Universal commensurability augmented Teichmüller space and moduli space

Guangming Hu¹ ; Hideki Miyachi² ; Yi Qi³

¹ Jinling Institute of Technology, College of Science
² Kanazawa University, College of Science and Engineering, School of Mathematics and Physics
³ Beihang University, School of Mathematics and Systems Science

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 897-907

Résumé

It is known that every unbranched finite covering $\alpha\colon\widetilde{S}_{g(\alpha)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq 2$ induces an isometric embedding $\Gamma_{\alpha}$ from the Teichmüller space $T(S)$ to the Teichmüller space $T(\widetilde{S}_{g(\alpha)})$. Actually, it has been showed that the isometric embedding $\Gamma_{\alpha}$ can be extended isometrically to the augmented Teichmüller space $\widehat{T}(S)$ of $T(S)$. Using this result, we construct a direct limit $\widehat{T}_{\infty}(S)$ of augmented Teichmüller spaces, where the index runs over all unbranched finite coverings of $S$. Then, we show that the action of the universal commensurability modular group $\operatorname{Mod}_{\infty}(S)$ 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 $\operatorname{Mod}_{\infty}(S)$ on $\widehat{T}_{\infty}(S)$ is dense. Finally, we also construct a direct limit $\widehat{M}_{\infty}(S)$ of augmented moduli spaces by characteristic towers and show that the subgroup $\operatorname{Caut}(\pi_{1}(S))$ of $\operatorname{Mod}_{\infty}(S)$ acts on $\widehat{T}_{\infty}(S)$ to produce $\widehat{M}_{\infty}(S)$ as the quotient.  

Détail

Keywords: Augmented Teichmüller space, commensurability modular group, augmented moduli space, characteristic tower