It is known that every finitely unbranched covering $\alpha:\widetilde{ S}_{ g(\alpha)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $\Gamma_{ \alpha}$ from the Teichm\"uller space $T(S)$ to the Teich\"uller space $T(\widetilde{ S}_{ g(\alpha)})$. Actually, it has been showed that the isometric embedding $\Gamma_{ \alpha}$ 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 $\widehat{ T}_{ \infty}(S)$ 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 $Mod_{ \infty}(S)$ 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 Teichm\"uller 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 $Caut(\pi_{ 1}(S))$ of $Mod_{ \infty}(S)$ acts on $\widehat{ T}_{ \infty}(S)$ to produce $\widehat{ M}_{ \infty}(S)$ as the quotient.
Categories: math.GT math.CV