Universal commensurability augmented Teichmüller space and moduli space

Abstract

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