Universal commensurability augmented Teichm\"uller space and moduli space
Abstract
It is known that every finitely unbranched covering α:Sg(α)→ S of a compact Riemann surface S with genus g≥2 induces an isometric embedding α from the Teichm\"uller space T(S) to the Teich\"uller space T(Sg(α)). Actually, it has been showed that the isometric embedding α can be extended isometrically to the augmented Teichm\"uller space T(S) of T(S). Using this result, we construct a directed limit T∞(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∞(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 the universal commensurability augmented Teichm\"uller space T∞(S) is dense. Finally, we also construct a directed limit M∞(S) of augmented moduli spaces by characteristic towers and show that the subgroup Caut(π1(S)) of Mod∞(S) acts on T∞(S) to produce M∞(S) as the quotient.