod_{\infty}(S)$ on the universal commensurability augmented Teichmüller space $\widehat{T}_{\infty}(S)$ is dense. Finally, we also construct a directed limit $\widehat{M}_{\infty}(S)$ of augmented moduli spaces by characteristic towers and show that the subgroup $Caut(π_{1}(S))$ of $Mod_{\infty}(S)$ acts on $\widehat{T}_{\infty}(S)$ to produce $\widehat{M}_{\infty}(S)$ as the quotient.
