Universal commensurability augmented Teichmüller space and moduli space
| Autor: | Hideki Miyachi, Guangming Hu, Yi Qi |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Teichmüller space
Physics Augmented Teichmüller space characteristic tower Articles Direct limit Commensurability (mathematics) commensurability modular group Moduli space augmented moduli space Combinatorics High Energy Physics::Theory Modular group Compact Riemann surface Isometric embedding Quotient |
| Zdroj: | Annales Fennici Mathematici |
| ISSN: | 2737-114X 2737-0690 |
| Popis: | 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 Teichmuller space \(T(S)\) to the Teichmuller space \(T(\widetilde{S}_{g(\alpha)})\). Actually, it has been showed that the isometric embedding \(\Gamma_{\alpha}\) can be extended isometrically to the augmented Teichmuller space \(\widehat{T}(S)\) of \(T(S)\). Using this result, we construct a direct limit \(\widehat{T}_{\infty}(S)\) of augmented Teichmuller 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. |
| Databáze: | OpenAIRE |
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