UNIVERSAL COMMENSURABILITY AUGMENTED TEICHMÜLLER SPACE AND MODULI SPACE

It is known that every unbranched finite covering (Formula Presented) of a compact Riemann surface S with genus g ≥ 2 induces an isometric embedding Γ_α from the Teichmüller space T (S) to the Teichüller space (Formula Presented). Actually, it has been showed that the isometric embedding Γ_α can be extended isometrically to the augmented Teichmüller space b (Formula Presented) of T (S). Using this result, we construct a direct limit b (Formula Presented) 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 Mod_∞(S) can extend isometrically on b (Formula Presented). Furthermore, for any X_∞ ∈ T_∞(S), its orbit of the action of the universal commensurability modular group Mod_∞(S) on (Formula Presented) is dense.