Universal commensurability augmented Teichmüller space and moduli space - Hu, Guangming; Miyachi, Hideki; Qi, Yi (Annales Fennici Mathematici, 2021)
It is known that every unbranched finite covering \(αS_g(α)→ S\) of a compact Riemann surface \(S\) with genus \(g≥ 2\) induces an isometric embedding \(Γ_α\) from the Teichmüller space \(T(S)\) to the Teichmüller space \(T(S_g(α))\). Actually, it has been showed that the isometric embedding \(Γ_α\) can be extended isometrically to the augmented Teichmüller space \(T(S)\) of \(T(S)\). Using this result, we construct a direct limit \(T_∞(S)\) 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 \(Mod_∞(S)\) can extend isometrically on \(T_∞(S)\). Furthermore, for any \(X_∞T_∞(S)\), its orbit of the action of the universal commensurability modular group \(Mod_∞(S)\) on \(T_∞(S)\) is dense. Finally, we also construct a direct limit \(M_∞(S)\) of augmented moduli spaces by characteristic towers and show that the subgroup \(Caut(π_1(S))\) of \(Mod_∞(S)\) acts on \(T_∞(S)\) to produce \(M_∞(S)\) as the quotient.