Universal commensurability augmented Teichmüller space and moduli space

Abstract

It is known that every finitely unbranched covering $\alpha :{\tilde{S}}_{g(\alpha )}\to S$ of a compact Riemann surface $S$ with genus $g\ge 2$ induces an isometric embedding ${\Gamma }_{\alpha }$ from the Teichmüller space $T(S)$ to the Teichüller space $T({\tilde{S}}_{g(\alpha )})$. Actually, it has been showed that the isometric embedding ${\Gamma }_{\alpha }$ can be extended isometrically to the augmented Teichmüller space $\hat{T}(S)$ of $T(S)$. Using this result, we construct a directed limit ${\hat{T}}_{\infty }(S)$ 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 $Mo{d}_{\infty }(S)$ can extend isometrically on ${\hat{T}}_{\infty }(S)$. Furthermore, for any $X_{\infty }\in T_{\infty }(S)$, its orbit of the action of the universal commensurability modular group $Mo{d}_{\infty }(S)$ on the universal commensurability augmented Teichmüller space ${\hat{T}}_{\infty }(S)$ is dense. Finally, we also construct a directed limit ${\hat{M}}_{\infty }(S)$ of augmented moduli spaces by characteristic towers and show that the subgroup $Caut({\pi }_1(S))$ of $Mo{d}_{\infty }(S)$ acts on ${\hat{T}}_{\infty }(S)$ to produce ${\hat{M}}_{\infty }(S)$ as the quotient.