Universal commensurability augmented Teichm\"uller space and moduli space

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 ${\mathrm{\Gamma }}_{\alpha }$ from the Teichm"uller space $T(S)$ to the Teich"uller 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 Teichm"{u}ller space $\widehat{T}(S)$ of $T(S)$. Using this result, we construct a directed limit ${\widehat{T}}_{\mathrm{\infty }}(S)$ of augmented Teichm"uller spaces, where the index runs over all finitely unbranched coverings of $S$. Then, we show that the action of the universal commensurability modular group $Mo{d}_{\mathrm{\infty }}(S)$ can extend isometrically on ${\widehat{T}}_{\mathrm{\infty }}(S)$. Furthermore, for any $X_{\mathrm{\infty }}\in T_{\mathrm{\infty }}(S)$, its orbit of the action of the universal commensurability modular group $Mo{d}_{\mathrm{\infty }}(S)$ on the universal commensurability augmented Teichm"uller space ${\widehat{T}}_{\mathrm{\infty }}(S)$ is dense. Finally, we also construct a directed limit ${\widehat{M}}_{\mathrm{\infty }}(S)$ of augmented moduli spaces by characteristic towers and show that the subgroup $Caut({\pi }_1(S))$ of $Mo{d}_{\mathrm{\infty }}(S)$ acts on ${\widehat{T}}_{\mathrm{\infty }}(S)$ to produce ${\widehat{M}}_{\mathrm{\infty }}(S)$ as the quotient.