Universal commensurability augmented Teichmüller space and moduli space

Authors

  • Guangming Hu Jinling Institute of Technology, College of Science
  • Hideki Miyachi Kanazawa University, College of Science and Engineering, School of Mathematics and Physics
  • Yi Qi Beihang University, School of Mathematics and Systems Science

Keywords:

Augmented Teichmüller space, commensurability modular group, augmented moduli space, characteristic tower

Abstract

It is known that every unbranched finite covering \(\alpha\colon\widetilde{S}_{g(\alpha)}\rightarrow S\) of a compact Riemann surface \(S\) with genus \(g\geq 2\) induces an isometric embedding \(\Gamma_{\alpha}\) from the Teichmüller space \(T(S)\) to the Teichmüller space \(T(\widetilde{S}_{g(\alpha)})\). Actually, it has been showed that the isometric embedding \(\Gamma_{\alpha}\) can be extended isometrically to the augmented Teichmüller space \(\widehat{T}(S)\) of \(T(S)\). Using this result, we construct a direct limit \(\widehat{T}_{\infty}(S)\) 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 \(\operatorname{Mod}_{\infty}(S)\) 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 \(\operatorname{Mod}_{\infty}(S)\) on \(\widehat{T}_{\infty}(S)\) is dense. Finally, we also construct a direct limit \(\widehat{M}_{\infty}(S)\) of augmented moduli spaces by characteristic towers and show that the subgroup \(\operatorname{Caut}(\pi_{1}(S))\) of \(\operatorname{Mod}_{\infty}(S)\) acts on \(\widehat{T}_{\infty}(S)\) to produce \(\widehat{M}_{\infty}(S)\) as the quotient.

 

Section
Articles

Published

2021-08-15

How to Cite

Hu, G., Miyachi, H., & Qi, Y. (2021). Universal commensurability augmented Teichmüller space and moduli space. Annales Fennici Mathematici, 46(2), 897–907. Retrieved from https://afm.journal.fi/article/view/110831